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Surgeryon contact 3-manifolds and Stein surfaces B. Ozbagci and A. I. Stipsicz Surgery on contact 3-manifolds and Stein surfaces B. Ozbagci A. I. Stipsicz J ´ ANOS BOLYAI MATHEMATICAL SOCIETY Budapest, F˝o u. 68., H–1027, Hungary c _ BOLYAI J ´ ANOS MATEMATIKAI T ´ ARSULAT Budapest, Hungary, 2004 ISBN: ??? ???? ??? Published by J ´ ANOS BOLYAI MATHEMATICAL SOCIETY Budapest, F˝o u. 68., H–1027, Hungary Assistant editor: ??? Printed in Hungary Budapest Contents Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1. Why symplectic and contact? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2. Results concerning Stein surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3. Some contact results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2. Topological surgeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1. Surgeries and handlebodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2. Dehn surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3. Kirby calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3. Symplectic 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1. Generalities about symplectic manifolds . . . . . . . . . . . . . . . . . . . . . 49 3.2. Moser’s method and neighborhood theorems . . . . . . . . . . . . . . . . . 55 3.3. Appendix: The complex classiﬁcation scheme for symplectic 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4. Contact 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1. Generalities on contact 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2. Legendrian knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3. Tight versus overtwisted structures . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5. Convex surfaces in contact 3-manifolds . . . . . . . . . . . . . . . . . . . . 85 5.1. Convex surfaces and dividing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2. Contact structures and Heegaard decompositions . . . . . . . . . . . . 96 6. Spin c structures on 3- and 4-manifolds . . . . . . . . . . . . . . . . . . . . . . 99 6.1. Generalities on spin and spin c structures . . . . . . . . . . . . . . . . . . . . 99 4 Contents 6.2. Spin c structures and oriented 2-plane ﬁelds . . . . . . . . . . . . . . . . . . 102 6.3. Spin c structures and almost-complex structures . . . . . . . . . . . . . . 105 7. Symplectic surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.1. Symplectic cut-and-paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.2. Weinstein handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.3. Another handle attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8. Stein manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.1. Recollections and deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2. Handle attachment to Stein manifolds . . . . . . . . . . . . . . . . . . . . . . . 125 8.3. Stein neighborhoods of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 9. Open books and contact structures . . . . . . . . . . . . . . . . . . . . . . . . 131 9.1. Open book decompositions of 3-manifolds . . . . . . . . . . . . . . . . . . . 131 9.2. Compatible contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.3. Branched covers and contact structures . . . . . . . . . . . . . . . . . . . . . . 150 10. Lefschetz fibrations on 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . 155 10.1. Lefschetz pencils and ﬁbrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 10.2. Lefschetz ﬁbrations on Stein domains . . . . . . . . . . . . . . . . . . . . . . . 162 10.3. Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 11. Contact Dehn surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.1. Contact structures on S 1 D 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.2. Contact Dehn surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.3. Invariants of contact structures given by surgery diagrams . . 191 12. Fillings of contact 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 12.1. Fillings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 12.2. Nonﬁllable contact 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 12.3. Topology of Stein ﬁllings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13. Appendix: Seiberg–Witten invariants . . . . . . . . . . . . . . . . . . . . . . 223 13.1. Seiberg–Witten invariants of closed 4-manifolds . . . . . . . . . . . . 223 13.2. Seiberg–Witten invariants of 4-manifolds with contact boun- dary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 13.3. The adjunction inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 14. Appendix: Heegaard Floer theory . . . . . . . . . . . . . . . . . . . . . . . . . . 235 14.1. Topological preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Contents 5 14.2. Heegaard Floer theory for 3- and 4-manifolds . . . . . . . . . . . . . . . 239 14.3. Surgery triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 14.4. Contact Ozsv´ath–Szab´o invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 249 15. Appendix: Mapping class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 15.1. Short introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 15.2. Mapping class groups and geometric structures . . . . . . . . . . . . . 264 15.3. Some proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Preface The groundbreaking results of the near past — Donaldson’s result on Lef- schetz pencils on symplectic manifolds and Giroux’s correspondence be- tween contact structures and open book decompositions — brought a top- ological ﬂavor to global symplectic and contact geometry. This topological aspect is strengthened by the existing results of Weinstein and Eliashberg (and Gompf in dimension 4) on handle attachment in the symplectic and Stein category, and by Giroux’s theory of convex surfaces, enabling us to perform surgeries on contact 3-manifolds. The main objective of these notes is to provide a self-contained introduction to the theory of surgeries one can perform on contact 3-manifolds and Stein surfaces. We will adopt a very topological point of view based on handlebody theory, in particular, on Kirby calculus for 3- and 4-dimensional manifolds. Surgery is a constructive method by its very nature. Applying it in an intricate way one can see what can be done. These results are nicely com- plemented by the results relying on gauge theory — a theory designed to prove that certain things cannot be done. We will freely apply recent results of gauge theory without a detailed introduction to these topics; we will be content with a short introduction to some forms of Seiberg–Witten theory and some discussions regarding Heegaard Floer theory in two Appendices. As work of Taubes in the closed, and Kronheimer–Mrowka in the manifold- with-boundary case shows, the analytic approach towards symplectic and contact topology can be very fruitfully capitalized when coupled with some form of Seiberg–Witten theory. On the other hand, Lefschetz pencils on symplectic, and open book decompositions on contact manifolds are well- suited for the newly invented contact Ozsv´ath–Szab´o invariants. Under some fortunate circumstances these dual viewpoints provide interesting re- sults in the subject. As a preview, Chapter 1 is devoted to the description of problems where the above discussed techniques can be applied. For setting up the topological background of surgeries on contact 3- manifolds and Stein surfaces we will ﬁrst examine the smooth surgery con- struction, with a special emphasis on 2-handle attachments to 4-manifolds 8 Preface and Dehn surgeries on 3-manifolds. This is done in Chapter 2. Then we turn to the symplectic cut-and-paste operation, which enables us to glue sym- plectic 4-manifolds along contact type boundaries. To put this operation in the right perspective, in Chapters 3 and 4 we ﬁrst brieﬂy review some parts of symplectic and contact topology in dimensions 4 and 3, respec- tively. We pay special attention to convex surfaces in contact 3-manifolds (Chapter 5), with an eye on its later applicability in contact surgery. Be- fore giving the general scheme of symplectic surgery in Chapter 7, we make a little digression and discuss spin c structures from a point of view suit- able for our later purposes. As a special case of the general gluing scheme, we will meet Weinstein’s construction for attaching symplectic 2-handles to ω-convex boundaries along Legendrian knots. After having these prepara- tions, we can turn to the discussion of the famous result of Eliashberg that shows how to attach a Stein 2-handle to the pseudoconvex boundary of a Stein domain along a Legendrian knot. For the convenience of the more topologically minded reader, in Chapter 8 a short recollection of rudiments of the theory of Stein manifolds is included. Once the gluing construction given, we can turn to its applications, including the search for Lefschetz ﬁ- bration structures on Stein domains, embeddability of Stein domains into closed surfaces with extra (symplectic or complex) structures, or the study of Stein ﬁllings of contact 3-manifolds (Chapters 10 and 12). In the contact setting, the most important technique for being able to do surgery is the convex surface theory developed by Giroux. After recalling relevant parts of this beautiful theory, and proving the neighborhood theorems we need in this subject, in Chapter 11 we will be able to do contact surgeries. With this construction at our disposal, now we can seek for applications: we will be able to draw explicit diagrams of many contact 3-manifolds, show ways to distinguish them and to determine the homotopy type of contact struc- tures given by various constructions. These results — together with various versions of gauge theories, including Seiberg–Witten theory and Heegaard Floer theory — provide ways to examine tightness and ﬁllability properties of numerous contact structures, which are given in Chapter 12. To make the presentation more complete, we include Chapter 9 on open book de- compositions and their relation to contact structures. The appearance of mapping class groups in these theories, together with some nice applications allows us to conclude the discussion with a short recollection of deﬁnitions and results in that ﬁeld. To guide the interested reader, we close this preface by listing some monographs discussing topics we only outline here. Handlebody theory and Preface 9 Kirby calculus, which is only sketched in Chapter 2, is discussed more thor- oughly in [66]. A more complete introduction to symplectic geometry and topology is provided by [111]. For additional reading on contact topology, the reader is advised to turn to [1, 2, 57]. Seiberg–Witten theory is covered by many volumes, including for example [119, 126, 149]. These notes are based on two lecture series given by the second author at the Banach Center (Warsaw, Poland) and at the University of Lille (France). He wants to thank these institutions for their hospitality. The ﬁnal form of the notes were shaped while the authors visited KIAS (Seoul, Korea); they wish to thank KIAS for its hospitality. The authors would like to thank Selman Akbulut, John Etnyre, Sergey Finashin, David Gay, Paolo Lisca, Gordana Mati´c and R´ obert Sz˝ oke for many enlightening conversations. Special thanks go to Hansj¨org Geiges for suggesting numerous corrections and improvements of an earlier version of the text. The second author also wants to express his thanks to his family — without their support this volume would not have come into existence. The ﬁrst author acknowledges support from the Turkish Academy of Sciences and from Ko¸ c University. The second author acknowledges partial support by OTKA T034885 and T037735. Istanbul and Budapest, 2004. Burak Ozbagci and Andr´ as Stipsicz 1. Introduction 1.1. Why symplectic and contact? The intense interest of 4-manifold topologists in symplectic geometry and topology might have the following explanation. The success of the classiﬁ- cation of higher (≥ 5) dimensional manifolds relies heavily on the famous “h-cobordism theorem”, in which the “Whitney trick” plays a fundamen- tal role. The Whitney trick asserts that (under favorable conditions) the algebraic and geometric intersection numbers of two submanifolds can be made equal by isotoping one of them. In other words, by isotopy we can get rid of “excess intersections”, which are present in the geometric picture but are invisible for algebra. After eliminating these intersections “algebra will govern geometry”, and the smooth classiﬁcation problem of manifolds can be translated into some (nontrivial) algebraic questions. Remark 1.1.1. The proof of the Whitney trick involves a map of a 2- dimensional disk into the manifold at hand. If we can achieve that this map is an embedding (with the appropriate normal bundle), we get a local model showing us the required isotopy. Once the dimension is high enough (at least 5), any map from the disk admits a perturbation such that the result is an embedding. In dimension four, however, the disk might have self-intersections, and we cannot get rid of those by simple dimension count. The key step of Freedman’s topological classiﬁcation theorem in dimen- sion four is to show that the Whitney trick does extend to dimension four provided we allow topological isotopies. In fact, as Donaldson’s theorems on the failure of the smooth h-cobordism theorem in dimension 4 show, in some examples the excess intersections persist if we allow smooth maps only. It is a standard fact that in a complex manifold complex submanifolds intersect (locally) positively, therefore no excess intersection points appear, 12 1. Introduction hence the above principles apply. Actually, if a 4-manifold X is only almost- complex (and this structure is much easier to ﬁnd, since its existence de- pends only on the homotopy type of X) then almost-complex submanifolds still intersect positively — with the usual restrictions of not sharing com- mon components, see [109]. The problem with almost-complex manifolds is that although the existence of the structure is guaranteed by some simple properties of the cohomology ring H ∗ (X 4 ; Z), it is very hard to show that almost-complex submanifolds exist (in general they do not), i.e., that the above principle ever comes into force. Now if the almost-complex manifold (X, J) also carries a compatible symplectic structure ω, then — accord- ing to fundamental results of Taubes — smooth properties of X already guarantee the existence of almost-complex (also known as J-holomorphic) submanifolds. Since this argument only provides a few almost-complex rep- resentatives, we cannot expect a complete solution for the classiﬁcation problem. The spectacular results built on Taubes’ work nevertheless show the above described principle in action. For this reason we chose to study symplectic 4-manifolds (and their topological counterparts, Lefschetz ﬁbra- tions) in more detail. According to Donaldson’s result, symplectic manifolds always decompose along a circle bundle into a union of a disk bundle and another piece which can be endowed with a Stein structure. Conversely, any Stein surface embeds into some closed symplectic 4-manifold. The analogy becomes even deeper if we study the topological counterparts of symplec- tic and Stein manifolds: these are Lefschetz ﬁbrations with closed or with bounded ﬁbers. Therefore it appears natural to study topological properties of symplectic and Stein manifolds together. When trying to perform surgeries in the symplectic or Stein category, we have to pay special attention to the structures induced on their 3- dimensional boundaries — this is how contact structures come into play. The topological counterpart of contact structures (which are open book decompositions on the 3-manifolds) ﬁts perfectly into this picture since open book decompositions can be interpreted as boundaries of Lefschetz ﬁbrations. (In the general case we allow achiral Lefschetz ﬁbrations as well.) The fascinating, and still not completely well-understood interplay of the above notions provides the leading theme of these notes. Topological questions regarding symplectic 4-manifolds and Lefschetz ﬁbrations are fairly well-treated in the literature ([66, 111]); in these in- stances we merely restrict ourselves to quoting the necessary results. For contact surgeries and open book decompositions the available sources are less complete, so in these cases a more thorough treatment of the relevant 1.2. Results concerning Stein surfaces 13 material is given. In the following we address the problem of understanding topological properties of Stein surfaces and contact 3-manifolds. In order to attack such a problem we need two major tools, which provide existence and nonexistence results. Complex geometry (e.g., complex surfaces, Mil- nor ﬁbers, links of singularities) provides a rich source of examples, giving the needed existence results. A more systematic way of studying the ex- istence problem is provided by the theory of handlebodies — initiated by Smale, Milnor and Kirby, and extended to the symplectic and Stein cate- gory by Weinstein, Eliashberg and Gompf. On the boundary, the handle attachment translates into contact surgery, showing existence of a variety of contact structures. By suitably generalizing the attachment scheme de- scribed by Weinstein (and incorporating achiral Lefschetz ﬁbrations into the theory), in fact all contact 3-manifolds can be treated in this way. On the other hand, gauge theory (more speciﬁcally, Seiberg–Witten theory and Ozsv´ath–Szab´o invariants) can be used to prove that manifolds or diﬀeo- morphisms with certain properties do not exist. Therefore Seiberg–Witten and Ozsv´ath–Szab´o invariants and Seiberg–Witten moduli spaces provide (in favorable cases) the needed nonexistence results. We can, for example, show that certain 4-manifolds do not carry any Stein structure, or speciﬁc contact 3-manifolds cannot be given as boundary of any Stein surface. 1.2. Results concerning Stein surfaces Before turning to the detailed discussion of various surgery constructions, we give a sample of results we would like to present in these notes. As it turns out, the existence of a Stein structure on a 4-manifold X considerably constrains its diﬀerential topology. The most apparent constraint can be summarized by the adjunction inequality given in Theorem 1.2.1. Closely related formulae appear in many other branches in 3- and 4-dimensional topology, and these type of results always play a central role in the theory at hand. (See Section 4.3 for the “contact version” of the adjunction formula.) Theorem 1.2.1 ([6]). If X is a 4-dimensional Stein manifold and Σ ⊂ X is a closed, connected, oriented, embedded surface of genus g in it, then [Σ] 2 +[ ¸ c 1 (X), [Σ] _ [ ≤ 2g −2 unless Σ is a sphere with [Σ] = 0 in H 2 (X; Z). 14 1. Introduction Remark 1.2.2. Note that if C is a (smooth) connected complex curve in a complex surface X then the Whitney product formula for Chern classes implies 2g(C) − 2 = [C] 2 − ¸ c 1 (X), [C] _ ; this equation is frequently called the adjunction equality. Its generalization for closed complex surfaces and smooth submanifolds Σ ⊂ X was proved ﬁrst by Kronheimer and Mrowka (in the case [Σ] 2 ≥ 0) and in general by Ozsv´ath and Szab´ o [133]. For more about the adjunction inequality see Section 13.3, where we will indi- cate how such formulae for a closed symplectic 4-manifold X follow from Seiberg–Witten theory, and describe the derivation of the above formula (for Stein surfaces) from the closed case. Notice that, for example, the in- equality shows that a Stein surface cannot contain a homologically essential, smoothly embedded sphere S with [S] 2 ≥ −1. Below we give some surprising corollaries of the above adjunction inequality; we hope that this demonstrates the power and diversity of the theorem. Simple nondiﬀeomorphic 4-manifolds The ﬁrst application gives an example of homeomorphic but nondiﬀeomor- phic 4-manifolds. Corollary 1.2.3 (Akbulut, [4]). The 4-manifolds X 1 , X 2 deﬁned by the knots K 1 , K 2 of Figure 1.1 are homeomorphic but nondiﬀeomorphic. −1 −1 −4 K K 1 2 Figure 1.1. Homeomorphic but nondiﬀeomorphic 4-manifolds 1.2. Results concerning Stein surfaces 15 The meaning of such Kirby diagrams will be explained in Chapter 2; for a more thorough treatment see [66]. Here we just note that the knots (together with the numbers) indicate how to glue a 4-dimensional 2-handle D 2 D 2 along ∂D 2 D 2 to D 4 in order to get X 1 and X 2 , respectively. Proof (sketch). Using some simple operations on the diagrams one can show that the 3-manifolds ∂X 1 and ∂X 2 are diﬀeomorphic (see [66, Fig- ure 11.4]). Now the signatures σ(X 1 ) and σ(X 2 ) are both equal to −1, the Euler characteristics are both equal to 2, and since ∂X 1 = ∂X 2 is an in- tegral homology sphere, the extension of Freedman’s famous theorem (see [51]) implies that X 1 and X 2 are homeomorphic. Next we show that X 1 carries a Stein structure. This follows from the theory of gluing symplectic handles (developed by Weinstein and Eliashberg), once we realize that K 1 can be represented by a Legendrian knot with Thurston–Bennequin num- ber equal to 0. (For this theory, the deﬁnitions of the above notions and Figure 1.2. Legendrian representative of the knot K1 constructions will be discussed in later chapters.) For such a Legendrian rep- resentative see Figure 1.2. Therefore, in order to distinguish X 1 from X 2 it is enough to prove that X 2 does not admit any Stein structure. This state- ment follows from the observation that the generator of H 2 (X 2 ; Z) (which has self-intersection −1) can be represented by a sphere. Such a represen- tative can be easily found once we get a disk D 2 ⊂ D 4 with ∂D 2 = K 2 — glue the core of the 2-handle to this disk. The existence of such a disk is shown by the “movie” of Figure 1.3. These pictures show how the disk in- tersects the spheres with radius r < 1 in the 4-disk D 4 as r grows from 0 to 1. These intersections start with two circles (which are boundaries of two disks) which get tangled as “time” passes (Figures 1.3 (1)–(2)), and then a ribbon is added to connect the disks, resulting an embedded disk in D 4 16 1. Introduction with boundary given by Figure 1.3(3). (Of course, in this process the value −4 (1) (2) (3) Figure 1.3. The movie showing the disk −4 plays no special role; it becomes important when proving the diﬀeomor- phism ∂X 1 ∼ = ∂X 2 .) Now the application of the adjunction inequality with g = 0 and [Σ] 2 = −1 would give −1 +[ ¸ c 1 (X 2 , J), [Σ] _ [ ≤ −2, a contradiction for any Stein structure J on X 2 . Therefore X 2 cannot carry any Stein structure, implying that X 1 and X 2 are nondiﬀeomorphic. Remark 1.2.4. The above example was found by Akbulut [4], using diﬀer- ent methods in the proof of nondiﬀeomorphism. This version of the proof is due to Akbulut and Matveyev [6]. 1.2. Results concerning Stein surfaces 17 Existence of Stein neighborhoods Theorem 1.2.5. Let S ⊂ CP 2 be a smoothly embedded sphere in CP 2 which is nontrivial in homology. Then there is no open set U containing a neighborhood of S which admits a Stein structure. Proof. The adjunction inequality of Theorem 1.2.1 implies that a homo- logically nontrivial sphere in a Stein surface has self-intersection ≤ −2. For [S] ,= 0 in H 2 (CP 2 ; Z) we have that [S] 2 > 0, providing the result. Remark 1.2.6. The same argument works for any smooth 2-dimensional submanifold Σ in a complex surface X with 2g(Σ) −2 < [Σ] 2 . Surprisingly enough, if the inequality [Σ] 2 +[ ¸ c 1 (X), [Σ] _ [ ≤ 2g(Σ) −2 does hold (notice that because of the absolute value this is, in fact, the union of two inequal- ities) then there is a Stein neighborhood U ⊂ CP 2 of Σ, see [49]. For the outline of this latter argument see also Section 8.3. This application of the adjunction inequality leads to the solution of a seemingly unrelated problem in complex analysis. Corollary 1.2.7 (Nemirovski, [125]). Suppose that S ⊂ CP 2 is a smoothly embedded sphere in CP 2 which is nontrivial in homology. If f is a holo- morphic function on some neighborhood of S then f is constant. Proof. Let us ﬁx a neighborhood U and a holomorphic function f on it. Consider the envelope of holomorphy of U, i.e., the maximal domain ˜ V containing U such that every holomorphic function on U extends holomor- phically to ˜ V . Denote this envelope of holomorphy by ˜ U. According to a result of Fujita, in our case ˜ U is either CP 2 or it is Stein. Since S ⊂ U ⊂ ˜ U, Theorem 1.2.5 shows that ˜ U cannot be Stein. Therefore ˜ U = CP 2 , hence all holomorphic functions on ˜ U (and so on U) are constant. Remark 1.2.8. Notice that if S is a complex submanifold generating H 2 (CP 2 ; Z) then the statement is obvious: if f is a holomorphic function on U then by restricting it to U ∩ (CP 2 − CP 1 ) ⊂ C 2 and applying a theorem of Hartogs we get an extension of f to CP 2 , implying that it is constant. The question answered by the above theorem was raised by Vitushkin, and similar results (for higher genus and immersed surfaces) are still in the focus of current research, see [49, 125]. 18 1. Introduction The four-ball genus of knots in S 3 Let K ⊂ S 3 be a given knot. The genus g(K) is deﬁned as min _ g(Σ) [ Σ ⊂ S 3 is a Seifert surface for K _ . For example, it is fairly easy to see that g(K) = 0 holds if and only if K is the unknot. The four-ball genus (or slice genus) g ∗ (K) can be deﬁned as min _ g(F) [ F ⊂ D 4 , ∂F = K _ , where F denotes a smoothly embedded connected surface in D 4 transverse to ∂D 4 . Obviously g ∗ (K) ≤ g(K), and as the proof of Corollary 1.2.3 showed, g ∗ (K) can be equal to 0 for a nontrivial knot K, e.g. for K 2 . (Knots with vanishing four-ball genus g ∗ are called smoothly slice.) The adjunction inequality provides a nontrivial lower bound for g ∗ (K) in the following way. Approximate K with a Legendrian knot L and glue a 2- handle to D 4 along K with surgery coeﬃcient one less than the contact framing of L. The resulting 4-manifold X will contain a surface ˆ F with g( ˆ F) = g ∗ (K), obtained by gluing the four-ball genus minimizing surface to the core of the 2-handle. Since X carries a Stein structure, and [ ˆ F] 2 and ¸ c 1 (X), [ ˆ F] _ admit expressions purely in terms of data of the Legendrian knot L as [ ˆ F] 2 = tb(L) − 1 and ¸ c 1 (X), [ ˆ F] _ = rot(L), the adjunction inequality gives a lower bound for g ∗ (K): tb(L) +[ rot(L)[ ≤ 2g ∗ (K) −1. For example: Corollary 1.2.9. The trefoil knot is not smoothly slice. Proof. The right-handed trefoil admits a Legendrian presentation with tb(L) = 1 and rot(L) = 0 (see Figure 1.4), hence the adjunction inequality translates as 0 ≤ 2g ∗ (K) − 2, implying 1 ≤ g ∗ (K). It is not hard to ﬁnd a genus-1 Seifert surface for K, therefore we see that the four-ball genus of the trefoil knot is 1. The unknotting number (or gordian number) u(K) of a knot K is deﬁned as the minimal number of crossing changes in any projection which untie the knot. Exercise 1.2.10. Show that u(K) ≥ g ∗ (K) for any knot K ⊂ S 3 . 1.2. Results concerning Stein surfaces 19 Figure 1.4. A right-handed Legendrian trefoil knot Notice that the inequality u(K) ≥ g ∗ (K) is not an equality in general; take for example the knot K 2 of Figure 1.1, which has g ∗ (K 2 ) = 0 but u(K 2 ) > 0 since it is not the unknot. Heegaard Floer theory provides knot invariants which can be fruitfully used to get new constraints on the 4-ball genus of a knot, see [129, 142]. Topological characterization of Stein domains According to a recent result of Loi and Piergallini [104], Stein domains admit a nice topological description in terms of Lefschetz ﬁbrations. Theorem 1.2.11 (Loi–Piergallini, [104]). If S is a complex 2-dimensional Stein domain then it admits a Lefschetz ﬁbration structure over D 2 . This result — similarly to Donaldson’s result on existence of Lefschetz pencils on closed symplectic 4-manifolds — brings a topological ﬂavour into the study of Stein domain. The original proof of Theorem 1.2.11 relies on an approach of presenting the 4-manifolds at hand as branched covers of D 4 along fairly complicated branch sets. A conceptually simpler proof of the same statement was given by Akbulut and the ﬁrst author [7], making use of the handle decomposition of a Stein domain and relating it to handle decompositions of 4-manifolds admitting Lefschetz ﬁbrations. The detailed description of this second approach will be given in Chapter 10. 20 1. Introduction 3-manifolds which are not Stein boundaries Our ﬁnal example in this section shows that the boundary of a Stein domain cannot be arbitrary. Theorem 1.2.12 (Lisca, [94]). Let E denote the boundary of the (+E 8 )- plumbing W (as shown by the plumbing diagram of Figure 1.5). There is 2 2 2 2 2 2 2 2 Figure 1.5. The (+E8)-plumbing no Stein domain S with ∂S = E. Proof. Using standard pull-apart arguments in Seiberg–Witten theory (see Chapter 13) it can be shown that if X = X 1 ∪ E X 2 and X is symplectic then b + 2 (X 1 ) = 0 or b + 2 (X 2 ) = 0. (This argument uses the fact that E admits a positive scalar curvature metric, since it is diﬀeomorphic to the Poincar´e homology sphere, with its standard orientation reversed, cf. Proposition 13.1.7(5.).) Now if S is a Stein domain with ∂S = E then S can be embedded into a closed symplectic 4-manifold X with b + 2 (X−S) > 0. In conclusion, from the above principle we get b + 2 (S) = 0. Therefore the closed 4-manifold Z = S ∪ E (−W) is negative deﬁnite. Since the intersection form of −W (which is the famous negative deﬁnite E 8 -form) does not embed into any diagonal intersection form, the intersection form Q Z cannot be diagonalized over the integers. This last consequence, however, contradicts Donaldson’s famous result about diagonalizability of deﬁnite intersection forms of smooth 4-manifolds, showing that S cannot exist. Remark 1.2.13. Analogous statements have been proved for the bound- aries of the (+E 7 )- and (+E 6 )-plumbings [96]. Results of this type will be discussed in Chapter 12 in more detail. We just note here that by ap- plying Seiberg–Witten invariants of manifolds with contact type boundary (see Section 13.2) it can be shown that these 3-manifolds (and many more of similar type) admit no symplectically ﬁllable contact structures [94, 101]. 1.3. Some contact results 21 1.3. Some contact results As we will see, contact structures on 3-manifolds fall into two very diﬀerent classes. Overtwisted structures were classiﬁed by Eliashberg, and the clas- siﬁcation scheme depends only on homotopic properties of the underlying 3-manifold. On the other hand, tight structures are expected to contain more geometric information about the manifold. The contact counterpart of the adjunction formula (frequently called the Bennequin inequality) char- acterizes tight structures. This inequality reads as follows: Suppose that Σ is an embedded surface-with-boundary in the contact 3-manifold (Y, ξ) with ∂Σ = L a Legendrian curve. (For the deﬁnitions of the notions used here, see Chapter 4.) Let tb Σ (L) ∈ Z denote the framing induced by the contact structure on L with respect to the framing Σ deﬁnes on L and rot Σ (L) the relative Euler number of ξ[Σ with ξ trivialized along ∂Σ by the tangents of L. Now Theorem 1.3.1 (Eliashberg, [26]). The inequality tb Σ (L) + ¸ ¸ rot Σ (L) ¸ ¸ ≤ −χ(Σ) is satisﬁed for all L and Σ if and only if the contact 3-manifold (Y, ξ) is tight. Contact structures and open books Just like Donaldson’s theory of symplectic Lefschetz pencils gives a topolog- ical characterization of symplectic 4-manifolds, recent work of Giroux gives a characterization of contact 3-manifolds in terms of open books. Giroux proved that there is a one-to-one correspondence between open books and contact structures on 3-manifolds up to some natural equivalence relations. More precisely, for a given closed 3-manifold Y the following holds: Theorem 1.3.2 (Giroux, [63]). (a) For a given open book decomposition of Y there is a compatible contact structure ξ on Y . Contact structures compatible with a ﬁxed open book decomposition are isotopic. (b) For a contact structure ξ on Y there is a compatible open book de- composition of Y . Two open book decompositions compatible with a ﬁxed contact structure admit common positive stabilization. 22 1. Introduction The reinterpretation of contact structures provided by this theorem enables us to treat them as topological objects. The nicest manifestation of this principle is probably the deﬁnition and application of contact Ozsv´ath– Szab´ o invariants discussed in Chapter 14. It is still an open (and very intriguing) question how the monodromy of the open book decomposition encodes tightness/ﬁllability properties of the corresponding compatible con- tact structure (cf. Chapter 9). As an example of results in this direction, we have the following theorem of Giroux: Theorem 1.3.3. An open book decomposition gives rise to a Stein ﬁllable contact structure if and only if it admits a positive stabilization for which the monodromy decomposes as a product of right-handed Dehn twists. Nonﬁllable contact 3-manifolds Suppose that (Y, ξ) is the boundary of a compact symplectic 4-manifold (X, ω) in the sense that ∂X = Y as oriented manifolds and ω[ ξ ,= 0. In this case we say that (Y, ξ) is (weakly) symplectically ﬁllable (or just ﬁllable), and (X, ω) is called a (weak) symplectic ﬁlling of (Y, ξ). The Bennequin inequality in (Y, ξ) now follows from the adjunction inequality for (X, ω), i.e., ﬁllable structures are always tight. The converse of this implication, however, does not hold: a contact manifold can be tight without being the appropriate boundary of any symplectic 4-manifold. The ﬁrst such structures were found by Etnyre and Honda [44]; we will give a variety of such contact 3-manifolds, cf. also [100, 101]. Next we give a sample of these results. Contact surgery provides a simple way for constructing contact 3- manifolds. Because of its topological character, the surgery diagram can be used very fruitfully to apply Heegaard Floer theory leading to the following Theorem 1.3.4 ([101]). The contact 3-manifold (Y, ξ) given by the surgery diagram of Figure 1.6 is tight but not ﬁllable. Proof (sketch). Nonﬁllability of this contact structure follows from the fact that the underlying 3-manifold Y is diﬀeomorphic to the boundary of the (+E 7 )-plumbing (see Exercise 2.3.5(e) together with Remark 1.2.13). Tightness follows from the fact that the contact Ozsv´ath–Szab´o invariant c(Y, ξ) is nonzero, although for overtwisted structures this invariant van- ishes. (For an outline of the deﬁnition of c(Y, ξ) ∈ ¯ HF(−Y ) see Chapter 14, for the computation in the above case, see Section 12.2.) 1.3. Some contact results 23 +1 Figure 1.6. Surgery diagram of a tight nonﬁllable contact 3-manifold This simple construction leads us to a plethora of similar examples — see Section 12.2. By a variant of these ideas we get Theorem 1.3.5 ([100]). For any n ∈ N there is a closed 3-manifold with at least n distinct tight, nonﬁllable contact structures. These examples will be given by Figure 12.6. Once again, tightness will be proved by (partially) determining the contact Ozsv´ath–Szab´o invariants, while we will show that the structures are nonﬁllable through determining homotopic properties of the contact structures via analyzing the diagram and then apply a version of Seiberg–Witten theory. This last step is a straightforward generalization of the proof of Theorem 1.3.4. Topology of Stein ﬁllings Another leading theme we will focus on in the study of a contact 3-manifold is trying to determine all its Stein ﬁllings. As we will see, for some simple 3- manifolds this problem can be solved, a prototype result (due to Eliashberg) gives the following Theorem 1.3.6 (Eliashberg). If W is a Stein ﬁlling of the standard contact 3-sphere (S 3 , ξ st ) then W is diﬀeomorphic to the 4-dimensional disk D 4 . Proof (sketch). Gauge theory as applied in the proof of Theorem 1.2.12 implies that b + 2 (W) = b − 2 (W) = 0. The surgered manifold Z = W ∪ (CP 2 −D 4 ) is a symplectic 4-manifold containing a symplectic sphere of self- intersection (+1), hence Z is symplectomorphic to CP 2 . Since a symplectic sphere representing the generator of H 2 (CP 2 ; Z) is isotopic to the complex 24 1. Introduction line CP 1 ⊂ CP 2 , we get that W is diﬀeomorphic to D 4 . For more details of this argument see Section 12.3. Similar strong classiﬁcation results of Stein ﬁllings have been obtained for a variety of 3-manifolds (see [96, 110] or Section 12.3), but the general description of all Stein ﬁllings of a contact 3-manifold is still missing. Here we restrict ourselves to two statements along these lines: Theorem 1.3.7 ([132]). There are contact 3-manifolds with inﬁnitely many nondiﬀeomorphic Stein ﬁllings. The proof of this theorem (see in Section 12.3) makes use of the connection between Stein structures and Lefschetz ﬁbrations. Using symplectic cut- and-paste technique and applying Seiberg–Witten theory we will get some restrictions on the topology of a Stein ﬁlling of a ﬁxed contact 3-manifold, for example Theorem 1.3.8 ([159]). For a given contact 3-manifold (Y, ξ) there exists a constant K (Y,ξ) such that if W is a Stein ﬁlling of (Y, ξ) then 3σ(W) + 2χ(W) ≥ K (Y,ξ) . In other words, the number c(W) = 3σ(W) + 2χ(W) for a Stein ﬁlling W of (Y, ξ) — which resembles the c 2 1 -invariant of a closed complex surface — is bounded from below. A little elaboration of the above result together with some speciﬁc cases gives evidence for the following Conjecture 1.3.9. For any contact 3-manifold (Y, ξ) there is a constant K such that if W is a Stein ﬁlling of (Y, ξ) then for its Euler characteristic χ(W) the inequality χ(W) ≤ K holds. 2. Topological surgeries After the short Prelude given in the introductory chapter we begin our dis- cussion by reviewing the smooth constructions behind contact and Stein surgeries. We assume that the reader is familiar with basics in diﬀeren- tial topology as given, for example, in [72]. Standard facts regarding sin- gular homology and cohomology theory will also be used without further explanation. The manifolds appearing in these notes are all assumed to be smooth (i.e., C ∞ -) manifolds, possibly with nonempty boundary. The gen- eral discussion of handlebodies will be followed by a short overview of Dehn surgeries in dimension three, and an outline of Kirby calculus concludes the chapter. For more details about the ideas and constructions sketched here, see [66]. 2.1. Surgeries and handlebodies The main construction behind all surgeries can be summarized by the follow- ing fairly simple scheme: Suppose that X 1 , X 2 are given n-dimensional man- ifolds with boundaries and Z i ⊂ ∂X i are (n −1)-dimensional submanifolds (with possibly nonempty boundary). For a diﬀeomorphism f : Z 1 →Z 2 we can glue the two manifolds X 1 and X 2 together along Z i via f, and get a new n-manifold X = X 1 # f(Z 1 )=Z 2 X 2 (with possibly nonempty boundary). In the following we will always assume that X i and Z i are compact (and then so is X), and that the X i are oriented. Note that an orientation of X i induces one for ∂X i and so orients Z i as well. In order to have a canonical orientation for X, we assume that f reverses orientation. Remark 2.1.1. In order to give a manifold structure to X we have to round oﬀ the corners created by gluing along Z i (which might have (n−2)- dimensional boundaries). This process is fairly straightforward in dimen- 26 2. Topological surgeries sion 2: we replace an angular corner by a region below a hyperbola, and by multiplying this picture with the extra dimensions, the same can be carried out in arbitrary dimensions, see [66]. The reason why the above construction works is that the boundaries ∂X i and so Z i admit “canonical” neighborhoods (by the collar neighborhood theorem), hence once the map f is ﬁxed, neighborhoods of Z i can be identiﬁed and so the smooth structures can be patched together. The same scheme will work for other structures (like symplectic, contact, and so on) once the right assumptions ensuring canonical neighborhoods have been made. The drawback of this general construction is that usually it is quite hard to describe and identify f, although — as we will see — in many cases the particular choice of the identiﬁcation is crucial. Here are a few simple examples of this operation: Examples 2.1.2. (a) Suppose that X 1 , X 2 are compact manifolds with boundaries ∂X 1 , ∂X 2 orientation reversing diﬀeomorphic via a smooth map f : ∂X 1 →∂X 2 . Then X = X 1 ∪ f X 2 is a closed manifold. (b) Suppose that X i are closed n-manifolds. Consider X i − int D n and glue them with an orientation reversing map f : S n−1 = ∂(X 1 −int D n ) → S n−1 = ∂(X 2 −int D n ) which extends to the disk D n . (This latter require- ment speciﬁes f up to isotopy.) The resulting manifold X = X 1 #X 2 is called the connected sum of the two manifolds X 1 and X 2 . For X i con- nected, the result can be proved to be independent of the choice of the disks. (c) Another special case of this general construction is when Z i ⊂ ∂X i (i = 1, 2) are both diﬀeomorphic to the (n − 1)-dimensional disk D n−1 . The resulting manifold X is usually denoted by X 1 ♮X 2 and is called the boundary connected sum of X 1 and X 2 along Z 1 and Z 2 . As in the previous case, the result can be proved to be independent of the choices provided the boundaries ∂X 1 and ∂X 2 are connected. (d) Suppose that X = X 1 ∪ Σ X X 2 and Y = Y 1 ∪ Σ Y Y 2 are closed manifolds (X i ⊂ X and Y i ⊂ Y , i = 1, 2, are compact codimension-0 submanifolds with boundaries and disjoint interiors). If there is an orientation preserving diﬀeomorphism f : ∂X 1 →∂Y 1 then we can use it to glue X 1 and Y 2 together along their boundaries to get Z = X 1 ∪Y 2 (and similarly V = Y 1 ∪X 2 ), see Figure 2.1. The choice of f is usually crucial in this construction. Exercises 2.1.3. (a) Let X 1 = X 2 = [0, 1] [0, 1] and Z 1 = Z 2 = ¦0¦ [0, 1] ∪ ¦1¦ [0, 1] ⊂ [0, 1] [0, 1]. Find f such that the resulting 2.1. Surgeries and handlebodies 27 X X Y Y 1 2 1 2 X Σ Y Σ Figure 2.1. Flipping X2 with Y2 manifold X f is diﬀeomorphic to S 1 [0, 1] and g such that the result X g is a M¨obius band. (b) Show that both S 3 and S 1 S 2 can be built by gluing two solid tori S 1 D 2 together (using diﬀerent identiﬁcations of the boundary tori). (c) More generally, show that every closed, oriented 3-manifold Y can be given as Y = H 1 ∪ H 2 where H 1 ∼ = H 2 are solid genus-g three-dimensional handlebodies with ∂H 1 ∼ = ∂H 2 = Σ g , where Σ g stands for the genus-g surface. (Hint: Use a Morse function.) Such a decomposition is usually called a Heegaard decomposition of Y . (d) Verify that ∂(X 1 ♮X 2 ) = ∂X 1 #∂X 2 . Notice that ∂(S k D n−k ) = ∂(D k+1 S n−k−1 ) = S k S n−k−1 , hence if S k ⊂ X n is a submanifold with trivial normal bundle νS k ∼ = S k D n−k then cutting out S k D n−k and gluing back in D k+1 S n−k−1 we get a new manifold. Once again, the chosen identiﬁcations do matter. A trivialization of νS k in X is called a framing of the submanifold. By ﬁxing a framing ϕ we get an embedding ˜ ϕ: S k D n−k → X. (If k ≤ 3 then the parametrization of S k ⊂ X is unique, otherwise we think of S k ⊂ X as given by the image of a map.) Now we can use ˜ ϕ[ ∂(S k ×D n−k ) to glue D k+1 S n−k−1 back in; the new manifold is the result of the surgery of X along S k (with the given framing). Notice that the connected sum operation of Example 2.1.2(b) is just a special case of this surgery scheme: Embed the disconnected manifold S 0 D n = ¦±1¦ D n into X 1 ∪ X 2 in such a way that ¦−1¦ D n ⊂ X 1 and ¦1¦ D n ⊂ X 2 and then do surgery on it, i.e., replace S 0 D n with the cylinder D 1 S n−1 . The eﬀect of the above surgery construction is the same as the following: Consider the (n + 1)- manifold X [0, 1] and attach an (n + 1)-dimensional (k + 1)-handle (or 28 2. Topological surgeries handle of index (k + 1)) D k+1 D n−k along the part ∂D k+1 D n−k of its boundary to ϕ(S k ) ⊂ X ¦1¦ with the speciﬁed framing — the image ϕ(S k ) and the framing completely determine the gluing map f. During this construction the tubular neighborhood ϕ(S k )D n−k sinks in the interior of the cobordism and D k+1 S n−k−1 appears on the surface. More generally, if X is an (n + 1)-manifold with boundary ∂X and ϕ: S k D n−k → ∂X is a given embedding, i.e., a framed sphere ϕ(S k ) is given, then we can glue the (n + 1)-dimensional (k + 1)-handle D k+1 D n−k to X using ϕ and get a new manifold. The repeated application of the above process (starting with a given closed n-manifold M and considering M [0, 1]) is called a (relative) handlebody built on M. Notice that M might be the empty manifold, in which case we get a handlebody. (In that case, to start the process, we ﬁrst glue a 0-handle D 0 D n+1 along ∂D 0 D n+1 = ∅ to the empty manifold.) It can be shown that any compact smooth manifold admits a handlebody decomposition, i.e., is diﬀeomorphic to a handlebody. (A relatively elementary proof of this statement can be found in [114, 115], where Morse theory is applied.) It is not hard to enumerate the possible framings an embedded sphere can have: ﬁx a framing and try to relate all the others to this ﬁxed one. Notice ﬁrst that framings need to be speciﬁed only up to homotopy. By assuming linearity on the ﬁbers (which can be achieved by an isotopy), any other framing deﬁnes a linear map at every point of the sphere (which linear map matches up the chosen bases in the ﬁber of the normal bundle), so at the end we get a map S k → GL n−k (R). Since homotopy does not change the framing, and GL n−k (R) retracts to O(n −k), we conclude that the diﬀerent framings of the k-dimensional sphere S k in an n-manifold are parametrized by π k _ O(n − k) _ . In particular, this shows that once n ≥ 2 the framing is unique if k = n − 1 or n. For k = 0 there are two possible framings, corresponding to the two components of O(n). One gives rise to an orientable, while the other to a nonorientable manifold. Since we restrict our attention to the study of orientable manifolds, we get uniqueness of framings even for k = 0. Notice that if we are dealing with oriented 3- and 4-manifolds (so n = 2 or 3), then there is only one more case to consider, namely when n = 3 and k = 1, i.e., when we glue 4- dimensional 2-handles to a 3-dimensional boundary. In this case we frame embedded circles in 3-manifolds, and the set of framings is parameterized by π 1 _ O(2) _ ∼ = π 1 (S 1 ) ∼ = Z. In this special case the normal D 2 -bundle can be regarded as a complex line bundle, hence it can be trivialized by a nowhere vanishing section. In conclusion, a framing of a knot K in a 3-manifold can 2.1. Surgeries and handlebodies 29 be most conveniently symbolized by an appropriate push-oﬀ of K. In order to set up an actual isomorphism between the set of framings and Z, we need to choose a preferred framing ﬁrst (which we will call the 0-framing). This choice, however, is canonical only in some special cases: for example, in S 3 or if the knot is null-homologous in the 3-manifold Y . Another instance of the existence of a canonical framing is provided by the situation when the knot is in Legendrian position in a contact 3-manifold, or if the knot is naturally contained in a surface (which induces a natural framing by pushing the knot oﬀ of itself inside the surface) — such a surface can be provided by a ﬁber of a ﬁbration, or a page of an open book decomposition, for example. Exercises 2.1.4. (a) For K ⊂ S 3 ﬁx a Seifert surface and consider the 0- framing to be the push-oﬀ of the knot along the Seifert surface. Show that this framing — called the Seifert framing — does not depend on the chosen Seifert surface, and that the isomorphism between the space of framings and Z is given by the linking number of K and the push-oﬀ of it along the framing. (For the linking number to make sense, ﬁx an orientation on K and orient any push-oﬀ accordingly.) (b) Generalize the uniqueness of the Seifert framing for any null-homologous knot in an arbitrary 3-manifold Y . (Hint: Argue that if two diﬀerent framings come from Seifert surfaces then their diﬀerence vanishes in the ﬁrst homology of the knot complement, contradicting the fact that the knot is null-homologous.) (c) Verify that the push-oﬀ K ′ of a null-homologous knot K ⊂ Y deﬁnes the Seifert framing if and only if the homology class [K ′ ] of K ′ vanishes in H 1 (Y −K; Z). Remark 2.1.5. When drawing the projection of a knot K ⊂ R 3 in R 2 (with the usual genericity assumptions and the conventions of over- and under-crossings), there is one more natural framing we can consider: it is the blackboard framing bb(K) induced by the particular projection. We get the blackboard framing bb(K) by pushing K oﬀ along a vector ﬁeld parallel to the plane of the projection, see Figure 2.2. Notice that this framing heavily depends on the chosen projection. The conversion between the Seifert framing and the blackboard framing is given by the linking number of K and its parallel push-oﬀ. Exercise 2.1.6. For a given knot K ⊂ S 3 compute the blackboard framing bb(K) of one of its projections with respect to the Seifert framing of the knot. 30 2. Topological surgeries Figure 2.2. The blackboard framing Conclude that w(K) = bb(K), where w(K) is the writhe of the projection. (The writhe is deﬁned as the signed sum of crossings in the projection. For this to make sense, we need to ﬁx an orientation on K, but w(K) can be proved to be independent of this choice of orientation. For the sign of a crossing see Figure 2.3.) _ + Figure 2.3. Positive and negative crossings As a consequence of our framing computation above, we get that in order to build 3-dimensional orientable handlebodies one only needs to keep track of the attaching spheres. For 1-handles these are essentially unique since S 0 = ¦−1¦ ∪ ¦1¦ embeds into a connected manifold uniquely up to isotopy. For 2-handles we get embedded circles in a genus-g surface — or equivalently, in the plane with 2g holes, which are glued together in pairs. For convenience, we identify S 2 with R 2 ∪ ¦∞¦ and draw the diagrams in its “ﬁnite” part R 2 . These pictures are called Heegaard diagrams. (For 2.2. Dehn surgery 31 an example see Figure 2.4. Here the shaded disks are identiﬁed with each other. In case of more pairs of shaded disks we connect the pairs to be identiﬁed with dotted lines. The curve of Figure 2.4 might seem to have many components, but after identifying the shaded disks it becomes a connected 1-manifold.) Finally, a 3-handle can be attached uniquely to Figure 2.4. A Heegaard diagram a 3-manifold with boundary diﬀeomorphic to S 2 . In dimension four the 2-handle attachment is somewhat more complicated; we will return to the detailed discussion of this question in Section 2.3. 2.2. Dehn surgery There is one more — purely 3-dimensional — construction we would like to discuss, frequently called Dehn or rational surgery. The basic idea is again pretty simple: consider a 3-manifold Y (for simplicity we assume that it is closed), and ﬁx a knot K ⊂ Y in it. By deleting a tubular neighborhood νK ( ∼ = S 1 D 2 ) of K and regluing it via a diﬀeomorphism f : ∂νK → ∂(Y −νK) we get a new 3-manifold. Obviously, the resulting manifold will depend on the chosen gluing map f. Notice that we reglue S 1 D 2 along a (2-dimensional) torus, and the self-diﬀeomorphisms f : T 2 →T 2 are well- understood: up to isotopy such an f is determined by the induced map f ∗ : H 1 (T 2 ; Z) → H 1 (T 2 ; Z) ∼ = Z 2 , i.e., (after ﬁxing a basis of H 1 (T 2 ; Z)) by a 2 2 integer matrix. Since f (and so f ∗ ) is invertible, the matrix is of determinant ±1; the fact that f reverses orientation implies that det f ∗ = −1. Consequently, after ﬁxing a basis of H 1 (T 2 ; Z), four integers specify f. Diﬀerent matrices might yield diﬀeomorphic 3-manifolds, for example, if for the maps f 1 , f 2 : ∂(S 1 D 2 ) →∂(Y −int νK) the composition 32 2. Topological surgeries f −1 2 ◦f 1 : ∂(S 1 D 2 ) →∂(S 1 D 2 ) extends to a diﬀeomorphism of S 1 D 2 then the surgered manifolds using f 1 or f 2 will be diﬀeomorphic. Actually, two of the four numbers already determine the surgery; we show this fact from a slightly diﬀerent point of view. Notice that S 1 D 2 can be thought of as the union of a 3-dimensional 2-handle and a 3-handle, and remember that the gluing of 3-handles is unique, while for 2-handles one only needs to specify the gluing circle, which is an embedded simple closed curve in ∂(Y − int νK). (Recall that in dimension three there is no framing issue.) Consequently the Dehn surgery is determined by a simple closed curve in ∂(Y − int νK), which can be given (up to isotopy) by ﬁxing its homology class a ∈ H 1 _ ∂(Y − int νK); Z _ ∼ = Z 2 . This is the curve which bounds the disk ¦pt.¦ D 2 in the surgered manifold. Deﬁnition 2.2.1. For a ﬁxed closed 3-manifold Y , knot K ⊂ Y and primitive element a ∈ H 1 _ ∂(Y −int νK); Z _ the manifold (Y −int νK) ∪ f (S 1 D 2 ) will be denoted by Y a (K), where f : ∂(S 1 D 2 ) →∂(Y −int νK) is speciﬁed by f ∗ _ ¦pt.¦ ∂D 2 ¸ = a ∈ H 1 _ ∂(Y −int νK); Z _ . The resulting manifold Y a (K) is called the Dehn surgery of Y along K with slope a. Notice that since _ ¦pt.¦∂D 2 ¸ is nondivisible and f ∗ is invertible, the chosen class a should be a primitive class. It can be proved that the simple closed curve representing such a homology class a is unique up to isotopy. A choice of a basis of H 1 _ ∂(Y −int νK); Z _ ∼ = Z 2 converts a into a pair of relatively prime integers. A canonical choice for one basis element is pro- vided by the meridian µ ∈ H 1 _ ∂(Y − int νK); Z _ of the knot K — i.e., a nontrivial primitive element which vanishes under the embedding of ∂νK into νK. By ﬁxing an orientation on K, the element µ is uniquely deter- mined by the requirement that it links K with multiplicity +1 (otherwise µ is determined only up to sign). Informally, µ is the homology class of the circle which is the boundary of a small normal disk to K (i.e., a small disk intersecting K transversely in a unique point), see Figure 2.5(a). The choice of a longitude λ (another basis element in H 1 _ ∂(Y − int νK); Z _ ) is, how- ever, not canonical. For an example see Figure 2.5(b). The longitude can be ﬁxed without further choices only if K admits a canonical framing. In fact, ﬁxing a longitude or a framing is equivalent, since the normal R 2 -bundle (regarded as a C-bundle) is trivialized by a nonvanishing section, i.e., by a longitude. Therefore, if K ⊂ Y is null-homologous (for example, Y = S 3 ) or if K is a Legendrian knot in a contact manifold, then using the canonical longitude λ the homology class a can be converted into a pair of relatively prime integers (p, q) by setting a = pµ+qλ, but in general the integers will 2.2. Dehn surgery 33 µ λ K (b) (a) K Figure 2.5. (a) the meridian and (b) an example of a longitude for the knot K depend on the choice of λ. An orientation of K ﬁxes an orientation for both µ and λ, and by reversing the orientation on K these elements switch signs. Therefore, although (p, q) depends on the chosen orientation, their ratio p q does not. Notice that ∞ = 1 0 is also allowed. Therefore after ﬁxing a lon- gitude if needed, the rational number p q ∈ Q ∪ ¦∞¦ encodes all the gluing information we need. For this reason Dehn surgery is also called rational surgery. Notice ﬁnally that by deﬁnition the diﬀerence of two framings λ 1 and λ 2 is some integral multiple of the class µ. Remark 2.2.2. The two integers p and q can be easily recovered from the matrix f ∗ , since it maps the meridian of ∂(S 1 D 2 ) into pµ + qλ. This shows that after the appropriate trivializations f ∗ = _ p p ′ q q ′ _ with pq ′ −p ′ q = −1. Lemma 2.2.3. Fix K ⊂ Y and a framing for K. If p q ∈ Z (i.e., q = ±1) then Y a (K) = Y p q (K) can be given by an ordinary surgery, i.e., by a (4- dimensional) 2-handle attachment. If p q = ∞ (i.e., q = 0) then Y a (K) = Y for any knot K. Proof. The coeﬃcient p q being an integer means that the curve representing a is simply a push-oﬀ of the knot K, therefore it determines a framing on it. 34 2. Topological surgeries This shows that integral Dehn surgery has the same eﬀect as 4-dimensional handle attachment. If q = 0 then a = µ, so we simply glue back the 2- handle of S 1 D 2 in the way it was before the surgery. Remark 2.2.4. Notice that the fact that p q is an integer is independent of the choice of λ, since µ is canonical, and λ just speciﬁes a “parallel” circle to K. Similarly, p q = ∞ is independent of the choice of λ since in this case a can be represented by a meridian. Alternatively, observe that for a = p 1 µ + q 1 λ 1 = p 2 µ + q 2 λ 2 we have that q 1 = q 2 and (provided λ 1 − λ 2 = kµ) that p 2 = p 1 + q 1 k. This argument shows again that the value of [q[ is independent of the chosen longitude. The following fundamental theorem asserts representability of 3-mani- folds by Dehn surgeries: Theorem 2.2.5 (Lickorish and Wallace, [93]). Every closed, oriented 3- manifold can be given as Dehn surgery on a link in S 3 . In fact, all rational numbers used in the surgery presentation can be assumed to be integers. Proof. Since the third cobordism group Ω 3 vanishes, for a 3-manifold Y there is an oriented 4-manifold W such that ∂W = Y . Surgering out the 1- and 3-handles we get a presentation of Y as the boundary of D 4 ∪ some 2-handles, hence as an integral surgery on a link. For a particular example of 3-manifolds we consider lens spaces. For this matter, for coprime integers p > q ≥ 1 take the group G p,q = __ z 0 0 z q _ ¸ ¸ ¸ z ∈ C, z p = 1 _ ⊂ U(2) and denote the factor S 3 /G p,q by L(p, q). (By viewing S 3 ⊂ C 2 as vectors of unit length, the action of U(2) on S 3 is obvious.) As Exercise 2.2.6 shows, L(p, q) is the result of − p q -surgery on the unknot. In fact, lens spaces are exactly those 3-manifolds which admit Heegaard decompositions along the 2-dimensional torus T 2 . Exercise 2.2.6. Show that the lens space L(p, q) is diﬀeomorphic to the result of a − p q -surgery on the unknot in S 3 . (Hint: See [148, Section 9B].) There are certain operations we can use to manipulate our surgery dia- grams without changing the resulting 3-manifold. In the following exercise we list those moves which will be useful in our subsequent discussion. 2.2. Dehn surgery 35 Exercises 2.2.7. (a) Verify the slam-dunk operation, i.e., that the two surgeries given by Figure 2.6 give diﬀeomorphic 3-manifolds. Here it is assumed that n ∈ Z and r ∈ Q ∪ ¦∞¦. (Hint: Perform surgery on K 2 ﬁrst and isotope K 1 into the glued-up solid torus T. Since ﬁrst we performed an integral surgery, K 1 will be isotopic to the core of T, hence when performing K r 2 n 1 K K n r 1 2 Figure 2.6. The slam-dunk operation the second surgery we cut T out again and reglue it. Therefore it can be done by one surgery; the coeﬃcient can be computed by ﬁrst assuming n = 0 and then adding n extra twists. For more details see [66, pp. 163–164].) (b) Turn a rational surgery in S 3 with coeﬃcient r into a sequence of integral surgeries. (Hint: Use the continued fraction expansion of r and apply (a) above. For the convention regarding continued fraction expansions see Section 11.1.) (c) Using the above result transform the Dehn surgery diagram of a lens space into an integral surgery on a linear chain of unknots. Using this diagram verify that L(p, q) = L(p, q ′ ) if qq ′ ≡ 1 (mod p). Warning 2.2.8. Notice that the slam-dunk operation can be performed only for n ∈ Z. Take, for example the 3-manifold given by the diagram of Figure 2.7. Applying a slam-dunk on the − 1 2 -framed circle we get L(2, 1) = RP 3 . But if we perform the illegal slam-dunk on the (−4)-framed circle, we get S 3 as a result. Exercises 2.2.9. (a) Verify the Rolfsen twist operation, i.e., that for n ∈ Z the two surgeries given by Figure 2.8 give diﬀeomorphic 3-manifolds. Here the framing of K is r = p q on the left and ( 1 r +n) −1 = p q+np on the right; the box with an n inside means n full twists (right-handed for n > 0 and left- handed for n < 0) and the surgery coeﬃcient on a component of the link 36 2. Topological surgeries RP 3 S 3 1 2 1 4 −4 −2 Figure 2.7. Warning with slam-dunks intersecting the spanning disk of K changes from r i to r i +n _ ℓk(K, K i ) _ 2 . The term ℓk(K, K i ) denotes the linking number of the two knots K and K i . (Hint: See [66, page 162].) q+np p n K q p Figure 2.8. Rolfsen twist (b) Verify that L(p, q) is diﬀeomorphic to L(p, q + np) for any integer n. (Hint: Introduce an ∞-framed normal circle to the − p q -framed unknot, perform Rolfsen twists and delete any ∞-framed surgery curve.) 2.2. Dehn surgery 37 (c) Show that surgery on the disjoint union of two framed links yields the connected sum of the two corresponding 3-manifolds. (We say that two links are disjoint if they can be separated by a plane.) (d) Verify that adding a disjoint unknot with surgery coeﬃcient (±1) or ∞ does not change the 3-manifold. (Hint: Show that (±1)-surgery along the unknot provides S 3 .) (e) Describe a diagram for −Y (the 3-manifold Y with opposite orientation) in terms of a diagram for Y . (Hint: Take the mirror image of the link presenting Y and multiply the framings by (−1).) In fact, there is a complete set of moves which determines when the resulting 3-manifolds are diﬀeomorphic: Theorem 2.2.10 (Kirby, [80]). Two links L, L ′ with rational coeﬃcients in S 3 determine diﬀeomorphic 3-manifolds through Dehn surgery if and only if L can be transformed into L ′ by a ﬁnite sequence of Rolfsen twists, isotopies and inserting and deleting components with coeﬃcient ∞. We have to note here that in particular cases it might be quite diﬃcult to ﬁnd the actual ﬁnite sequence of moves transforming one surgery picture of a given 3-manifold into another. Before turning to the 4-dimensional case, we show a way to read oﬀ the ﬁrst homology of the 3-manifold at hand from its rational surgery diagram. For this matter, suppose that Y is given by rational surgery on the n-component link L = (K 1 , . . . , K n ) ⊂ S 3 with surgery coeﬃcients p i q i with respect to the meridians µ i and longitudes λ i , where these latter provide the Seifert framings for K i in S 3 . It is not hard to see that H 1 (S 3 −∪ n i=1 int νK i ; Z) = Z n , freely generated by the homology classes of the meridians: simply use the long exact homology sequence of the pair (S 3 , S 3 −∪ n i=1 int νK i ). Next, as the surgery procedure dictates, we add a 3-dimensional 2- and a 3-handle to every T 2 -boundary component of S 3 −∪ n i=1 int νK i . Notice that if Σ i is a Seifert surface for K i (containing the longitude λ i ) then it provides the relation λ i = j=i ℓk(K i , K j )µ j , where ℓk(K i , K j ) stands for the linking number of the two knots in S 3 . Now each 2-handle provides a relation among the µ i ’s: by deﬁnition p i µ i + q i λ i becomes zero after the surgery (since this is the curve which bounds the core of the new 2-handle). Therefore we conclude Theorem 2.2.11. If Y is given by Dehn surgery along (K 1 , . . . , K n ) ⊂ S 3 with surgery coeﬃcients p i q i (i = 1, . . . , n) then H 1 (Y ; Z) can be presented 38 2. Topological surgeries by the meridians µ i (i = 1, . . . , n) as generators and the expressions p i µ i +q i j=i ℓk(K i , K j )µ j = 0 as relators. Corollary 2.2.12. If Y is given by p q -surgery along a knot K ⊂ S 3 then H 1 (Y ; Z) ∼ = Z p . (Here Z 0 is interpreted as Z.) We say that a 3-manifold Y is an integral homology sphere if H ∗ (Y ; Z) = H ∗ (S 3 ; Z); equivalently if H 1 (Y ; Z) = 0. The 3-manifold Y is a rational ho- mology sphere if H ∗ (Y ; Q) = H ∗ (S 3 ; Q); in other words, if ¸ ¸ H 1 (Y ; Z) ¸ ¸ < ∞. Alternatively, Y is a rational homology sphere if and only if its ﬁrst Betti number b 1 (Y ) is zero. Exercises 2.2.13. (a) Show that the 3-manifold S 3 r (K) we get by r-surgery on K ⊂ S 3 is an integral homology sphere if and only if r = 1 k for some k ∈ Z. (b) Suppose now that Y is given by (n 1 , . . . , n k )-surgery on the link L = (K 1 , . . . , K k ) ⊂ S 3 (n i ∈ Z). Verify that Y is an integral homology sphere if and only if the determinant of the linking matrix of L is ±1. Show that Y is a rational homology sphere if this determinant is nonzero. The diagonal entries of the linking matrix are given by the surgery coeﬃcients. (Hint: Use the long exact sequence for the pair of the 4-manifold X given by the 4-dimensional 2-handle attachment along L and the 3-manifold Y = ∂X.) 2.3. Kirby calculus Suppose that X n is a given smooth n-dimensional manifold. By choosing an appropriate Morse function on X we see that it admits a handlebody decomposition and we can always assume that our handlebody is built by attaching handles in the order with increasing index to the 0-handle D n . In this section we will focus on the n = 4 case. If X 4 is closed then (according to a result of Laudenbach and Po´enaru [91]) the gluing of the union of 3- and 4-handles (which union is diﬀeomorphic to ♮ k S 1 D 3 for some k) is unique. Therefore, in order to present closed 4-manifolds, we may restrict our attention to the discussion of 4-dimensional 2-handlebodies, i.e., 2.3. Kirby calculus 39 handlebodies involving handles with index ≤ 2. In addition, a Stein surface always admits a handle decomposition involving 0-, 1- and 2-handles only, hence the study of 2-handlebodies is suﬃcient for the purposes of these notes. The attaching of a 1-handle (at least if we assume orientability, which we always do) is unique up to isotopy. There are two common ways of picturing the attachment of a 1-handle to the boundary S 3 of the unique 0-handle D 4 . (For convenience we identify S 3 with R 3 ∪ ¦∞¦ and use only its “ﬁnite part” R 3 ). We can draw a pair of D 3 ’s in R 3 , indicating where the feet of the 1-handle are attached, or alternatively we can draw an unknot with a dot on it, symbolizing that we consider the 4-manifold D 4 − ¦a neighborhood of a spanning disk for the above unknot in D 4 ¦, i.e., a dotted circle refers to the compact 4-manifold _ D 2 − ν¦p¦ _ D 2 , where ν¦p¦ denotes a small tubular neighborhood of a point p ∈ int D 2 . Obviously, in both ways we get a 4-manifold diﬀeomorphic to S 1 D 3 . We will follow the latter convention, therefore the subhandlebody X 1 =union of 1-handles ∼ = ♮ k (S 1 D 3 ) will be symbolized by a k-component unlink in S 3 with a dot on every component: the unknots in S 3 simply denote the boundaries of the disks D 2 ¦p i ¦ (i = 1, . . . , k) deleted from D 4 . The 2-handles are attached along a framed link in ∂(♮ k S 1 D 3 ). By the above convention this link can be regarded as lying in S 3 , therefore (using the Seifert framings) the surgery coeﬃcients can be naturally converted into integers. A 2-handle passes through a 1-handle exactly when its attaching circle links with the dotted circle of the 1-handle. Such a link presentation of the 4-dimensional 2-handlebody is called a Kirby diagram. Remark 2.3.1. One can easily convert a handle picture using the ﬁrst convention into the dotted circle notation. To do this, ﬁrst isotope all attaching circles away from the region between the two feet D 3 ⊂ R 3 of the 1-handle. Then delete the embedded 3-balls, connect the attaching circles of the 2-handles and link them with a dotted circle. An example for this procedure is given by Figure 2.9. The ﬁrst convention (which uses the attaching balls of the 1-handle) is probably conceptually clearer, but when manipulating the diagram of an explicitly given 4-manifold, the dotted circle notation — introduced by Akbulut in [3] — is much more convenient. Exercises 2.3.2. (a) Verify that Figure 2.10 gives a diagram for D 2 T 2 . Visualize the ﬁbration on the diagram. (b) Show the equivalence of Figure 2.10 with the diagram of Figure 2.11. A given 4-manifold might admit many diﬀerent Kirby diagrams. Since any two Morse functions can be joined by a path of functions, by analyzing 40 2. Topological surgeries (1) (2) (3) Figure 2.9. Converting 1-handle into dotted circle 0 Figure 2.10. D 2 ×T 2 2.3. Kirby calculus 41 0 Figure 2.11. An alternative diagram for D 2 ×T 2 the changes during such a path one can prove that two diagrams repre- sent the same manifold if and only if they can be connected by repeated applications of the following moves: • isotopies of the link in S 3 , • handle slides and • adding/deleting cancelling 1/2- and 2/3-handle pairs. (A pair of handles is cancelling if their union amounts to a connected sum with D 4 .) In the diagram we visualize a 2-handle slide corresponding to circles K 1 , K 2 by connect summing K 1 to a push-oﬀ of K 2 corresponding to its framing along an arbitrary band. The new surgery coeﬃcient K ′ 1 becomes the sum of the old coeﬃcients of the two knots ± twice their linking number — the sign depends on whether the connecting band respects or disrespects a chosen orientation on K 1 and K 2 . One can slide 1-handles over each other as 0- framed 2-handles, and a 2-handle slides over a 1-handle by treating the latter as a 0-framed 2-handle. When sliding a 1-handle over an other 1-handle we must be careful with the choice of the band, since the resulting dotted circles should still form an unlink. A 1-handle/2-handle pair cancels if the 2-handle intersects the spanning disk of the 1-handle in a single point; in this case ﬁrst we slide oﬀ all the 2-handles geometrically linking the dotted circle in question (using, for example, the cancelling 2-handle) and then erase the 1/2-handle pair from the picture. The process can of course be reversed 42 2. Topological surgeries by introducing a pair of knots geometrically linking once (and one is the unknot); then by putting a dot on the unknot and an arbitrary surgery coeﬃcient to the other knot the 4-manifold remains the same. Finally, a 2-handle can be cancelled against a 3-handle if (possibly after handleslides) it can be represented by a 0-framed unknot disjoint from the rest of the picture. Notice that on an unknot (disjoint from the other dotted circles) we can have surgery coeﬃcient 0 or a dot — such a change corresponds to surgery along the sphere given by the 0-framed unknot. The operations listed above obviously do not change the boundary ∂X of a 4-manifold X given by a diagram. Changing a dot to 0-framing (or vice versa) changes the 4-manifold but leaves the boundary intact. Besides these moves, we can also insert or delete a (±1)-framed unknot disjoint form the rest of the picture — which corresponds to adding and removing a copy of CP 2 or CP 2 — without changing the boundary of the 4-manifold. For more details about these operations see [66]. The art of manipulating diagrams using the above rules and understanding the structure of smooth 3- and 4- manifolds in this way is frequently called Kirby calculus. Here we restricted ourselves to outline the very basics of this theory, and highlighted only the aspects which are important in our contact geometric studies. For a more complete treatment of Kirby calculus the reader is advised to turn to [66]. Example 2.3.3. The sequence of moves given by [66, Figure 11.14] provides a proof for the fact that the 4-manifolds X 1 , X 2 of Corollary 1.2.3 have diﬀeomorphic boundaries. Suppose that X admits a handlebody decomposition with a single 0- handle and some 1- and 2-handles. The homology groups H i (X; Z) and H i (∂X; Z) can be easily read oﬀ from a diagram corresponding to such a handle decomposition; this method will be discussed in the following. Consider the Abelian groups C 1 and C 2 freely generated by [K ′ 1 ], . . . , [K ′ t ] and [K 1 ], . . . , [K n ], corresponding to the t dotted circles and the n attaching circles of the 2-handles respectively, and deﬁne the map ϕ: C 2 →C 1 by [K] → t i=1 ℓk(K, K ′ i )[K ′ i ] on the generators and extend linearly. As for CW-complexes, we get C 1 / imϕ ∼ = H 1 (X; Z) and ker ϕ ∼ = H 2 (X; Z). This latter identity follows from C 3 = 0, which is the consequence of the absence of 3-handles. Now the universal coeﬃcient theorem and Poincar´e duality allows us to com- pute all homologies and cohomologies of X. In fact, the ring structure of 2.3. Kirby calculus 43 H ∗ (X, ∂X; Z) can also be read oﬀ from the picture. We restrict ourselves to the case when there are no 1-handles in the decomposition: if the homo- logy classes α 1 , . . . , α n ∈ H 2 (X; Z) are represented by Seifert surfaces of K i together with the cores of the 2-handles (i = 1, . . . , n) then we can easily see that PD(α i ) ∪PD(α j ) = ℓk(K i , K j ); as before, PD(α i ) ∪PD(α i ) = n i , the framing of K i . In other words, in the basis PD(α 1 ), . . . , PD(α n ) of H 2 (X, ∂X; Z) ∼ = H 2 (X; Z) the intersection form of X is represented by the linking matrix of the link ¦K i ¦ n i=1 . Here PD denotes the Poincar´e duality isomorphism between H 2 (X; Z) and H 2 (X, ∂X; Z). By considering surfaces F i in D 4 with ∂F i = K i and gluing the core disks to them we might ﬁnd lower genus representatives of the homology class α i ∈ H 2 (X; Z). The geo- metric intersections of these F i ’s are, however, harder to visualize. Next we discuss the computation of H i (∂X; Z). First perform surg- eries along the 1-handles, i.e., replace the dots on the dotted circles by 0. This transforms X into a simply connected 4-manifold Z but leaves ∂X unchanged. Notice that Z is the union of a 0-handle and m(= t + n) 2- handles which are attached (after renaming) along the knots K 1 , . . . , K m with framings n 1 , . . . , n m . According to the above said, H 2 (Z; Z) is freely generated by the closed surfaces Σ i we get by gluing an orientable Seifert surface of the knot K i and the core of the 2-handle together. After ﬁxing an orientation on K i these surfaces are canonically oriented: ﬁx the ori- entation making K i the oriented boundary of the Seifert surface. Let D i denote a small meridional disk to K i . It is fairly straightforward to see that H 2 (Z, ∂Z; Z) is generated by the relative homologies represented by [D i ] (i = 1, . . . , m). Here we choose an orientation on these disks in such a way that K i intersects D i positively when we use the orientation on K i ﬁxed above. The long exact homology sequence of the pair (Z, ∂Z) reduces to 0 →H 2 (∂Z; Z) →H 2 (Z; Z) ϕ 1 −→H 2 (Z, ∂Z; Z) ϕ 2 −→H 1 (∂Z; Z) →0 (since H 3 (Z, ∂Z; Z) ∼ = H 1 (Z; Z) = 0 and H 1 (Z; Z) = 0 by the simple connectivity of Z). As Theorem 2.2.11 shows, the map ϕ 1 is given by ϕ 1 _ [Σ i ] _ = n i [D i ] + j=i ℓk(K i , K j )[D j ], while ϕ 2 is simply ϕ 2 _ [D i ] _ = [∂D i ] = µ i , where µ i denotes the homology class of the linking normal circle of the knot K i oriented in such a way that their linking number is (+1). The exact sequence (with the maps described above) provides an explicit presentation for both H 1 (∂Z; Z) = H 1 (∂X; Z) ∼ = H 2 (∂X; Z) and H 2 (∂Z; Z) = H 2 (∂X; Z) ∼ = H 1 (∂X; Z). 44 2. Topological surgeries With introducing more notation, in fact we can picture cobordisms involving only 1- and 2-handles. To this end, consider the cobordism W from Y 1 to Y 2 . First present Y 1 as ∂X 4 for some 4-manifold X and draw a diagram for X. Next, add the knots corresponding to the handles of W, and distinguish the two sets of curves by putting the framings of the link producing X into brackets; for a simple example see Figure 2.12. There < −2 > −1 Figure 2.12. A relative Kirby diagram of a cobordism from RP 3 to S 3 is one rule we have to obey with handleslides and cancellations in such a cobordism: handles in X cannot be slid over handles in the cobordism W and handles in X cannot be cancelled against handles in W. On the other hand, we can obviously slide handles of W over handles of X. It is only a little more complicated to investigate homologies in cobor- disms. Suppose that W is a given cobordism from Y 1 to Y 2 . Fix a 4-manifold X with ∂X = Y 1 , and suppose that it is given by attaching 2-handles to D 4 along a framed link L. For the sake of simplicity, suppose furthermore that W is given by a single 2-handle attachment to Y 1 . Denote the 4-manifold X ∪ W by X ′ . Exercises 2.3.4. (a) Determine the homology class in H 2 (X ′ ; Z) generating H 2 (W, ∂W; Z). (Hint: Consider a primitive homology class α ∈ H 2 (X ′ ; Z) such that Q X ′ (α, β) = 0 for all β ∈ H 2 (X; Z) ⊂ H 2 (X ′ ; Z).) (b) Determine the self-intersection Q W (α, α) of this generator. (c) Find a surface in W representing the above α ∈ H 2 (W; Z). (Hint: Use the above computation to represent α ∈ H 2 (X ′ ; Z) with a surface. By adding extra handles make sure that the surface is disjoint from the cores of all the 2-handles deﬁning X. Now show that the surface is in W.) Notice that diﬀerent presentations of Y 1 as ∂X 4 might provide diﬀerent estimates on the genus of a surface representing α. (d) Go through the above computations for the cobordism provided by Figure 2.12. Find a torus of square (−2) in this cobordism. 2.3. Kirby calculus 45 (e) Let K ⊂ S 3 be a given knot with 4-ball genus g s (K). Perform n-surgery along K and denote the resulting 3-manifold S 3 n (K) by Y . Let K ′ be a meridian to K and deﬁne the cobordism W by attaching a 2-handle along K ′ with surgery coeﬃcient k. Let α ∈ H 2 (W, ∂W; Z) denote a generator. Compute the self-intersection of α and give an estimate for the genus of a surface representing it in W. We conclude this section with a few examples and exercises. The 3- manifold Y given by Figure 2.13 is called a Seifert ﬁbered 3-manifold with ......... ......... ......... ......... n g r k r 1 1 1 r 1 2 Figure 2.13. A Seifert ﬁbered 3-manifold Seifert invariants (g, n; r 1 , . . . , r k ) (g, n ∈ N, r i ∈ Q). Notice that the dotted circles form an unlink in the diagram. If r i ≥ 1 then we say that this set of invariants is in standard form. Note that by applying Rolfsen twists any such diagram can be transformed into standard form. When g = n = 0, the 3-manifold with Seifert invariants (g, n; r 1 , . . . , r k ) is usually denoted by M(r 1 , . . . , r k ). Notice that according to this convention the surgery coeﬃcients are negative reciprocals of the given data. 46 2. Topological surgeries Exercises 2.3.5. (a) Determine the intersection matrix of the 4-manifold X given by Figure 2.14. (This manifold is frequently called the Gompf nucleus.) What is H 1 (∂X; Z)? − n 0 Figure 2.14. Kirby diagram for the nucleus Nn (b) Verify that Figure 2.15 gives a 4-manifold X diﬀeomorphic to the disk n Figure 2.15. Disk bundle over a genus-3 surface with Euler number n bundle π: D 3,n → Σ 3 over the genus-3 surface Σ 3 with Euler number n. Draw the diagram of D g,n for an arbitrary positive integer g and n ∈ Z. Compute the intersection form, signature and Euler characteristic for D g,n . (c) By inverse slam-dunks ﬁnd a 4-manifold X such that ∂X = M(g, n; r 1 , . . . , r k ). (Hint: Use the continued fraction expansions of r i ∈ Q, cf. Exer- cise 2.2.7(b).) 2.3. Kirby calculus 47 (d) Verify that the boundary of the (+E 7 )-plumbing (a truncation of the long leg of the diagram of Figure 1.5) is diﬀeomorphic to the 3-manifold we get by doing (+2)-surgery on the right-handed trefoil knot — see Fig- ure 2.16. (Hint: Adapt [66, Figure 12.9] to the present problem.) Figure 2.16. Right-handed trefoil knot (e) Prove that (+5)-surgery on the right-handed trefoil knot is a lens space. (Hint: Use the exercise above and truncate the long leg of the (+E 7 )- plumbing.) (f ) Show that r-surgery on the right-handed trefoil knot gives the Seifert ﬁbered manifold M( − 1 2 , 1 3 , − 1 r−6 ). Use this fact to reprove (c) above. Determine the Seifert invariants of the result of (+6)- and (+7)-surgeries on the trefoil knot. (g) Generalize the above result to a (2, 2n +1) torus knot T (2,2n+1) . (Hint: S 3 r (T (2,2n+1) ) is diﬀeomorphic to M( − 1 2 , n 2n+1 , − 1 r−4n−2 ), cf. [102].) Another family of 3–manifolds is provided by the Brieskorn spheres Σ(p, q, r) (p, q, r ∈ N). Such a 3–manifold can be most conveniently deﬁned as the oriented boundary of the compactiﬁed Milnor ﬁber V (p, q, r), where V (p, q, r) = _ (x, y, z) ∈ C 3 [ x p +y q +z r = ǫ, [x[ p +[y[ q +[z[ r ≤ 1 _ for 0 < ǫ small. In other words, Σ(p, q, r) can be identiﬁed with the link of the isolated singularity ¦x p +y q +z r = 0¦. By perturbing the equation we rather consider the smoothing of this singularity — the introduction of the perturbing term ǫ leaves the topology of the link Σ(p, q, r) unchanged. 48 2. Topological surgeries It can be shown that the smooth Milnor ﬁber V (p, q, r) admits a plumb- ing description, and the 3–manifolds Σ(p, q, r) are Seifert ﬁbered manifolds. The computation of the Seifert invariants from the triple (p, q, r) ∈ N 3 can be rather involved. In order to ﬁx our convention, we remark here that we orient Σ(2, 3, 5) (i.e., the Poincar´e homology sphere) as the boundary of the negative deﬁ- nite E 8 -plumbing, which is the same as (−1)-surgery on the left-handed tre- foil knot. Consequently, (+1)-surgery on the right-handed trefoil provides −Σ(2, 3, 5), which is the boundary of the positive deﬁnite E 8 -plumbing. This orientation convention is consistent with complex geometry — the Poincar´e sphere with its natural orientation is the oriented boundary of the compact- iﬁed Milnor ﬁber V (2, 3, 5), where we equip this latter 4-manifold with the orientation naturally induced by its complex structure. Performing (+1)-surgery on the left-handed trefoil knot we get Σ(2, 3, 7) and (−1)-surgery on the right-handed trefoil gives −Σ(2, 3, 7). Examples 2.3.6. (a) As it follows from the above discussion, −Σ(2, 3, 5) = M( − 1 2 , 1 3 , 1 5 ) and in a similar vein −Σ(2, 3, 4) = M( − 1 2 , 1 3 , 1 4 ) and −Σ(2, 3, 7) = M( − 1 2 , 1 3 , 1 7 ). (b) In general, however, the transition from Σ(p, q, r) to M(r 1 , r 2 , r 3 ) is less simple, for example −Σ(2, 3, 6n −1) = M( − 1 2 , 1 3 , − n 6n−1 ). 3. Symplectic 4-manifolds In this section we recall some general facts about symplectic manifolds. Then we give a short discussion of Moser’s method, which is applied in the proof of numerous fundamental statements discussed in the text. The chap- ter concludes with a short review on what is known about the classiﬁcation of symplectic 4-manifolds. For a more detailed treatment of symplectic geo- metry and topology the reader is advised to turn to [111]; here we restrict our attention mostly to the 4-dimensional case. 3.1. Generalities about symplectic manifolds Deﬁnition 3.1.1. A 2-form ω on the smooth n-manifold X is a symplectic form if ω is closed (i.e., dω = 0) and nondegenerate (i.e., for any nonzero tangent vector v there is w with ω(v, w) ,= 0). The pair (X, ω) is called a symplectic manifold. Since any antisymmetric form is degenerate on an odd dimensional vector space, a symplectic manifold is necessarily even dimensional. Examples 3.1.2. (a) For R 2n with coordinates (x 1 , y 1 , . . . , x n , y n ) the 2- form ω st = n i=1 dx i ∧ dy i is symplectic, called the standard symplectic structure on R 2n . (b) The above form is invariant under translations, hence deﬁnes a sym- plectic form on the 2n-torus T 2n = R 2n /Z 2n . (c) Let g denote the Fubini–Study metric on the complex projective space CP n . Then ω FS (u, v) = g(iu, v) is a symplectic form on CP 2 . 50 3. Symplectic 4-manifolds (d) If (X i , ω i ) are symplectic (i = 1, 2) then their product X 1 X 2 with any of the pulled back forms π ∗ 1 ω 1 ± π ∗ 2 ω 2 is a symplectic manifold. (The map π i : X 1 X 2 →X i denotes the projection to the i th factor.) (e) A volume form on an oriented surface is a symplectic form. Exercises 3.1.3. (a) Show that the nondegeneracy of ω is equivalent to the nonvanishing of ω n = ω ∧ . . . ∧ ω (n times). Notice that in this way ω provides an orientation for X; for oriented manifolds we require the two orientations to agree. (b) Show that the sphere S n admits a symplectic structure only if n = 2. (c) Prove that S 1 S 3 does not carry any symplectic structure. Show the same for CP 2 . Here CP 2 denotes the complex projective plane with its natural (complex) orientation reversed. (Hint: Note that dω = 0 implies that [ω] represents a cohomology class in H 2 (X; R), and [ω] n > 0 follows from nondegeneracy and the compatibility with the given orientation. Use compactness of the above manifolds.) (d) Prove that a smooth projective variety (i.e., a complex manifold with a holomorphic embedding into some complex projective space) admits a symplectic structure. (e) Verify that for any smooth manifold V the cotangent bundle T ∗ V with the 2-form dλ is a symplectic manifold, where the Liouville 1-form λ is deﬁned as λ p (v) = p(π ∗ v) for p ∈ T ∗ V , v ∈ T p (T ∗ V ) and π: T ∗ V →V . It turns out that symplectic manifolds are “close” to complex manifolds in the sense that their tangent bundles can be equipped with complex structures. For this to make sense we need a deﬁnition: Deﬁnition 3.1.4. A linear map J : TX →TX is an almost-complex struc- ture if J 2 = −id TX . An almost-complex structure is said to be compatible with a given symplectic structure ω if ω(Ju, Jv) = ω(u, v) and for u ,= 0 we have ω(u, Ju) > 0, that is g(u, v) = ω(u, Jv) is a Riemannian metric. If ω and J are compatible then (X, ω, J, g) is called an almost-K¨ ahler manifold. For any symplectic structure ω there exists a compatible almost-complex structure J, moreover the space of such J’s is contractible. (This state- ment can be proved ﬁberwise.) In conclusion, the tangent bundle TX of a symplectic manifold (X, ω) carries a complex structure. In fact, all compat- ible almost-complex structures are homotopic to one another, therefore the Chern classes c i (X, ω) ∈ H 2i (X; Z) are well-deﬁned. 3.1. Generalities about symplectic manifolds 51 Deﬁnition 3.1.5. A submanifold Σ of an almost-K¨ahler manifold (X, ω, J, g) is symplectic if the restriction ω[ TΣ is a symplectic form on Σ. The submanifold is J-holomorphic (or pseudo-holomorphic) if TΣ is J- invariant, that is, v ∈ TΣ ≤ TX implies Jv ∈ TΣ. The submanifold L ⊂ X is Lagrangian if ω[ L = 0. Finally, L ⊂ X is totally real if T l L ∩ JT l L = ¦0¦ for all l ∈ L. Example 3.1.6. Recall that for a symplectic manifold (X, ω) the product X X with ω (−ω) is a symplectic manifold. It is not hard to see that the submanifolds X ¦pt.¦ and ¦pt.¦ X are symplectic submanifolds of _ XX, ω (−ω) _ while the diagonal _ (x, x) ∈ X [ x ∈ X _ is Lagrangian. Exercises 3.1.7. (a) Show that Σ ⊂ (X, ω) is symplectic if and only if ω[ TΣ is nondegenerate. (b) Suppose that ω and J are compatible. Show that a J-holomorphic submanifold is symplectic. Find a counterexample for the converse. (c) Show that a submanifold Σ is symplectic if and only if there is a compatible J for which it is J-holomorphic. (d) Show that if L is Lagrangian and J is an ω-compatible almost-complex structure then L is totally real. (Hint: Show that if V is a complex subspace of T x X then ω[ V ,= 0.) Following the holomorphic analogy, J-holomorphic curves in an almost- complex 4-manifold intersect positively, more precisely: Theorem 3.1.8 ([109]). Suppose that the surfaces Σ 1 and Σ 2 are J- holomorphic submanifolds of the almost-complex 4-manifold X. If Σ 1 and Σ 2 do not share a component then [Σ 1 ] [Σ 2 ] ≥ 0, with equality if and only if the submanifolds are disjoint. One of the most important formulae in the study of symplectic manifolds is the following adjunction equality, which is just a simple manifestation of the Whitney product formula for characteristic classes: Theorem 3.1.9. If Σ 2 ⊂ X 4 is a symplectic submanifold then −χ(Σ) = [Σ] 2 − ¸ c 1 (X), [Σ] _ . Proof. Notice that in order for c 1 (X) to make sense we need to ﬁx an ω- compatible almost-complex structure. If Σ is symplectic, one can choose a compatible J such that Σ becomes a J-holomorphic curve. Then the splitting TX[ Σ = TΣ⊕νΣ (as complex bundles) gives ¸ c 1 (X), [Σ] _ = ¸ c 1 (TΣ), [Σ] _ + ¸ c 1 (νΣ), [Σ] _ . 52 3. Symplectic 4-manifolds The identity ¸ c 1 (TΣ), [Σ] _ = ¸ e(TΣ), [Σ] _ = χ(Σ) for the Euler characteristic is fairly straightforward, while ¸ c 1 (νΣ), [Σ] _ = [Σ] 2 needs only a little argument realizing that a push-oﬀ of Σ gives rise to a section of νΣ →Σ. One of the goals of symplectic topology is to understand which man- ifolds admit symplectic structures and if they do, how many inequivalent structures do they carry. Using Gromov’s h-principle it can be shown that every open 2n-manifold admits a symplectic structure (see [33, 56]); the question is more subtle for closed manifolds. In order to understand top- ological properties of symplectic 4-manifolds, ﬁrst we have to understand obstructions to the existence of symplectic structure and describe construc- tions of symplectic manifolds. According to the following proposition, the existence of an almost-complex structure depends only on the homotopy type of a 4-manifold. Recall that the existence of a symplectic structure implies the existence of an almost-complex structure. Proposition 3.1.10 (Wu, [175]). A closed, oriented 4-manifold X carries an almost-complex structure if and only if there is a class h ∈ H 2 (X; Z) such that h ≡ w 2 (X) (mod 2) and h 2 = 3σ(X) + 2χ(X). In particular, a simply connected, closed 4-manifold X is almost-complex if and only if b + 2 (X) is odd. Remark 3.1.11. If X is not closed then the 4-dimensional cohomology class h 2 ∈ H 4 (X; Z) might have noncompact support, hence it might not be integrable on X. Therefore ¸ h 2 , [X] _ might not be deﬁned. If X is compact with nonempty boundary ∂X and the restriction of h to ∂X is torsion then h 2 can be deﬁned as a rational number as follows: the multiple nh will vanish on the boundary ∂X, hence the square (nh) 2 has compact support, so the expression h 2 = 1 n 2 (nh) 2 ∈ Q is a well-deﬁned quantity. The diﬀerence h 2 −3σ(X) −2χ(X), however, is not necessarily zero anymore for an almost-complex structure. It provides an invariant of the oriented 2- plane ﬁeld induced by the complex tangencies on ∂X; for more about this topic see Section 6.2. We note here that since the congruence h ≡ w 2 (X) (mod 2) always admits a solution (which implies, in particular, the existence of a spin c structure on X), every nonclosed 4-manifold carries an almost- complex structure. 3.1. Generalities about symplectic manifolds 53 A ﬁner obstruction to the existence of ω is given by the following theorem of Taubes. (For the brief deﬁnition of the Seiberg–Witten function SW X and more on Taubes’ work see Chapter 13.) Theorem 3.1.12. If X carries a symplectic structure ω then for the Seiberg–Witten invariant SW X we have that SW X _ ±c 1 (X, ω) _ = ±1. Moreover, if K ∈ H 2 (X; Z) and SW X (K) ,= 0 then ¸ ¸ K [ω] ¸ ¸ ≤ ¸ ¸ c 1 (X, ω) [ω] ¸ ¸ , with equality if and only if K = ±c 1 (X, ω). It can be shown, for example, that SW 3CP 2 ≡ 0, hence although 3CP 2 ad- mits an almost-complex structure, it cannot be equipped with a symplectic structure. Above we saw obstructions to the existence of symplectic structures, in the following we will describe some constructions to produce symplectic manifolds. As we already mentioned, all K¨ ahler surfaces are symplectic. One of the most eﬀective ways of constructing symplectic manifolds is the symplectic normal connected sum operation, which we will describe in Section 7.1. Another source of examples comes from surface bundles over surfaces, since we have Theorem 3.1.13 (Thurston, [166]). If the 4-manifold X admits a ﬁbration X → Σ g such that the ﬁber has genus diﬀerent from 1 then X admits a symplectic structure. Remark 3.1.14. If the ﬁber is a torus, similar result cannot be expected, since S 1 S 3 admits a torus ﬁbration over the sphere: multiply the Hopf ﬁbration S 3 → S 2 by a circle. On the other hand, as a theorem of Geiges [55] shows, torus bundles over tori are all symplectic. A generalization of Theorem 3.1.13 to more general Lefschetz ﬁbrations will be discussed in Section 10.1. These constructions give partial results regarding the existence of a sym- plectic structure on a given smooth 4-manifold. Such studies are usually called “geographic” questions of symplectic 4-manifolds. For overviews of various aspects of such geographic questions see [14, 144, 152, 156]. The next problem is: how many symplectic structures can a 4-manifold carry. Such investigations are usually called “botany”. To make the question pre- cise, we have to clarify what do we mean by distinct symplectic structures. 54 3. Symplectic 4-manifolds Deﬁnition 3.1.15. Let X be a given 4-manifold and ω 0 , ω 1 two symplectic forms on it. The forms ω 0 and ω 1 are said to be deformation equivalent if there is a smooth path of symplectic forms interpolating between them. The form ω 0 is the pullback of ω 1 if there is a diﬀeomorphism f : X → X such that f ∗ ω 1 = ω 0 . Finally, ω 0 and ω 1 are equivalent if they lie in the same equivalence class under the equivalence relation generated by the above two relations. Theorem 3.1.16 ([112, 150, 170]). For any n ∈ N there is a simply connected 4-manifold X n which carries at least n inequivalent symplectic structures. The construction of the manifolds and the symplectic structures uses Gompf’s symplectic normal connected sum operation (given in Theorem 7.1.10). By computing c 1 of the various symplectic structures it is easy to show that they are deformation inequivalent (since the c 1 ’s are distinct). By proving that the diﬀerent c 1 ’s lie in diﬀerent orbits of Diff + (X) it fol- lows that the symplectic structures are inequivalent. In this last step either the divisibilities of the integral cohomology classes show the nonexistence of certain diﬀeomorphisms [150], or the Seiberg–Witten equations pose re- strictions on the action of Diff + (X) on H 2 (X; Z), see [112, 170]. The spectacular success of the results of Taubes on Seiberg–Witten invariants of symplectic 4-manifolds indicates that appropriate extensions of these techniques might lead to new results for a much broader class of 4-manifolds. Such a potential extension was initiated by Taubes [165] by considering singular symplectic forms, that is, closed 2-forms nondegenerate only away from a subset of the given closed 4-manifold X. Exercise 3.1.17. Suppose that for a given 4-manifold X the condition b + 2 (X) ≥ 1 holds. Show that there exists a closed 2-form ω which is nondegenerate away from the closed 1-manifold Z = ¦x ∈ X [ ω x = 0¦ of its zeros. (Hint: Fix a metric on X and consider harmonic representatives of a second cohomology class of positive square. Choose generic metric.) The analysis for setting up a correspondence between J-holomorphic curves in X−Z (with appropriate boundary conditions) and Seiberg–Witten solu- tions on X is much more complicated than in the symplectic case, and it is in the focus of current research. A fairly explicit way of ﬁnding a singular symplectic form on a closed 4-manifold X with b + 2 (X) ≥ 1 is given by Gay and Kirby [54]. This procedure makes use of symplectic surgery in the spirit it is discussed in later chapters. 3.2. Moser’s method and neighborhood theorems 55 3.2. Moser’s method and neighborhood theorems In this section we shortly outline the circle of ideas usually referred to as Moser’s method. Using this method we can prove that symplectic manifolds “have no local invariants”. This last statement can be interpreted in two ways: (i) a symplectic manifold is locally standard, or (ii) a small deforma- tion of the symplectic structure produces symplectomorphic manifold. The ﬁrst statement actually generalizes to neighborhoods of special submani- folds, while the second holds for deformations keeping the cohomology class deﬁned by the symplectic form ﬁxed (see Moser’s Stability Theorem 3.2.1). The main idea can be easily summarized: Suppose that X is compact and ω t ∈ Ω 2 (X) is a family of symplectic forms with exact derivative: d dt ω t = dσ t . We claim that in this case there is a family Ψ t ∈ Diff(X) of diﬀeomorphisms such that Ψ ∗ t ω t = ω 0 . The diﬀeomorphisms Ψ t can be constructed via the ﬂow of the family of vector ﬁelds X t they induce by d dt Ψ t = X t ◦ Ψ t , Ψ 0 = id . The key point is that if Ψ ∗ t ω t = ω 0 then for X t we have 0 = d dt Ψ ∗ t ω t = Ψ ∗ t _ d dt ω t +ι Xt dω t +dι Xt ω t _ = Ψ ∗ t d(σ t +ι Xt ω t ), since dω t = 0 and d dt ω t = dσ t . Therefore a vector ﬁeld X t satisfying σ t = −ι Xt ω t will be appropriate for our purposes. This equation is, indeed, easy to solve for X t since ω t is nondegenerate. Then solving d dt Ψ t = X t ◦ Ψ t for Ψ t we get the family Ψ t with the desired property. (This last step can be achieved without any problem provided the manifold X is compact; the general case needs some more care.) Applying this principle, one can deduce the following (see [111]): Theorem 3.2.1 (Moser’s Stability Theorem). Suppose that ω t _ t ∈ [0, 1] _ is a family of symplectic forms on the closed manifold X and [ω t ] = [ω 0 ]. Then there is an isotopy ϕ t such that ϕ 0 = id X and ϕ ∗ t ω t = ω 0 . Proof. In order to apply the above principle we need to show the existence of a family σ t with d dt ω t = dσ t . Since [ω t ] is constant, we obviously get that d dt ω t is exact; applying Hodge theory, for example, a smooth family of appropriate σ t can be chosen. 56 3. Symplectic 4-manifolds More interestingly, Moser’s method shows that symplectic manifolds are locally the same. This principle rests on the following result: Theorem 3.2.2. Suppose that X is a smooth manifold with Y ⊂ X a compact submanifold and ω 1 , ω 2 ∈ Ω 2 (X) two closed 2-forms which are equal and nondegenerate on T y X for all y ∈ Y . Then there are open neighborhoods N 1 , N 2 of Y in X and a diﬀeomorphism ψ: N 1 → N 2 such that ψ[ Y = id and ψ ∗ ω 2 = ω 1 . Proof (sketch). By applying Moser’s argument described above, the the- orem reduces to ﬁnding a 1-form σ ∈ Ω 1 (N 1 ) with dσ = ω 2 − ω 1 and σ[ T Y X = 0. In fact, with such σ the family ω t = (1 −t)ω 1 +tω 2 = ω 1 +tdσ will be symplectic on a neighborhood of Y . This follows from the fact that nondegeneracy is an open condition, while dω t = (1 − t)dω 1 + tdω 2 = 0. Therefore the argument of Moser provides an appropriate vector ﬁeld X t which vanishes along Y . By possibly shrinking N 1 , this implies the exis- tence of ψ and N 2 with the properties given in the theorem. For the explicit construction of σ see [111, page 95]. Applying this theorem for Y = ¦pt.¦ we get Darboux’s theorem: Theorem 3.2.3 (Darboux). For a point x ∈ X in the symplectic manifold (X, ω) there is a chart U ⊂ X containing x such that _ U, ω[ U _ is symplecto- morphic to some open set V ⊂ R 2n equipped with the standard symplectic form ω st [ V . Remark 3.2.4. One can deﬁne symplectic manifolds by requiring that ev- ery point admits a neighborhood symplectomorphic to some open set in (R 2n , ω st ) and the transition functions between such charts respect the sym- plectic structures on the charts. This approach turns out to be equivalent to Deﬁnition 3.1.1. In fact, symplectic structures are standard not only around points, but also near symplectic and Lagrangian submanifolds. In the following we formulate these theorems only for 4-dimensional symplectic manifolds. Theorem 3.2.5 (Symplectic neighborhood theorem, Weinstein [172]). Sup- pose that (X i , ω i ) is a symplectic 4-manifold with 2-dimensional closed sym- plectic submanifolds Σ i ⊂ X i for i = 1, 2. Suppose furthermore that there is an isomorphism F : ν(Σ 1 ) →ν(Σ 2 ) of the normal bundles ν(Σ i ) → Σ i cov- ering a symplectomorphism f : _ Σ 1 , ω 1 [ Σ 1 _ → _ Σ 2 , ω 2 [ Σ 2 _ . Then f extends to a symplectomorphism on some tubular neighborhoods of the surfaces Σ i . 3.2. Moser’s method and neighborhood theorems 57 Proof (sketch). The symplectomorphism f guarantees that ω 1 and f ∗ ω 2 coincide on TΣ 1 ⊂ TX 1 . By choosing appropriate neighborhoods we can assume that the two structures are equal in the normal direction as well using the isomorphism of the normal bundles. Then an application of Theorem 3.2.2 yields the result. With a small modiﬁcation of this argument we get: Theorem 3.2.6 (Lagrangian neighborhood theorem, Weinstein [172]). Let (X, ω) be a symplectic 4-manifold and L ⊂ X a compact Lagrangian sub- manifold. Then there is a neighborhood V ⊂ X of L in X and a neigh- borhood U ⊂ T ∗ L of the zero-section in the cotangent bundle of L and a diﬀeomorphism φ: U → V such that φ ∗ ω = −dλ and φ[ L = id, where λ is the Liouville form on T ∗ L. Recall that the Liouville form on T ∗ L is deﬁned by λ p (v) = p(π ∗ v) for π: T ∗ L → L and v ∈ T p (T ∗ L). For more details of the proofs see [111] or McDuﬀ’s lectures in [34]. Notice that for symplectic submanifolds we need the existence of a diﬀeomorphism F : ν(Σ 1 ) →ν(Σ 2 ) and a symplecto- morphism f : _ Σ 1 , ω 1 [ Σ 1 _ → _ Σ 2 , ω 2 [ Σ 2 _ ; the topology around a symplectic submanifold Σ is not unique and symplectic structures might be diﬀerent for diﬀeomorphic Σ’s. (For example, the volume _ Σ ω n/2 is an invariant.) The isomorphism type of the normal bundle ν(Σ) of Σ ⊂ X 4 is determined by the self-intersection number [Σ] 2 ∈ Z. After possibly rescaling ω 2 on X 2 there exists a symplectomorphism f : _ Σ 1 , ω 1 [ Σ 1 _ → _ Σ 2 , ω 2 [ Σ 2 _ once Σ 1 and Σ 2 are diﬀeomorphic, that is, the genera g(Σ 1 ) and g(Σ 2 ) are equal. In conclusion, the assumptions of Theorem 3.2.5 can be checked from the topology of the situation. In the Lagrangian case, on the other hand, ω[ L = 0 holds, so the topology of L already determines its neighborhood, as the following exercise shows. Exercise 3.2.7. Show that for L 2 ⊂ X 4 Lagrangian we have [L] 2 = −χ(L). (Hint: Fix a compatible almost-complex structure J and show that JT p L is the orthogonal complement of T p L with respect to the metric g J induced by ω and J.) In conclusion, two orientable Lagrangian 2-manifolds in a symplectic 4-manifold admit symplectomorphic neighborhoods if and only if the genera of the surfaces are equal. 58 3. Symplectic 4-manifolds 3.3. Appendix: The complex classification scheme for symplectic 4-manifolds In the following we give a short overview about the present status of the smooth classiﬁcation of closed symplectic 4-manifolds. The classiﬁcation scheme tries to imitate the classiﬁcation results obtained for compact com- plex surfaces (for a detailed description of the latter see [10]), hence ﬁrst we introduce the notion of minimality and Kodaira dimension of symplec- tic 4-manifolds. Since this discussion falls outside the main theme of this volume, we mainly give the statements without proofs. Deﬁnition 3.3.1. A symplectic 4-manifold X is minimal if it does not contain a symplectic submanifold S ⊂ X diﬀeomorphic to the 2-sphere S 2 with [S] [S] = −1. Remark 3.3.2. A detailed analysis of the Seiberg–Witten invariants shows that minimality is equivalent to requiring that X does not contain any smoothly embedded 2-sphere S with [S] [S] = −1. A symplectic 4-manifold can always be blown up in a point by imitating the corresponding complex operation (for an extended discussion see [111, page 233]); i.e., if X admits a symplectic structure then so does its blow- up X ′ = X#CP 2 . In this latter symplectic 4-manifold the generator of the H 2 (CP 2 ; Z)-factor can be represented by a symplectic sphere of square (−1). Using the symplectic normal connected sum operation (for the de- tailed description see Section 7.1), we can prove the converse: if S ⊂ X is a symplectic sphere with square (−1) then X is the blow-up of another symplectic manifold. This implies Lemma 3.3.3. A symplectic 4-manifold X can be written as Y #nCP 2 where Y is a minimal symplectic manifold. Y is called a minimal model of X. Proof. If S ⊂ X is a symplectic sphere of self-intersection (−1) then X = X 1 #CP 2 since νS is diﬀeomorphic to CP 2 − int D 4 . Taking the symplectic sum of X and CP 2 along S and CP 1 ⊂ CP 2 (as it is discussed in Theorem 7.1.10) we ﬁnd that X 1 is symplectic. Repeating the above process completes the proof. Notice that each step reduces b 2 (X) by 1, hence this procedure will terminate after ﬁnitely many steps. 3.3. Appendix: The complex classiﬁcation scheme for symplectic 4-manifolds 59 Remark 3.3.4. The minimal model is not necessarily unique; for example CP 2 #2CP 2 ∼ = S 2 S 2 #CP 2 admits diﬀerent minimal models (CP 2 and S 2 S 2 ) according to the order of blow-downs. For a related discussion see Remark 3.3.9. Let us assume that (X, ω) is a minimal symplectic 4-manifold. Following the complex analogy, its Kodaira dimension is deﬁned as follows: Fix an ω-compatible almost-complex structure J and consider c 1 (X, ω) = c 1 (X, J). Deﬁnition 3.3.5. • If c 1 (X, ω)[ω] > 0 or c 2 1 (X, ω) < 0 then the Kodaira dimension κ(X) of X is −∞. • In case c 1 (X, ω)[ω] = 0 we say that X is of Kodaira dimension 0. • For c 1 (X, ω)[ω] < 0 and c 2 1 (X, ω) = 0 we deﬁne κ(X) = 1. • Finally, if c 1 (X, ω)[ω] < 0 and c 2 1 (X, ω) > 0 then κ(X) = 2. • If X is nonminimal then κ(X) is deﬁned as the Kodaira dimension of its minimal model. Theorem 3.3.6 (Liu, [103]). If (X, ω) is a minimal symplectic 4-manifold with c 2 1 (X, ω) < 0 then X is a ruled surface, that is, an S 2 -bundle over a Riemann surface. It follows that κ(X) is deﬁned for any minimal symplectic 4-manifold X, and it is well-deﬁned, since by Theorem 3.3.6 the above cases are mutually disjoint. In principle κ(X) might depend on the minimal model chosen, since X min might not be unique. Proposition 3.3.7. If the symplectic 4–manifold has a minimal model X min with κ(X min ) ≥ 0 then this minimal model is unique up to diﬀeomorphism. Therefore the Kodaira dimension κ(X) of any symplectic 4-manifold is well- deﬁned. Note that the quantity c 2 1 (X, ω) is equal to 3σ(X) + 2χ(X), hence depends only on the topology of X. As a consequence, it can be shown that κ(X) depends only on the oriented diﬀeomorphism type of X. As it turns out, we have a fairly good understanding of the topology of symplectic 4-manifolds with κ = −∞: Theorem 3.3.8 (Liu, [103]). If X is minimal and κ(X) = −∞ then X is diﬀeomorphic either to CP 2 or to a ruled surface. 60 3. Symplectic 4-manifolds Remark 3.3.9. All these manifolds carry complex, in fact, K¨ ahler struc- tures. According to the classiﬁcation of complex surfaces, these are all the K¨ ahler surfaces with (complex) Kodaira dimension −∞ [10]. Suppose that we blow up a ﬁber of a ruled surface X → Σ g . When constructing the minimal model of this symplectic 4-manifold, we can choose which (−1)- sphere to blow down: the exceptional sphere of the blow-up or the proper transform of the ﬁber. It is not very hard to see that the result of one blow-down is spin, while the other is not. Therefore the minimal model of the blown-up 4-manifold is not unique. It can be shown that further blow- ups of these symplectic 4-manifolds are the only ones admitting nonunique minimal models. The next theorem follows from Taubes’ correspondence between Seiberg– Witten and Gromov–Witten invariants (see Chapter 13 for the statement). Theorem 3.3.10 ([111]). If κ(X) = 0 and b + 2 (X) > 1 then c 1 (X, ω) = 0. In the case b + 2 (X) = 1, the assumption κ(X) = 0 implies 2c 1 (X, ω) = 0. Examples of such manifolds are provided by the K3-surface, T 2 -bundles over T 2 (which are all symplectic by the quoted result of Geiges) and the Enriques surface. This latter manifold is the quotient of a K3-surface by an appropriate free Z 2 -action, therefore its fundamental group is Z 2 and the ﬁrst Chern class is a nonzero torsion element of order two. (For a con- struction see [66].) Note that according to Theorem 3.1.12 the assumption c 1 (X, ω) = 0 implies that for a manifold X with b + 2 (X) > 1 there is a unique basic class (i.e., K ∈ H 2 (X; Z) with SW X (K) ,= 0), and this unique class is equal to 0. According to a result of Morgan and Szab´ o, a simply connected 4-manifold X with SW X (0) odd is homeomorphic to the K3-surface, hence it can be proved that Theorem 3.3.11 (Morgan–Szab´ o, [120]). If (X, ω) is a simply connected symplectic 4-manifold with κ(X) = 0 then X is homeomorphic to the K3- surface. Remark 3.3.12. Complex surfaces with Kodaira dimension 0 are classiﬁed: besides torus bundles over the torus or the sphere there are the K3-surfaces and Enrique surfaces. For more detail see [10, pp. 188–189]. It seems reasonable to expect that all symplectic 4-manifolds with κ = 0 admit a genus-1 Lefschetz ﬁbration, hence these are essentially torus bundles, the K3-surface and the Enriques surface. Much less is known about symplectic manifolds with κ = 1 in general. For example, 3.3. Appendix: The complex classiﬁcation scheme for symplectic 4-manifolds 61 Theorem 3.3.13 (Gompf, [64]). If G is a ﬁnitely presented group then there is a symplectic 4-manifold (X, ω) with κ(X) = 1 and π 1 (X) ∼ = G. In the simply connected case, however, the homeomorphism type of (mini- mal) symplectic 4-manifolds with κ(X) = 1 is understood: Theorem 3.3.14. If X is a minimal simply connected symplectic 4-mani- fold with κ(X) = 1 then X is homeomorphic to an elliptic surface. All complex surfaces of Kodaira dimension 1 are elliptic surfaces and can be constructed from E(1) = CP 2 #9CP 2 and torus bundles using ﬁber sum and an additional operation called logarithmic transformation. For additional discussion on the topology of elliptic surfaces see [66]. If the symplectic 4-manifold (X, ω) has κ(X) = 2 then we say that it is of general type. Such examples are provided by complex surfaces of general type (since these complex manifolds are all algebraic, hence K¨ ahler). Recall that for a complex surface the condition κ(X) = 2 implies c 2 1 (X) ≤ 3c 2 (X), the famous Bogomolov–Miyaoka–Yau inequality. (For a related discussion see [153].) This inequality, or some similar relation between c 2 1 and c 2 is conjectured to hold for symplectic 4-manifolds of general type. We do not know too much about the topology of symplectic 4-manifolds of general type. The following conjecture (usually attributed to Gompf) would provide a strong topological restriction: Conjecture 3.3.15. If a symplectic 4-manifold (X, ω) satisﬁes κ(X) ≥ 0 then for its Euler characteristic χ(X) ≥ 0 holds. Notice that ruled surfaces might have negative Euler characteristic (depend- ing on the genus of the base), but κ ≥ 0 excludes them in the conjecture. The following lemma provides a tool for studying χ(X) of symplectic 4- manifolds with b + 2 (X) = 1: Lemma 3.3.16. If (X, ω) is a symplectic 4-manifold and b + 2 (X) = 1 then either κ(X) = −∞ or b 1 (X) ∈ ¦0, 2¦. Proof. By the existence of an almost-complex structure we get that 1 − b 1 + b + 2 = 2 − b 1 is even, therefore b 1 (X) is even. Now by κ(X) ≥ 0 and Theorem 3.3.8 we have that 0 ≤ c 2 1 (X min , ω) = 3σ(X min ) + 2χ(X min ) = 3 _ b + 2 (X min ) −b − 2 (X min ) _ + 2 _ 2 −2b 1 (X min ) +b + 2 (X min ) +b − 2 (X min ) _ = 9 −b − 2 (X min ) −4b 1 (X min ), showing that 4b 1 (X min ) = 4b 1 (X) ≤ 9, which yields the result. 62 3. Symplectic 4-manifolds Conjecture 3.3.17. There is no symplectic 4-manifold X with b + 2 (X) = 1, b 1 (X) = 2 and b − 2 (X) = 0. (Notice that such a 4-manifold would provide a counterexample to Conjecture 3.3.15.) 4. Contact 3-manifolds This chapter is devoted to the recollection of basic facts about contact manifolds. As before, we start with the general case, but very quickly specialize to 3-manifolds. To understand the topology of contact 3-mani- folds we consider submanifolds and the contact structures near them. The contact version of Darboux’s theorem says that every point in a contact 3-manifold has a neighborhood which is standard regardless of the contact structure. Then we consider knots which are always tangent or always transverse to the contact planes and examine their classical invariants. It turns out that the contact structures near these types of knots are essentially unique. For an arbitrary surface embedded in a contact 3-manifold we look at the characteristic foliation induced by the contact structure to extract information. It is typical to move a surface by a small isotopy to modify its characteristic foliation to get a generic picture and/or to eliminate certain type of singularities. As it turns out, the characteristic foliation determines the contact structure near the surface. A more complete treatment of the ideas and theorems collected here can be found in e.g. [1, 39, 56, 57]. 4.1. Generalities on contact 3-manifolds Deﬁnition 4.1.1. Suppose that Y is a given (2n+1)-dimensional manifold. A 1-form α ∈ Ω 1 (M) is a contact form if α ∧ (dα) n is nowhere zero. The 2n-dimensional distribution ξ ⊂ TM is a contact structure if locally it can be deﬁned by a contact 1-form α as ξ = ker α. Example 4.1.2. The standard contact structure ξ st on R 2n+1 can be given in the coordinates (x 1 , y 1 , . . . , x n , y n , z) as ker (dz + n i=1 x i dy i ). The com- plex tangents to S 2n−1 ⊂ C n also form a contact structure. 64 4. Contact 3-manifolds According to a classical result of Frobenius, the plane ﬁeld ξ = ker α is integrable if and only if α ∧ dα = 0. Integrability is equivalent to be- ing closed under Lie bracket, hence ξ = ker α is integrable if and only if α _ [v 1 , v 2 ] _ = 0 whenever α(v i ) = 0 (i = 1, 2). Recall that dα _ [v 1 , v 2 ] _ = L v 2 α(v 1 ) −L v 1 α(v 2 ) +α _ [v 1 , v 2 ] _ , hence if ξ is integrable then dα vanishes on ξ = ker α. So ξ is “maximally nonintegrable” if dα is nondegenerate on ξ, i.e., α∧(dα) n ,= 0. Therefore the contact condition can be interpreted as “maximally nonintegrable”. In other words, α is a contact form if dα is a symplectic form on the hyperplane distribution ker α. As we will see, Dar- boux’s theorem generalizes to the contact setting (Theorem 4.1.13), hence in conjunction with Remark 3.2.4 it can be shown that contact structures can be given by patching open subsets of R 2n+1 together with transition func- tions respecting ξ st . The existence of contact structures on open manifolds (similarly to the symplectic case) follows from an appropriate h-principle see [33, 56], for example. The question becomes more subtle on closed (odd dimensional) manifolds. From now on we will assume that Y is a 3-manifold, that is, in the above deﬁnition n = 1. In the following we describe a few examples of contact structures on R 3 , T 3 and S 3 just to illustrate how contact structures may look like in 3-manifolds. Exercise 4.1.3. Verify that α 1 = dz + xdy and α 2 = dz − y dx deﬁne contact structures on R 3 . Visualize the contact planes ξ i = ker α i , for i = 1, 2. (Hint: For the latter see Figure 4.1.) Examples 4.1.4. (a) The form α 3 = dz + r 2 dθ (with polar coordinates (r, θ) on the (x, y)-plane) gives a contact structure ξ 3 = ker α 3 on R 3 . To check this, realize that in (x, y, z)-coordinates the form α 3 is equal to dz +xdy −y dx, and so α 3 ∧dα 3 = 2 dx∧dy ∧dz = 2r dr ∧dθ ∧dz. One can easily see that the contact planes are spanned by ¦ ∂ ∂r , r 2 ∂ ∂z − ∂ ∂θ ¦. These planes are horizontal (i.e., parallel to the xy-plane) along the z-axis and as we move out along any ray perpendicular to the z-axis the planes will twist in a clockwise manner. The contact structure ξ 3 is obviously invariant under translation in the z-direction and under rotation in the (x, y)-plane. Notice that the planes will not twist “too quickly” as the twisting angle is an increasing function of r which monotone converges to π 2 as r tends to ∞. (b) Similarly, the form β = cos r dz + r sin r dθ is a contact 1-form on R 3 . To check this we calculate dβ = −sinr dr ∧ dz + (sin r +r cos r) dr ∧ dθ, 4.1. Generalities on contact 3-manifolds 65 x y z Figure 4.1. The contact planes of the standard contact structure on R 3 β ∧ dβ = −r sin 2 r dθ ∧ dr ∧ dz + (sin r cos r +r cos 2 r) dz ∧ dr ∧ dθ = _ 1 + sin r cos r r _ r dr ∧ dθ ∧ dz, and it is easy to see that 1 + sin r cos r r > 0 for r > 0. Again, as in the previous example, ker β admits the same symmetries as ξ 3 and the contact planes (spanned by ¦ ∂ ∂r , r tan r ∂ ∂z − ∂ ∂θ ¦) are horizontal along the z-axis and they will twist in a clockwise manner as we move out along any ray perpendicular to the z-axis. This time, however, the contact planes will make inﬁnitely many full twists as r goes to ∞. More generally, the 1-form β n = cos f n (r) dz +sin f n (r) dθ, where f n (r) is a strictly monotone function equal to r 2 near r = 0 and asymptotic to nπ + π 2 (as r → ∞) provides a contact form on R 3 for every nonnegative integer n. For n = 0 this form gives the standard contact structure on R 3 . 66 4. Contact 3-manifolds (c) Let us identify the 3-torus T 3 with R 3 /Z 3 . For any positive integer n the 1-form sin(2πnx) dy + cos(2πnx) dz deﬁned on R 3 induces a contact structure θ n on T 3 . (d) Consider the smooth map f : R 4 → R deﬁned by f(x 1 , y 1 , x 2 , y 2 ) = x 2 1 + y 2 1 + x 2 2 + y 2 2 . Let p denote the point (x 1 , y 1 , x 2 , y 2 ). It is clear that S 3 = f −1 (1) and T p S 3 = ker df p = ker(2x 1 dx 1 +2y 1 dy 1 +2x 2 dx 2 +2y 2 dy 2 ). By identifying R 4 with C 2 we can deﬁne a complex structure J on each tangent space of R 4 by J ∂ ∂x i = ∂ ∂y i and J ∂ ∂y i = − ∂ ∂x i for i = 1, 2. Let ξ be the plane ﬁeld of complex tangencies of J along S 3 , i.e., ξ p = T p S 3 ∩ J(T p S 3 ). We claim that ξ is a contact structure on S 3 . To show this we will ﬁnd a contact 1-form α on S 3 such that ξ = ker α. Consider the 1-form df ◦ J on R 4 . By evaluating on the basis vectors ¦ ∂ ∂x 1 , ∂ ∂y 1 , ∂ ∂x 2 , ∂ ∂y 2 ¦ it is easy to check that −df p ◦ J = 2x 1 dy 1 − 2y 1 dx 1 + 2x 2 dy 2 − 2y 2 dx 2 . Moreover we have J(T p S 3 ) = ker(−df ◦ J) since J 2 = −id. Let α = − 1 2 (df ◦ J) ¸ ¸ S 3 . It is clear that ξ = ker α. It is straightforward to check that α ∧ dα is nonzero on S 3 . We will check this only at a point p = (x 1 , y 1 , x 2 , y 2 ) on S 3 where x 1 ,= 0, y 1 ,= 0, y 2 ,= 0. A basis for the tangent space T p S 3 can be chosen as _ ∂ ∂x 1 − x 1 y 1 ∂ ∂y 1 , ∂ ∂x 2 − x 2 y 2 ∂ ∂y 2 , ∂ ∂x 1 − x 1 y 2 ∂ ∂y 2 _ . Now it is easy to see that α ∧ dα > 0 on this basis. Hence we conclude that α = (x 1 dy 1 −y 1 dx 1 +x 2 dy 2 −y 2 dx 2 ) ¸ ¸ S 3 is a contact form. We deﬁne ξ = ker α as the standard contact structure on S 3 and denote it by ξ st . (e) For a more subtle source of examples consider a complex manifold (X, J) with a function ϕ: X → R such that the symmetric 2-form g ϕ (u, v) = −dJ ∗ dϕ(u, Jv) is a Riemannian metric. (Here J denotes multiplication by i on TX and J ∗ is the induced map on T ∗ X.) Then the 1-form α ϕ given by α ϕ (v) = −dJ ∗ dϕ(∇ gϕ ϕ, v) deﬁnes a contact form on ϕ −1 (a) for a regular value a of ϕ. We will return to these examples in Chapter 8. Exercise 4.1.5. Show that the 1-form β n deﬁned in Example 4.1.4(b) is a contact form for every nonnegative integer n. Deﬁnition 4.1.6. Two contact 3-manifolds (Y, ξ) and (Y ′ , ξ ′ ) are called con- tactomorphic if there is a diﬀeomorphism f : Y → Y ′ such that f ∗ (ξ) = ξ ′ . If ξ = ker α and ξ ′ = ker α ′ , this is equivalent to the existence of a nowhere zero function g: Y →R such that f ∗ α ′ = gα. Two contact structures ζ and ζ ′ on a manifold Y are said to be isotopic if there is a contactomorphism h: (Y, ζ) →(Y, ζ ′ ) which is isotopic to the identity. 4.1. Generalities on contact 3-manifolds 67 In fact, two contact structures on a closed manifold are isotopic if and only if they are homotopic through contact structures by Theorem 4.1.16 below. Notice also that there exist contact structures which are contactomorphic but not isotopic, see Exercise 11.3.12(c). Exercises 4.1.7. (a) Prove that α 1 = dz +xdy, α 2 = dz −y dx and α 3 = dz +xdy −y dx = dz +r 2 dθ deﬁne contactomorphic contact structures on R 3 . (Hint: For identifying the contact structures given by α 1 and α 3 , use the diﬀeomorphism ϕ(x, y, z) = (x, y 2 , z+ xy 2 ) or φ(x, y, z) = ( x+y 2 , y−x 2 , z+ xy 2 ).) (b) Let p be any point in S 3 . Show that (S 3 −¦p¦, ξ st [ S 3 −{p} ) is contacto- morphic to (R 3 , ξ st ). (Hint: See [57] for a complete solution.) In case ξ can be deﬁned by a global 1-form, we say that the contact structure is coorientable. This implies that ξ is orientable as a 2-plane ﬁeld on Y . In fact, the 2-plane ﬁeld underlying the contact structure ξ is orientable if and only if ξ can be deﬁned by a global 1-form. Given an oriented 3-manifold Y , we say that ξ = ker α is a positive contact structure on Y if the orientation of Y coincides with the orientation given by α ∧dα. Notice that the orientation induced on Y by α ∧ dα is independent of the contact form α deﬁning the contact structure ξ. In the following we will always assume that the contact structures at hand are positive and cooriented. We orient the normal direction to the contact planes by α, or equivalently the contact planes are oriented by dα. If we choose −α as our contact form then the normal orientation and hence the orientation of the contact planes will be reversed but −α ∧ d(−α) will induce the same orientation on the 3-manifold. Deﬁnition 4.1.8. Suppose that the contact structure ξ is given as ker α for the contact 1-form α ∈ Ω 1 (Y ). The vector ﬁeld R α on Y satisfying dα(R α , .) = 0 and α(R α ) = 1 is called the Reeb vector ﬁeld of α. In other words, at each point p ∈ Y , the Reeb vector ﬁeld points in the direction where the skew-symmetric 2-form dα p (of rank 2) degenerates in the tangent space T p Y and it is uniquely determined by the normalization condition α(R α ) = 1. Notice that R α is transverse to the contact planes and depends on the contact 1-form, not just on the contact structure. In general the Reeb vector ﬁeld R fα will be very diﬀerent from R α for a nowhere vanishing function f : Y →R. Exercises 4.1.9. (a) Suppose that β a 1-form on Y such that β(R α ) = 0. Show that there is a unique vector ﬁeld X with X(p) ∈ ξ p on Y such that 68 4. Contact 3-manifolds ι X dα = β. (Hint: Choose X = β(e 2 )e 1 −β(e 1 )e 2 dα(e 1 ,e 2 ) with respect to some (local) frame e 1 , e 2 of ξ.) (b) Find the Reeb vector ﬁelds for the contact forms deﬁned in Exam- ples 4.1.4(a) and (b). (Hint: The answers are R α 3 = ∂ ∂z , and R β = (r sin r +cosr +r cot r cos r) −1 ∂ ∂θ + + (1 +r cot r) _ r sin r + (1 +r cot r) cos r _ −1 ∂ ∂r .) Next we turn to the study of submanifolds in contact 3-manifolds. Deﬁnition 4.1.10. Suppose that (Y, ξ) is a given contact 3-manifold. A knot K ⊂ Y isLegendrian if the tangent vectors TK satisfy TK ⊂ ξ, i.e., α(TK) = 0 for the contact 1-form α deﬁning ξ. The knot K is transverse if TK is transverse to ξ along the knot K, i.e., if α(TK) ,= 0. The contact framing of a Legendrian knot is deﬁned by the orthogonal of ξ along K. (In other words, push K oﬀ in the normal direction to ξ.) Equivalently, we can take the framing obtained by pushing K oﬀ in the direction of a nonzero vector ﬁeld transverse to K which stays inside the contact planes. This framing is the Thurston–Bennequin framing of the Legendrian knot. Remark 4.1.11. If K is null-homologous in (Y, ξ) then it admits a natural 0-framing provided by any embedded surface Σ ⊂ Y with ∂Σ = K, cf. Exercise 2.1.4(b). In this case the Thurston–Bennequin framing can be converted into an integer tb(K) ∈ Z: measure the Thurston–Bennequin framing with respect to the Seifert framing, i.e., the natural 0-framing. Notice that the 0-framing does not depend on the chosen surface Σ, therefore the resulting number tb(K) will be independent of Σ. Also notice that by the coorientation of ξ a transverse knot T ⊂ (Y, ξ) comes with a natural orientation: choose the nonzero tangent vector v to be positive if α(v) > 0 for the contact 1-form α ∈ Ω(Y ) providing the given coorientation for ξ. Similarly to the symplectic case, contact structures are the same locally — in either sense. As before, these theorems rest on the following result. Theorem 4.1.12. Let Y be a given 3-manifold with N ⊂ Y a com- pact subset. Consider contact structures ξ 0 , ξ 1 on Y which coincide as cooriented contact structures on N, i.e., ξ 0 [ N = ξ 1 [ N as oriented 2-plane ﬁelds. Then there exists a neighborhood U of N and a contactomorphism ϕ: _ U, ξ 0 [ U _ → _ U, ξ 1 [ U _ which is isotopic to id U rel N. 4.1. Generalities on contact 3-manifolds 69 Proof (sketch). The proof makes use of Moser’s method again (see Sec- tion 3.2). Assume that α 0 , α 1 are contact forms giving rise to the contact structures ξ 0 , ξ 1 . Since ξ 0 [ N = ξ 1 [ N , there exists a function f : N → R + such that α 1 = fα 0 on the compact set N. Then consider the family α t = (1 − t)α 0 + tα 1 of 1-forms on Y . To see that α t is a contact form on N for every t ∈ [0, 1], we calculate that α t ∧ dα t = _ (1 −t) 2 + 2f(1 −t)t ¸ α 0 ∧ dα 0 +t 2 α 1 ∧ dα 1 > 0. In addition, one can show that there is a neighborhood U of N such that α t is a contact form on U for every t. (Notice that α t is not necessarily contact on the entire Y .) We would like to represent the map ϕ in the theorem as the time–1 map of a family of diﬀeomorphisms ϕ t with (ϕ t ) ∗ ξ 0 = ξ t and ϕ 0 = id. Equivalently, we start with the equation ϕ ∗ t α t = λ t α 0 and diﬀerentiate it with respect to t, yielding d dt (ϕ ∗ t α t ) = ϕ ∗ t _ d dt α t + L Xt α t _ = d dt λ t α 0 = d dt λ t λ t ϕ ∗ t α t . (The ﬁrst equality is an exercise in diﬀerential forms and its proof can be found, for example, in [12].) As before, X t denotes the family of vector ﬁelds induced by ϕ t , i.e., d dt ϕ t = X t ◦ ϕ t . Taking µ t = d dt (log λ t ) ◦ ϕ −1 t , the above equation gets the form ϕ ∗ t _ d dt α t +d _ α t (X t ) _ +ι Xt dα t _ = ϕ ∗ t (µ t α t ), which is solved by X t ∈ ker α t provided (4.1.1) d dt α t +ι Xt dα t = µ t α t . Now consider the Reeb vector ﬁeld R αt of the contact 1-form α t and plug it into the above equation to get d dt α t (R αt ) = µ t . This deﬁnes the function µ t , hence Equation (4.1.1) above can be uniquely solved for X t , since dα t is nondegenerate on ker α t . (Recall that this non- degenracy is equivalent for α t being a contact 1-form.) Now X t integrates to the desired ﬂow ϕ t . 70 4. Contact 3-manifolds Applying this principle for N = ¦pt.¦ we get Theorem 4.1.13 (Darboux’s theorem for contact structures). For every y ∈ Y in the contact 3-manifold (Y, ξ) there is a neighborhood U ⊂ Y such that _ U, ξ[ U _ is contactomorphic to _ V, ξ st [ V _ for some open set V ⊂ R 3 . As in the symplectic case, similar argument extends to special sub- manifolds. For stating the relevant results, consider the contact structure ζ 1 = ker _ cos(2πφ) dx − sin(2πφ)dy _ and ζ 2 = ker(cos rdφ + r sin r dθ) on S 1 R 2 . (Here φ is the coordinate in the S 1 -direction, while (x, y) are Cartesian and (r, θ) are polar coordinates on R 2 .) Exercise 4.1.14. Show that S 1 ¦0¦ is Legendrian for ζ 1 and transverse for ζ 2 . By taking N = S 1 , the neighborhood theorems 3.2.5 and 3.2.6 now translate to Theorem 4.1.15 (Contact neighborhood theorems). If K ⊂ (Y, ξ) is a Legendrian knot then there are neighborhoods U 1 ⊂ Y of K and U 2 ⊂ S 1 R 2 of S 1 ¦0¦ such that _ U 1 , ξ[ U 1 _ and _ U 2 , ζ 1 [ U 2 _ are contactomorphic via a contactomorphism mapping K to S 1 ¦0¦. If K is transverse, then some neighborhood of it is contactomorphic to some neighborhood of S 1 ¦0¦ in (S 1 R 2 , ζ 2 ) — again K is mapped to S 1 ¦0¦. For a detailed proof of the Legendrian case — which will be more important from our present point of view — see [57] or [1, Section 2.2]. The proof relies on Theorem 4.1.12 after ﬁnding a map f taking K to S 1 ¦0¦ in such a way that f ∗ maps ξ[ K to ζ i [ S 1 ×{0} . Notice that since both the 1-manifold K and its normal bundle N(K) is topologically unique, no topological assumption is needed for the neighborhood theorems to hold. Another important corollary of the principle of Theorem 4.1.12 is Gray’s stability theorem, the contact version of Moser’s stability Theorem 3.2.1 — the other manifestation of the principle that contact structures admit no local invariants. Theorem 4.1.16 (Gray, [71]). If α t (t ∈ [0, 1]) is a smooth family of contact forms on a closed 3-manifold Y then there is an isotopy ϕ t of Y such that ϕ 0 = id and ϕ ∗ t α t = λ t α 0 for some smooth family of smooth functions λ t : Y →R + . In particular, (ϕ t ) ∗ ξ 0 = ξ t for ξ t = ker α t . Remark 4.1.17. Notice that the theorem deals with contact structures as opposed to contact forms; in general one cannot ﬁnd ϕ t satisfying ϕ ∗ t α t = α 0 . 4.1. Generalities on contact 3-manifolds 71 There is an intimate relationship between symplectic and contact man- ifolds, which will not be discussed in full detail here. (See Deﬁnition 12.1.1 as an example of this relationship.) In our cut-and-paste construction we will frequently refer to a symplectic manifold Symp(Y, ξ) associated to a contact manifold (Y, ξ) — called the symplectization of (Y, ξ). In order to deﬁne Symp(Y, ξ), choose a contact form α for ξ and consider Symp(Y, ξ) = ¦v ∈ T ∗ m Y [ v = tα m for some t > 0¦. It is easy to see that Symp(Y, ξ) is diﬀeomorphic to Y (0, ∞) and that for any other contact 1-form β with ker β = ξ the 1-form β or −β is simply a section of Symp(Y, ξ). (This trivially follows from the fact that if ker α = ker β then α = fβ for some f ∈ C ∞ (Y ) with f > 0 or f < 0.) By taking ω = d(λ[ Symp(Y,ξ) ) we get a closed 2-form on Symp(Y, ξ) — here λ stands for the Liouville 1-form on T ∗ Y deﬁned as λ p (v) = p _ π ∗ (v) _ for p ∈ T ∗ Y, v ∈ T p (T ∗ Y ) and projection π: T ∗ Y → Y . We claim that ω is a symplectic form, that is, ω ∧ ω ,= 0. Exercise 4.1.18. By considering the contact 1-form α as a map α: Y → T ∗ Y show that α ∗ λ = α. Using the same simple idea verify that on α(Y ) the forms π ∗ λ and α coincide. Therefore tπ ∗ λ = λ[ Symp(Y,ξ) , hence ω = d(tπ ∗ λ) = dt ∧ π ∗ λ + tπ ∗ dλ. Now since dλ ∧ dλ = 0, we get that ω ∧ ω = 2t _ dt ∧ π ∗ (λ ∧ dλ) _ ,= 0; showing that ω deﬁnes a symplectic structure on Symp(Y, ξ). Remark 4.1.19. An alternative way to describe the symplectic 2-form ω on Symp(Y, ξ) is to take the 1-form µ = tα on Y (0, ∞) and deﬁne ω as dµ = t dα +dt ∧ α; the result is clearly the same. Notice that the resulting symplectic form is exact. Exercise 4.1.20. Show that L ⊂ (Y, ξ) is Legendrian if and only if LR ⊂ Symp(Y, ξ) is Lagrangian. 72 4. Contact 3-manifolds 4.2. Legendrian knots In order to have a better understanding of the topological constructions we will introduce in Chapter 8, we discuss a way to visualize Legendrian knots in the standard contact S 3 (or, equivalently, in R 3 ) equipped with the standard contact structure ξ st = ker(dz+xdy). See [40] for more on Legendrian knots. Consider a Legendrian knot L ⊂ (R 3 , ξ st ) and take its front projection, i.e., its projection to the yz-plane. Notice that the projection has no vertical tangencies (since − dz dy = x ,= ∞), and for the same reason at a crossing the strand with smaller slope is in front. A straightforward computation (see [57]) shows that L can be C 2 -approximated by a Legendrian knot for which the projection has only transverse double points and (2, 3)-cusp singularities (see Figure 4.2). Vice versa, a knot projection with these (a) (b) Figure 4.2. Cusp singularity of the projection properties (that is, cusps instead of vertical tangencies and no crossings depicted by Figure 4.3(a)) gives rise to a unique Legendrian knot in (R 3 , ξ st ) — deﬁne x from the projection as x = − dz dy . Since any projection can be isotoped to satisfy the above properties, we can easily show that every knot can be isotoped to Legendrian position. (This knot is, however, far from being unique up to Legendrian isotopy.) Lemma 4.2.1. Any knot K ⊂ S 3 can be isotoped to a Legendrian knot. Proof. Consider a generic projection of K ⊂ R 3 ⊂ S 3 onto the yz-plane. Isotope the knot near the ﬁnitely many points where dz dy = ∞ by adding cusps. At each crossing make sure that the strand with more negative slope crosses in front by adding “zig-zags” if necessary (see Figure 4.3). The Legen- drian knot can be recovered from a projection with these properties. 4.2. Legendrian knots 73 (a) (b) Figure 4.3. Introducing new zig-zags at an illegal crossing Remark 4.2.2. In fact, any knot K in a contact 3-manifold can be C 0 - approximated by a Legendrian knot; for the proof of this statement see [57], for example. The contact framing tb(L) of a knot L can be computed as follows. (Recall that we measure the contact framing with respect to the Seifert framing in S 3 .) Deﬁne w(L) (the writhe of L) as the sum of signs of the double points (see Figure 4.4) — for this to make sense we need to ﬁx an orientation on the knot, but the answer will be independent of this choice, cf. also Exercise 2.1.6. Lemma 4.2.3. If c(L) is the number of cusps, then the Thurston–Bennequin framing tb(L) given by the contact structure is equal to w(L) − 1 2 c(L) with respect to the framing given by a Seifert surface. 74 4. Contact 3-manifolds _ + Figure 4.4. Positive and negative crossings Proof. The equality tb(L) = w(L) − 1 2 c(L) can be seen by noting that ∂ ∂z is transverse to ξ = ker(dz + xdy) hence tb(L) is just the linking number ℓk(L, L ′ ) where L ′ is a small vertical push-oﬀ of L. Now Figure 4.5 shows that the canonical framing diﬀers from the blackboard framing by a left half-twist for each cusp, and this veriﬁes our formula for tb(L). Recall from Exercise 2.1.4 that the blackboard framing of a knot diﬀers from the framing given by the Seifert surface by the writhe of the knot projection at hand. Notice that c(L) is always even, since any cusp pointing right is followed by one pointing left and vice versa. Consequently c(L) = 2c r (L) = 2c l (L) where c r (L) and c l (L) denote the number of right and left cusps, resp. Another invariant, the rotation number rot(L) can be deﬁned by trivializing ξ st along L and then taking the winding number of TL. For this invariant to make sense we need to orient L, and the result will change sign when reversing orientation. Since H 2 (S 3 ; Z) = 0, this number will be independent of the chosen trivialization. Lemma 4.2.4. For the rotation number we have rot(L) = 1 2 _ c d (L)−c u (L) _ where c d (L) (and c u (L)) denotes the number of down (and up) cusps in the projection. Proof. Notice that the vector ﬁeld ∂ ∂x gives rise to a trivialization of ξ st , hence the rotation number can be computed as the winding number with respect to this vector ﬁeld. In conclusion, we have to count how many times the tangent of L passes ∂ ∂x as we traverse L. Deﬁne l ± (resp. r ± ) as the number of left (resp. right) cusps where the knot L is oriented upward/downward. Then the above principle shows that rot(L) = l − −r + . Doing the same count with − ∂ ∂x we get that rot(L) = r − − l + , and taking the average of the two expressions gives the result. 4.2. Legendrian knots 75 contact blackboard Figure 4.5. Contact and blackboard framings Exercise 4.2.5. Show that for a Legendrian knot L ⊂ (S 3 , ξ st ) the sum tb(L) + rot(L) ∈ Z is always odd. The Thurston–Bennequin invariant and the rotation number admit nat- ural generalization to any (homologically trivial) Legendrian knot L in any contact 3-manifold (Y, ξ): Suppose that for an embedded orientable (com- pact) surface Σ ⊂ Y we have ∂Σ = L. Then the contact framing can be measured with respect to the framing on L induced by Σ — the resulting number tb Σ (L) is the Thurston–Bennequin invariant of L with respect to Σ. (As we already pointed out, this quantity is independent of Σ.) By considering the SO(2)-bundle ξ[ Σ with the trivialization along ∂Σ given by the tangents of L (after ﬁxing an orientation on it), we get a relative Euler number e _ ξ[ Σ _ ∈ Z, which is called the rotation number of L with respect to Σ. Equivalently, since ξ is trivial over Σ we can ﬁx a trivialization which indeed induces a trivialization of ξ over ∂Σ = L. Also ﬁx a vector ﬁeld 76 4. Contact 3-manifolds v of tangents to L inducing the given orientation of L. Then the winding number of v along L with respect to the ﬁxed trivialization of ξ on L is the rotation number of L. The rotation number of L depends on the orienta- tion of L and will change sign when the orientation of L is reversed. It also might depend on the chosen surface Σ ⊂ Y . Exercise 4.2.6. Find a contact structure (Y, ξ), a Legendrian knot L ⊂ (Y, ξ) and surfaces Σ 1 , Σ 2 such that rot Σ 1 (L) ,= rot Σ 2 (L). (Hint: Start with a contact structure ξ and closed surface Σ such that ¸ e(ξ), [Σ] _ ,= 0 and ﬁnd L on Σ separating it.) Recall that any knot K ⊂ (Y, ξ) can be C 0 -approximated by a Leg- endrian knot. It has been extensively studied recently to what extent a Legendrian knot is determined by the knot type in R 3 and the two “classi- cal” invariants (the Thurston–Bennequin number and the rotation number). It has been proved [31] that if the Legendrian knot L is smoothly isotopic to the unknot then the above classical invariants determine L ⊂ (S 3 , ξ st ) up to Legendrian isotopy. In particular, up to Legendrian isotopy there is a unique knot L which is smoothly the unknot and has tb(L) = −1, rot(L) = 0. This L is usually called the Legendrian unknot — see Figure 4.2(a). Similar re- sults have been achieved for torus knots and ﬁgure eight knots [41, 47]. The answer to the above question in general is negative, though: according to results of Chekanov [15, 37] there are Legendrian knots which have the same classical invariants but are not Legendrian isotopic. For further reading on this topic see [40]. 4.3. Tight versus overtwisted structures Special to dimension three, contact structures fall into two distinct cate- gories. Deﬁnition 4.3.1. (a) An embedded disk D ⊂ (Y, ξ) is an overtwisted disk in the contact 3-manifold (Y, ξ) if ∂D = L is a Legendrian knot with tb D (L) = 0, i.e., if the contact framing of L coincides with the framing given by the disk D. (b) The contact manifold (Y, ξ) is overtwisted if it contains an overtwisted disk; (Y, ξ) is called tight otherwise. The contact structure ξ on Y is universally tight if its pull-back to the universal cover of Y is tight. If 4.3. Tight versus overtwisted structures 77 ξ becomes overtwisted when pulled back to some ﬁnite cover of Y then it is called virtually overtwisted. Remark 4.3.2. A contact structure covered by a tight contact structure is tight. Exercises 4.3.3. (a) Show that the contact form β in Example 4.1.4(b) deﬁnes an overtwisted contact structure. (Hint: The disk _ z = ε(π 2 −r 2 ), r ≤ π _ is an overtwisted disk for suﬃciently small [ε[.) (b) More generally, show that β n in Example 4.1.4(b) is overtwisted for n ≥ 1. (Hint: For n ≥ 1 consider the overtwisted disk ¦z = 0, r ≤ r 0 ¦ for f n (r 0 ) = π.) (c) Prove the assertion in Remark 4.3.2. According to a fundamental result of Eliashberg [24], overtwisted contact structures on closed 3-manifolds can be classiﬁed using homotopy theory, since Theorem 4.3.4 (Eliashberg, [24]). Two overtwisted contact structures are isotopic if and only if they are homotopic as oriented 2-plane ﬁelds. More- over, every homotopy class of oriented 2-plane ﬁelds contains an overtwisted contact structure. In summary, the classiﬁcation of overtwisted contact structures reduces to a homotopy theoretic problem which is not very hard to solve. We will return to the discussion of the homotopy classiﬁcation of oriented 2-plane ﬁelds in Section 6.2. In fact, using contact surgery we will verify the second assertion of the theorem, usually attributed to Lutz and Martinet. Notice that so far we do not have any example of tight contact structures. In general it is very hard to show that there is no overtwisted disk present in a given contact 3-manifold. This fact gives particular interest to the following result. Theorem 4.3.5 (The Bennequin inequality, [11]). If L is a Legendrian knot in (R 3 , ξ st ) or in (S 3 , ξ st ) and Σ ⊂ Y is a Seifert surface for L then tb(L) + ¸ ¸ rot(L) ¸ ¸ ≤ −χ(Σ). Since an overtwisted disk D has χ(D) = 1 and tb D (L) = 0, this theorem implies Corollary 4.3.6. The standard contact structures (S 3 , ξ st ) and (R 3 , ξ st ) are tight. 78 4. Contact 3-manifolds As we will see later, the examples given in Example 4.1.4(e) are all tight. Theorem 4.3.5 admits a natural generalization. Theorem 4.3.7 (Eliashberg). The contact 3-manifold (Y, ξ) is tight if and only if for all Σ ⊂ Y with ∂Σ = L Legendrian we have tb Σ (L)+ ¸ ¸ rot Σ (L) ¸ ¸ ≤ −χ(Σ). This inequality resembles to the adjunction inequality we saw in Theo- rem 1.2.1, so informally tight contact structures are those which obey the appropriate adjunction inequality. Later we will see that the analogy be- tween the adjunction inequality of Theorem 1.2.1 and the above inequality is even deeper. Another inequality of the same spirit states that Theorem 4.3.8 (Eliashberg, [26]). If e(ξ) denotes the Euler class of a tight contact structure ξ then ¸ ¸ ¸ e(ξ), [Σ] _¸ ¸ ≤ −χ(Σ) for any closed embedded surface Σ ,= S 2 and ¸ e(ξ), [S 2 ] _ = 0. Notice that since [Σ] 2 = 0 in H 2 (Y ; Z), this formula can again be regarded as an analogue of the adjunction inequality for 4-manifolds. Once again, this inequality fails to hold for overtwisted structures, in general. In order to sketch the proofs of these inequalities, we need a tool for studying contact structures near surfaces. Notice that by the nonintegrability of the plane ﬁeld ξ, a surface Σ generically intersects the plane ﬁeld (through the tangent planes TΣ) in lines. Deﬁnition 4.3.9. Fix a contact structure ξ on Y . For a surface Σ ⊂ Y we can consider ξ ∩TΣ, and for generic Σ this intersection is a line ﬁeld except at ﬁnitely many points (where Σ is tangent to ξ, hence ξ ∩ TΣ = ξ = TΣ). Integrating ξ∩TΣ we get a foliation of Σ with singularities at the tangencies, called the characteristic foliation T Σ of Σ in (Y, ξ). Examples 4.3.10. (a) Consider the unit sphere S in the contact manifold (R 3 , ξ 3 ) where ξ 3 = ker(dz +r 2 dθ) as in Example 4.1.4(a). Since the contact planes are horizontal along the z-axis, they are tangent to S at the points (0, 0, ±1), and hence the characteristic foliation on S has singularities at these points. By visualizing the contact planes as they slowly twist while moving out along any ray perpendicular to the z-axis one can see that (0, 0, ±1) are the only singular points and each leaf of the characteristic foliation will “spiral” around the sphere connecting the two singular points as shown in Figure 4.6. (b) Consider the disk D of radius π in the (r, θ)-plane in (R 3 , β) as in Exam- ple 4.1.4(b). Recall that the contact planes are spanned by ¦ ∂ ∂r , r tan r ∂ ∂z − 4.3. Tight versus overtwisted structures 79 Figure 4.6. Characteristic foliation on S 2 ⊂ (R 3 , ξst) ∂ ∂θ ¦. So it is clear that the center of the disk and each point on the boun- dary of D (where r = π) is a singular point. Each leaf of the characteristic foliation is a line segment connecting the center of the disk to a boundary point. This gives an example of a nongeneric characteristic foliation on a surface. Notice that D is an overtwisted disk since tb D (∂D) = 0. Now imagine that we slightly push up (or push down) the interior of D with- out moving its boundary to obtain a new disk D ′ . Notice that the planes tangent to D ′ along its boundary are no longer horizontal. It is clear that the boundary of D ′ becomes a closed leaf of the characteristic foliation with only one singularity in the center of D ′ , see Figure 4.7. Figure 4.7. The overtwisted disk, before and after 80 4. Contact 3-manifolds Deﬁnition 4.3.11. Consider the eigenvalues λ 1 , λ 2 of the linearization of the ﬂow at a generic isolated singular point p. We deﬁne the index of p to be equal to +1 if λ 1 λ 2 > 0 and −1 if λ 1 λ 2 < 0. A generic isolated singular point of index +1 (resp. −1) is called an elliptic (resp. hyperbolic) singular point. We depict a generic elliptic and a hyperbolic point in Figure 4.8. (a) (b) Figure 4.8. Isolated (a) elliptic and (b) hyperbolic singular points By a vague analogy we can think of elliptic points as maxima and minima of a Morse function on a surface, while hyperbolic points correspond to saddle points. This analogy gets even deeper when recognizing that for a generic ﬂow hyperbolic points cannot be connected by a leaf — similar to the saddle points of a Morse–Smale function. In addition, we can assign a sign to each (isolated) singular point p of the characteristic foliation: The singularity is positive (resp. negative) if the orientation of ξ p agrees (resp. disagrees) with the orientation of T p Σ. Notice that this makes sense once Σ and ξ are both oriented. In Example 4.3.10(a) both singular points are elliptic with opposite signs. See Section 8.3 for similar notions in dimension four. The characteristic foliation T Σ can be oriented as follows: If p is a nonsingular point of a leaf L of T Σ , then we choose v ∈ T p L so that (v, n) is an oriented basis for T p Σ, where n ∈ T p Σ is an oriented normal vector to ξ p . With this choice of orientation a positive elliptic point becomes a source and a negative elliptic point becomes a sink. 4.3. Tight versus overtwisted structures 81 In order to understand the topology of contact 3-manifolds we need to have a good grasp on how to cut and paste contact structures along surfaces. It turns out that the characteristic foliation determines the contact structure near the surface. The following result can be obtained as an application of Theorem 4.1.12; for a proof see for example [57]. Theorem 4.3.12. If Σ i ⊂ (Y i , ξ i ) (i = 1, 2) embedded surfaces are diﬀeo- morphic through a diﬀeomorphism f : Σ 1 → Σ 2 which preserves the char- acteristic foliations then f extends to a contactomorphism on some neigh- borhood of Σ 1 . Using the concept and count of positive and negative elliptic/hyperbolic points we can outline proofs of Theorems 4.3.8 and 4.3.7. Proof of Theorem 4.3.8 (sketch). Suppose that Σ is a closed, embedded, connected, oriented surface in a contact 3-manifold. We assume that the characteristic foliation T Σ is generic, i.e., the singular points are isolated and no two hyperbolic points are connected by a leaf. We can express ¸ e(ξ), [Σ] _ and χ(Σ) = ¸ e(TΣ), [Σ] _ in terms of the number of various types of singular points of T Σ . Let e ± and h ± denote the number of ± elliptic/hyperbolic points of T Σ . Fix a vector ﬁeld w which directs T Σ . Now it easily follows from the Poincar´e–Hopf theorem that χ(Σ) = (e + +e − ) −(h + +h − ), since each elliptic (resp. hyperbolic) point is a zero for w of index +1 (resp. −1). To calculate ¸ e(ξ), [Σ] _ we need to count the oriented intersection number of a generic section of the bundle ξ[ Σ with the zero section by considering them as embedded oriented surfaces in the total space of the bundle ξ[ Σ . We choose the section of ξ[ Σ given by w which also gives a section of the tangent bundle TΣ of Σ. Notice that to calculate χ(Σ) = ¸ e(TΣ), [Σ] _ we count the oriented intersection number of the zero section of the tangent bundle TΣ with a generic section (e.g., given by w ). The count of oriented intersection number of sections to calculate ¸ e(ξ), [Σ] _ will diﬀer from the calculation of χ(Σ) exactly at those intersection points where the orientations of the contact planes disagree with the orientations of the tangent planes. So we need a sign reversal in the count exactly at the negative singular points of T Σ to derive the formula ¸ e(ξ), [Σ] _ = (e + −e − ) −(h + −h − ). By adding the above equations we get ¸ e(ξ), [Σ] _ +χ(Σ) = 2(e + −h + ). 82 4. Contact 3-manifolds It is a theorem of Giroux (called the Elimination lemma) that if an elliptic and hyperbolic point of the same sign are connected by a leaf of the char- acteristic foliation on a surface then there is an isotopy of the surface such that both singular points disappear. (For the corresponding phenomenon in dimension four see Section 8.3.) Conversely we can always create a pair of elliptic and hyperbolic points of the same sign on a given leaf. Therefore we can assume that there is no closed leaf in T Σ . We will call the new surface we obtain after such isotopies Σ again, and clearly ¸ e(ξ), [Σ] _ and χ(Σ) will not change under these isotopies. Notice that until now we have not used the tightness of the contact structure. Suppose that p is a positive elliptic point on the surface Σ. Now let O p be the union of all leaves limiting to p and let D p be the closure of it. Suppose that D p is an embedded disk so that ∂D p = D p −O p . Then all the singular points of T Σ on D p other than p will be on ∂D p . Since T Σ is oriented, there is no positive elliptic point on ∂D p and no two elliptic points can be adjacent. This is because a positive elliptic point is a source and a negative elliptic point is a sink, so a leaf connecting two elliptic points is directed form the positive to the negative. Therefore the arcs on ∂D p between elliptic points are divided by hyperbolic points and, by the assumption we made at the beginning of the proof about T Σ , no two hyperbolic points are adjacent. Suppose that there is no posi- tive hyperbolic point on ∂D p . Then we can eliminate all the singular points on ∂D p using the Elimination lemma and thus D p becomes an overtwisted disk which cannot exist in a tight contact manifold. Hence there has to be a positive hyperbolic point q on ∂D p . But then we can eliminate the posi- tive elliptic point p using this positive hyperbolic point q. The diﬃcult part of the proof is to show that we can eliminate a positive elliptic point even if D p is not embedded. For details of this part of the proof the reader is advised to turn to [26, 39]. By completing this last step we conclude that e + = 0 can be assumed, trivially implying ¸ e(ξ), [Σ] _ ≤ −χ(Σ). Moreover by subtracting the above equations and eliminating the negative elliptic points we prove that − ¸ e(ξ), [Σ] _ ≤ −χ(Σ). In conclusion we get the inequality ¸ ¸ ¸ e(ξ), [Σ] _¸ ¸ ≤ −χ(Σ). Deﬁnition 4.3.13. Let γ be an arbitrary transverse knot in a contact 3- manifold bounded by a Seifert surface Σ. We deﬁne the self-linking number sl Σ (γ) of γ as the linking number of γ and γ ′ , where γ ′ is a push-oﬀ obtained by a nonzero vector ﬁeld in the contact planes. That is, sl Σ (γ) is the oriented 4.3. Tight versus overtwisted structures 83 intersection number of γ ′ with Σ. If γ ⊂ R 3 or S 3 then sl Σ (γ) can be shown to be independent of Σ; in this case we drop Σ from the notation. Given a Legendrian knot L, we can construct two copies of L by pushing L in opposite directions in a suﬃciently small annulus neighborhood of L to obtain positive and negative transverse push-oﬀs L ± of L. If L ⊂ (R 3 , ξ st ) then it is easy to obtain the front projections of the transverse push-oﬀs L ± from the front projection of a Legendrian knot L: For L + just smooth out the upward cusps and replace downward cusps by negative kinks. See Figure 4.9. (For details regarding projections of transverse knots Figure 4.9. From Legendrian to transverse knot see [40, 57].) By using these projections and the fact that the self-linking number of a transverse knot in (R 3 , ξ st ) is equal to its writhe in its front projection, we get that for a Legendrian knot L ⊂ (R 3 , ξ st ) sl(L + ) = w(L) −c d (L) = w(L) − 1 2 _ c d (L) +c u (L) _ − 1 2 _ c d (L) −c u (L) _ = tb(L) −rot(L). This equation holds for a null-homologous transverse knot in an arbitrary contact 3-manifold, leading us to the equation sl Σ (L ± ) = tb(L) ∓rot Σ (L). Proof of Theorem 4.3.7. Notice that one direction of this equivalence is clear: an overtwisted disk provides a surface Σ violating the inequality. To prove Eliashberg’s theorem ﬁx Σ with ∂Σ = L a Legendrian knot and 84 4. Contact 3-manifolds consider the positive and negative transverse push-oﬀs L ± of the Legendrian knot L. We can interpret the self-linking number of a transverse knot γ as a relative Euler number and by the use of the method of the proof of Theorem 4.3.8 we derive the equation sl Σ (γ) = −(e + −h + ) + (e − −h − ). In conclusion, we get a relation between sl Σ (γ) and the number of diﬀerent types of singular points of the characteristic foliation on the surface Σ bounded by γ. Combining this result with χ(Σ) = (e + +e − ) −(h + +h − ) we get sl Σ (L ± ) + χ(Σ) = 2(e − − h − ). By using the Elimination lemma and the tightness of the contact structure as in the proof of Theorem 4.3.8 we can assume that e − = 0 and thus sl Σ (L ± ) ≤ −χ(Σ), clearly implying Eliashberg’s inequality. (See [39] and [40] for further details.) We close this section by remarking that Legendrian knots in over- twisted contact structures might have arbitrarily high Thurston–Bennequin invariants. More precisely, if L ⊂ (Y, ξ) is homologically trivial and _ Y − L, ξ[ Y −L _ is overtwisted then for every n there is L ′ smoothly iso- topic to L such that tb Σ (L ′ ) = n. It turns out that by taking enough copies of the boundary of the overtwisted disk and connect sum them we get an un- knot with the desired property and then the general statement easily follows by an additional connect sum. For Legendrian knots with tight complement the situation is more complicated. 5. Convex surfaces in contact 3-manifolds When trying to do surgery on contact 3-manifolds we need to understand contact structures in neighborhoods of embedded surfaces. As we already pointed out in Chapter 4, for a given surface Σ ⊂ (Y, ξ) the characteristic foliation T Σ determines the contact structure near Σ. But it is not easy to describe or relate characteristic foliations. It turns out that the same information can be captured by certain conﬁgurations of curves on the surface at hand once the surface is in a special position with respect to the contact structure. This theory has been developed and fruitfully applied by Giroux and Honda in various circumstances in 3-dimensional contact geometry. For the sake of completeness, in this Chapter we recall the fundamental deﬁnitions and results regarding convex surfaces and dividing sets. These statements will be used in our study of contact Dehn surgery in Chapter 11. For a more detailed introduction to the subject see [43, 76]. 5.1. Convex surfaces and dividing sets Deﬁnition 5.1.1. A vector ﬁeld v on a contact manifold (Y, ξ) is called contact if its ﬂow ϕ t preserves the contact planes, i.e., (ϕ t ) ∗ ξ = ξ. The following lemma gives a convenient characterization of contact vector ﬁelds on contact 3-manifolds. Lemma 5.1.2. Let (Y, ξ) be a contact 3-manifold, where ξ = ker α for some contact 1-form α. A vector ﬁeld v on Y is contact if and only if L v α = fα for some smooth function f : Y →R. Exercises 5.1.3. (a) Show that for each smooth function H: Y →R there is a unique vector ﬁeld V H ∈ ξ such that v H = HR α + V H is a contact 86 5. Convex surfaces in contact 3-manifolds vector ﬁeld, where R α denotes the Reeb vector ﬁeld for α. (Hint: V H is the unique vector ﬁeld in ξ satisfying ι V H dα = dH(R α )α − dH, cf. also Exercise 4.1.9(a).) (b) Verify that for each contact vector ﬁeld v there is a unique smooth function H: Y →R such that v = v H . Inspired by the higher dimensional analogue we make the following Deﬁnition 5.1.4. A smooth surface Σ ⊂ (Y, ξ) is convex if there is a contact vector ﬁeld v transverse to Σ. If ∂Σ ,= ∅ then we also require that ∂Σ is Legendrian. Remark 5.1.5. Notice that the direction of the contact vector ﬁeld v in the deﬁnition is irrelevant, therefore there is no distinguished side of Σ. In that respect the term “convex” is unfortunate, since there is no concavity present. It can be shown that Σ ⊂ (Y, ξ) is convex if and only if it has a neighborhood N = νΣ = ΣI such that ξ[ N is invariant in the I-direction. Consequently, in the neighborhood νΣ = ΣI of the convex surface Σ the contact 1-form α can be written as f dt +β, where f is a function, β is a 1-form on Σ and t denotes the I-coordinate. Proposition 5.1.6 (Giroux, [61]). Any closed surface admits a C ∞ -small perturbation which puts it into convex position. Remark 5.1.7. In [76] it was shown that this is also true for a surface with boundary as long as the surface has Legendrian boundary and the twisting of the contact planes with respect to the surface is not positive. We should point out that even though every surface in a contact 3-manifold can be perturbed into a convex surface it is the existence of non-convex surfaces which makes the theory interesting. In Example 5.1.11 we will describe a non-convex surface. Deﬁnition 5.1.8. Suppose that Σ ⊂ (Y, ξ) is a convex surface with the contact vector ﬁeld v. Deﬁne Γ = _ x ∈ Σ [ v(x) ∈ ξ x _ ⊂ Σ as the dividing set of Σ. As the following proposition shows, the dividing set (generically) is a multi- curve, i.e., a properly embedded smooth 1-manifold, possibly disconnected and possibly with boundary. We will often refer to this set Γ of ﬁnite union of disjoint simple closed curves and properly embedded arcs on Σ as the dividing curves. 5.1. Convex surfaces and dividing sets 87 Proposition 5.1.9 (Giroux, [61]). The dividing set Γ is a 1-dimensional submanifold of the surface Σ transverse to the characteristic foliation T Σ and Σ − Γ = Σ + ∪ Σ − where the ﬂow of a vector ﬁeld w which directs T Σ expands (contracts) a volume form Ω on Σ + (on Σ − , resp.) and w points outward from Σ + along Γ = ∂Σ + . Proof. We choose coordinates x ∈ Σ and t in the I-direction for the I- invariant neighborhood νΣ = Σ I of the convex surface Σ. In these coordinates the 1-form α deﬁning ξ can be expressed as α = f dt +β. The vertical vector ﬁeld v = ∂ ∂t is a contact vector ﬁeld for Σ since ∂ ∂t is clearly transverse to Σ and L ∂ ∂t α = 0. Then for a point x ∈ Σ we have α x ( ∂ ∂t ) = 0 if and only if f(x) = 0, and therefore Γ = f −1 (0). Now the contact condition 0 < α ∧ dα = (β +f dt) ∧ (dβ +df ∧ dt) = β ∧ df ∧ dt +f dt ∧ dβ = dt ∧ (β ∧ df +fdβ) implies that β∧df+fdβ > 0. (Notice that β∧dβ = 0 on ΣI.) In particular, f(x) = 0 implies that β ∧ df ,= 0 and hence df ,= 0. Consequently, Γ is a submanifold of Σ, transversely cut out by f. Let u be a vector tangent to Γ. Then df(u) = 0 and thus β ∧df ,= 0 implies that β(u) ,= 0. That is, u is not in TT Σ = ker β and it follows that Γ is transverse to T Σ . Let w be a vector ﬁeld which directs T Σ . The vector ﬁeld w can be deﬁned by the equation ι w Ω = α[ Σ = β for a volume form Ω on Σ. Notice that w vanishes exactly at the zeros of β. Moreover if we take a diﬀerent volume form on Σ we get a positive multiple of w directing T Σ . We deﬁne the region Σ + (resp. Σ − ) as the set of points on Σ where the normal orientation of ξ agrees (resp., disagrees) with the orientation of the contact vector ﬁeld v. Equivalently, Σ + (resp. Σ − ) is the subsurface where f > 0 (resp. f < 0). To see this ﬁrst notice that f changes sign at Γ: Consider the oriented basis (w, u) of Σ at a point x ∈ Γ. Then (β ∧ df)(w, u) > 0 implies that df(w) < 0. Now it is easy to calculate the spanning vectors for the contact planes and we can see that the normal orientation of the planes agree with v = ∂ ∂t if and only if f > 0. Furthermore the vector ﬁeld w points outward from the boundary of Σ + . To see that the ﬂow of w expands Ω on Σ + we observe that L w f Ω = dι w f Ω +ι w f dΩ = dι w f Ω = d( β f ) = 1 f 2 (fdβ +β ∧ df) > 0. The choice of a contact vector ﬁeld is not unique; nevertheless we have Proposition 5.1.10 (Giroux, [61]). The isotopy class of the dividing curves is independent of the choice of the contact vector ﬁeld. 88 5. Convex surfaces in contact 3-manifolds The following example of a non-convex torus is given in [41], cf. also [43]. Example 5.1.11. Consider the contact structure on Y = R 2 S 1 induced by the contact structure ξ 3 = ker(dz+r 2 dθ) on R 3 (with cylindrical coordinates (r, θ, z)) through the identiﬁcation z ∼ z+1. Let k be a positive real number. We will show that the torus T = T k = ¦(r, θ, z) ∈ Y [ r = k¦ is not convex. Recall that the contact planes of the given contact structure ξ 3 are spanned by ¦ ∂ ∂r , r 2 ∂ ∂z − ∂ ∂θ ¦ (see Example 4.1.4(a)). Therefore at any point p on T the intersection of the tangent plane to T and the contact plane ξ 3 is given by the line generated by the vector k 2 ∂ ∂z − ∂ ∂θ . Here we can view this line in the (θ, z)-plane when we consider T as obtained by the identiﬁcations z ∼ z +1 and θ ∼ θ +2π. Thus we conclude that the characteristic foliation on T is linear as shown in Figure 5.1(a). Suppose that T is convex. Then the contact 1-form on Y can be written as f dt +β in a vertically invariant neighborhood of T as explained above, where f is a smooth function and β is a 1-form on T. The form β is given by (dz + r 2 dθ)[ T = dz + k 2 dθ and hence dβ = 0. On the other hand the contact condition implies that β ∧df +fdβ > 0 as shown in the proof of Proposition 5.1.9. It follows that df(w) < 0 for some vector ﬁeld w directing the characteristic foliation on T which is a contradiction since the function f on T has to be periodic in θ and z and thus f can not be decreasing along a linear foliation on T. Exercises 5.1.12. (a) Perturb the torus T = T k in the example above into a convex torus in (Y, ξ 3 ). (Hint: First consider the two disjoint annuli in the complement of two orbits of the characteristic foliation on T. Then push slightly one of the annuli (ﬁxing its boundary) towards the z-axis while pushing the other one slightly in the opposite direction to get a smooth embedded torus. Show that the dividing curves look like the dashed lines in Figure 5.1(b).) (b) Show that the unit sphere S 2 in (R 3 , ξ 3 ) is convex. Determine the dividing set on S 2 . (Hint: Try the vector ﬁeld v = z ∂ ∂z + r 2 ∂ ∂r .) Deﬁnition 5.1.13. Let L be a Legendrian curve on a convex surface Σ in a contact 3-manifold (Y, ξ). Then tw(L, Σ) denotes the twisting number of the contact planes ξ along L measured with respect to the surface framing on L. Notice that tw(L, Σ) gives tb(L) if Σ is a Seifert surface for L. Exercise 5.1.14. Suppose that L is a Legendrian curve on a convex surface Σ which is transverse to the dividing set Γ. Show that tw(L, Σ) = − 1 2 (Γ∩L). (Hint: Fix a contact vector ﬁeld v for the convex surface Σ. The twisting 5.1. Convex surfaces and dividing sets 89 z z (a) (b) θ θ Figure 5.1. (a) Linear foliation on the non-convex torus T and (b) the dividing set (dashed lines) on its convex perturbation of ξ with respect to Σ is the twisting of ξ relative to v. Observe that each point in Γ ∩ L contributes − 1 2 to tw(L, Σ).) Proposition 5.1.15. Suppose that Σ is a closed convex surface in a contact manifold (Y, ξ). Then ¸ e(ξ), [Σ] _ = χ(Σ + ) −χ(Σ − ). Proof. In Theorem 4.3.8 we showed that ¸ e(ξ), [Σ] _ = (e + −h + )−(e − −h − ). It follows by deﬁnitions that the positive (resp. negative) singular points will be in Σ + (resp. Σ − ). Then using the Poincar´e–Hopf theorem for a vector ﬁeld on a manifold which is transverse to the boundary we get χ(Σ + ) = e + −h + and χ(Σ − ) = e − −h − . If T is any singular foliation on the surface Σ then a multicurve Γ on Σ is said to divide T if the pair (T, Γ) satisﬁes the properties proved in Proposition 5.1.9, where T Σ is replaced by T. The power of studying the dividing set comes from the fact that Γ (rather than the full characteristic foliation) already determines the contact structure near Σ: Theorem 5.1.16 (Giroux’s ﬂexibility, [61]). If T is another singular fo- liation on Σ divided by Γ then there is an isotopy Ψ s : Σ → N = νΣ _ s ∈ [0, 1] _ , Ψ 0 = id Σ and Ψ s [Γ = id Γ such that Ψ s (Σ) is convex for all s and T Ψ 1 (Σ) = Ψ 1 (T). 90 5. Convex surfaces in contact 3-manifolds Therefore, by ﬁxing Γ, any foliation divided by Γ can be thought of as the characteristic foliation; in conclusion Γ determines the germ of the contact structure along Σ. The next lemma shows a connection between convex surfaces and Legendrian knots on them. First we need the following deﬁnition. Deﬁnition 5.1.17. A properly embedded 1-submanifold C of a convex surface Σ is nonisolating if C is transverse to Γ and the closure of every component of Σ¸(Γ ∪ C) intersects Γ. Lemma 5.1.18 (Legendrian Realization Principle, [79, 76]). If C is non- isolating on a convex surface Σ then C can be made Legendrian, i.e., there exists an isotopy ψ s : Σ → N = νΣ _ s ∈ [0, 1] _ , ψ 0 = id Σ such that ψ s (Σ) is convex for all s ∈ [0, 1], ψ 1 (Γ Σ ) = Γ ψ 1 (Σ) , and ψ 1 (C) is Legendrian. Remark 5.1.19. The nonisolating condition guarantees that C can be extended to a singular foliation divided by Γ. Then by Theorem 5.1.16 we can realize this foliation on Σ as the characteristic foliation and hence C becomes Legendrian after an isotopy of (Σ, C) ﬁxing Γ. The set Γ ⊂ Σ of dividing curves can in principle be very complicated. A constraint on Γ is posed by the following result of Giroux: Theorem 5.1.20 (Giroux’s criterion). Suppose that Σ ⊂ (Y, ξ) is a convex surface (possibly with Legendrian boundary) and Σ ,= S 2 . Then Σ has a tight neighborhood if and only if Γ contains no homotopically trivial component. If Σ = S 2 then νΣ is tight if and only if Γ consists of a single component. Proof (sketch). We give a proof for the “only if” direction. Suppose that Γ contains at least two components one of which is homotopically trivial. Let γ denote the homotopically trivial curve which bounds a disk D. Let γ ′ be a curve parallel to γ such that γ ′ ∩ Γ = ∅. Then γ ′ is nonisolating on Σ and hence by the Legendrian realization principle we can make γ ′ Legendrian (so that it stays disjoint from γ). This implies that the surface framing of γ ′ agrees with its contact framing by Exercise 5.1.14. Thus the disk bounded by γ ′ on Σ is an overtwisted disk by deﬁnition. Now suppose that Γ has only one component γ which is homotopically trivial. Take a homotopically essential non-separating simple closed curve δ ⊂ Σ−Γ Σ and use the “folding” method of Honda [76] to introduce a pair of dividing curves parallel to δ. Then repeat the previous argument to ﬁnd an overtwisted disk. 5.1. Convex surfaces and dividing sets 91 Exercises 5.1.21. (a) Use Theorem 4.3.8 to show that if the dividing set on a closed convex surface Σ ,= S 2 in a contact 3-manifold (Y, ξ) consists of only one homotopically trivial curve then (Y, ξ) is overtwisted. (Hint: Observe that ¸ e(ξ), [Σ] _ = χ(Σ + ) −χ(Σ − ) = ±2g.) (b) Use Giroux’s criterion to prove Theorem 4.3.8. (Hint: Put the given Σ in a tight contact 3-manifold (Y, ξ) into convex position. It is clear that χ(Σ) = χ(Σ + ) +χ(Σ − ). Compare this fact with the equation ¸ e(ξ), [Σ] _ = χ(Σ + ) − χ(Σ − ) of Proposition 5.1.15 and observe that χ(Σ − ) ≤ 0 when Σ ,= S 2 . If Σ = S 2 then Σ −Γ Σ is the disjoint union of two disks. See [43] for further details.) In the following we focus on the special case of Σ = T 2 . Exercise 5.1.22. Suppose that a convex torus T 2 has a tight neighborhood (e.g., it is embedded in a tight contact 3-manifold). Then show that the dividing set Γ on T 2 consists of 2n parallel circles (n ≥ 1). By ﬁxing an identiﬁcation of T 2 with R 2 /Z 2 , the slope of these parallel curves is called the slope of the torus at hand. Of course, there is no canonical choice of identiﬁcation of T 2 with R 2 /Z 2 in general. In particular cases, however, there are natural directions to choose — for example if T 2 = ∂(S 1 D 2 ) or T 2 = ∂νK is the boundary of the neighborhood of a Legendrian knot then the meridian µ provides an obvious direction. Example 5.1.23. Consider N = R 2 (R/Z) ≃ R 2 S 1 with the 1-form α = cos(2πz) dx −sin(2πz) dy in the coordinates (x, y) for R 2 and z for R/Z. First we check that α is a contact form on N: since dα = 2π sin(2πz) dx ∧ dz + 2π cos(2πz) dy ∧ dz, we have α ∧ dα = 2π cos 2 (2πz) dx ∧ dy ∧ dz + 2π sin 2 (2πz) dx ∧ dy ∧ dz = 2π dx ∧ dy ∧ dz. The contact form α on N induces a contact form on the solid torus N d = _ (x, y, z) [ x 2 + y 2 ≤ d 2 _ for d > 0. We claim that ∂N d is a convex torus. To this end, consider the vector ﬁeld v = x ∂ ∂x + y ∂ ∂y . It is clear that v is 92 5. Convex surfaces in contact 3-manifolds transverse to ∂N d . To show that v is a contact vector ﬁeld, we check that L v α = α (see Lemma 5.1.2). For the given α we calculate ι v α = ι _ x ∂ ∂x +y ∂ ∂y _ α = ι x ∂ ∂x α +ι y ∂ ∂y α = xcos(2πz) −y sin(2πz), dι v α = cos(2πz) dx −2πxsin(2πz) dz −sin(2πz) dy −2πy cos(2πz) dz, and ι v dα = ι x ∂ ∂x dα +ι y ∂ ∂y dα = 2πxsin(2πz) dz + 2πy cos(2πz) dz. Then it follows by Cartan’s formula L v α = ι v dα +dι v α that L v α = α. The central circle C = _ (x, y, z) ∈ N d [ x = y = 0 _ acquires a canonical contact framing. This framing can be given by the longitude λ that is obtained by pushing C along a vector ﬁeld which is transverse to C and stays inside the contact planes ζ = ker α. If we choose the vector ﬁeld that is orthogonal to C, it is easy to calculate that λ = ¦ _ d sin(2πz), d cos(2πz), z _ ¦. Since ∂N d is a convex torus in a contact 3-manifold, there are dividing curves on ∂N d induced by the contact vector ﬁeld v. By deﬁnition, the dividing curves consist of the points x ∈ ∂N d such that v(x) ∈ ζ(x), i.e, when α x _ v(x) _ = 0. The solution of the equation _ cos(2πz) dx −sin(2πz) dy _ _ x ∂ ∂x +y ∂ ∂y _ = xcos(2πz) −y sin(2πz) = 0 or equivalently the equation z = 1 2π tan −1 _ x y _ can be given by the set Γ = ¦ _ ±d sin(2πz), ±d cos(2πz), z _ ¦, which consists of two parallel copies of the longitude λ. Consequently, with the trivialization of ∂N d by λ and the meridian µ the slope of the dividing curves comes out to be equal to ∞. Here µ will correspond to the x- and λ to the y-axis; hence the slope being p q means that Γ is parallel to the curve pλ +qµ. In fact, we can visualize the contact planes as follows: The planes are horizontal at z = 0 and start twisting as z is increasing and they become horizontal again when z = 1. So the characteristic foliation consists of two singular lines of slope = ∞ and parallel nonsingular leaves of slope ,= ∞. (Notice that this characteristic foliation is not generic.) 5.1. Convex surfaces and dividing sets 93 Remark 5.1.24. In general, on S 1 D 2 only the meridian µ is canonical, hence the slope of ∂(S 1 D 2 ) is well-deﬁned only up to an action of SL 2 (Z) leaving µ ﬁxed, i.e., of the action _ 1 m 0 1 _ — the Dehn twists changing the framing. It is not hard to see that using this equivalence any nonzero slope can be transformed into the form − p q with (p, q) = 1 and p > q ≥ 0; moreover this form is unique: just notice that under the action of the above matrix the slopes p q and p pm+q are equivalent. Exercise 5.1.25. Find slopes equivalent to 2 3 and 1. For topologically simple 3-manifolds the dividing curves may determine the entire contact structure. The following is a fundamental result which is essential for the classiﬁcation of tight contact structures. Theorem 5.1.26 (Eliashberg). Assume that there exists a contact struc- ture ξ on a neighborhood of ∂D 3 which makes ∂D 3 convex with connected dividing set. Then there exists a unique extension of ξ to a tight contact structure on the 3-disk D 3 up to an isotopy which ﬁxes the boundary. Exercise 5.1.27. Using Theorem 5.1.26 show that the 3-sphere S 3 admits (up to isotopy) a unique tight contact structure. The exercise above can be solved by a simple-minded approach to ﬁnd an upper bound on the number of tight contact structures on a given 3- manifold. In order to calculate an upper bound we cut the 3-manifold along convex surfaces until we end up with a disjoint union of 3-disks. At each step we keep track of all possible conﬁgurations of dividing curves on these surfaces along which we cut our 3-manifold. We will apply this strategy below to ﬁnd an upper bound for the number of tight contact structures on the solid torus for the case when the boundary slope of the dividing curves is equal to 1 n . We will ﬁrst state a basic lemma called the “edge rounding” which is frequently used to transfer dividing sets between two convex surfaces meeting along a Legendrian curve. Exercise 5.1.28. Let Σ i be a convex surface with dividing set Γ i for i = 1, 2. Assume that ∂Σ 2 is a Legendrian curve in Σ 1 . Let A = Γ 1 ∩ ∂Σ 2 and B = Γ 2 ∩∂Σ 2 . Then between two adjacent points of A there is a point in B and between two adjacent points of B there is a point in A. (Hint: Consider the unique geometric model of contact structures in a neighborhood of the Legendrian curve.) 94 5. Convex surfaces in contact 3-manifolds Lemma 5.1.29 (Edge rounding, [76]). Let Σ i be a convex surface with the dividing set Γ i for i = 1, 2 and assume that ∂Σ 1 = ∂Σ 2 is Legendrian. Then using the standard local model around ∂Σ 1 we can glue Σ 1 to Σ 2 by rounding the edge ∂Σ 1 = ∂Σ 2 to get a smooth surface Σ so that the dividing curves Γ i connect up as shown in Figure 5.1.29 to form a dividing set Γ on Σ. Σ 1 Σ Σ 2 Figure 5.2. Connecting up the dividing curves while rounding an edge Theorem 5.1.30. Suppose that ξ 1 and ξ 2 are two tight contact structures on S 1 D 2 with two parallel dividing curves on the convex boundary ∂(S 1 D 2 ) having slope equal to 1 n for some n ∈ Z. Then ξ 1 and ξ 2 are isotopic. Proof (sketch). Notice ﬁrst that 1 n and 1 m are equivalent boundary slopes for any m, n ∈ Z and −1 = 1 −1 also represents this class. Hence it suﬃces to classify the tight contact structures for any one of these slopes. It is clear that a meridian on the convex surface ∂(S 1 D 2 ) is nonisolating and therefore we can isotope this meridian into Legendrian position by the Legendrian Realization Principle. Notice that the twisting tw(∂D, D) of the contact planes along ∂D with respect to a spanning disk D of the meridian is negative. Thus D can be isotoped to a convex disk by Remark 5.1.7. Then tightness of the contact structures at hand implies by Giroux’s criterion 5.1. Convex surfaces and dividing sets 95 (Theorem 5.1.20) that the dividing set Γ D on the disk D contains no closed components, hence Γ D is a single arc connecting two points a 1 and a 2 on ∂D. Let b 1 , b 2 ∈ ∂D denote the points of the intersection of D with the dividing set on the convex boundary of the solid torus. Now we have a convex torus intersecting a convex disk along a Legendrian curve and we know the dividing sets on these surfaces. Hence by Exercise 5.1.28, b 1 is positioned between a 1 and a 2 while b 2 is positioned between a 2 and a 1 on the (oriented) circle ∂D. Next we cut S 1 D 2 along D and smooth the corners by rounding the edges using Lemma 5.1.29. Notice that when we remove a neighborhood νD of D from S 1 D 2 we get a 3-disk D 3 such that the dividing set on its boundary is connected. Now Eliashberg’s Theorem 5.1.26 concludes the proof: near the boundary and near the spanning disk D the contact structures ξ 1 and ξ 2 are isotopic (shown by the dividing curves), and the complement of νD in S 1 D 2 is D 3 with connected dividing set on its boundary. Therefore Theorem 5.1.26 extends the above isotopy to S 1 D 2 , ﬁnishing the proof. The case of general boundary slope follows by the same line of argument: By considering the disk D, however, there are more possible conﬁgurations for the dividing curves on it, since the dividing curves on ∂(S 1 D 2 ) will intersect ∂ _ ¦pt.¦D 2 _ in more points: if the slope is r = p q then ∂D inter- sects the dividing set of ∂(S 1 D 2 ) in 2p points. Every conﬁguration gives a potential tight contact structure, and so this argument gives a (poten- tially weak) upper bound for the number of tight structures. In fact, many of the diﬀerent conﬁgurations correspond to isotopic tight structures. In order to get the classiﬁcation, Honda followed a slightly diﬀerent path, and manipulated the set of dividing curves on the boundary slope by applying “bypasses”. For details see [76]. Remark 5.1.31. Notice that we assumed that the boundary slope is dif- ferent from zero. The reason is that there is no tight contact structure on S 1 D 2 with boundary slope zero: in this case ∂ _ ¦pt.¦ D 2 _ is disjoint from the dividing curves of the boundary, therefore ¦pt.¦D 2 (after having been isotoped to have Legendrian boundary) provides an overtwisted disk. 96 5. Convex surfaces in contact 3-manifolds 5.2. Contact structures and Heegaard decompositions In this section we review a construction of Torisu [168] associating a unique contact structure to an open book decomposition of a 3-manifold. Torisu’s result is based on the work of Giroux on convex contact structures. We follow an alternative line of proof which is based on the discussion in Sec- tion 5.1. It turns out that Torisu’s contact structure is compatible with the given open book decomposition in the sense of Giroux. (See Chapter 9 for relevant deﬁnitions regarding open book decompositions and their relation to contact structures.) Suppose that (L, π) is a given open book decom- position on a closed 3-manifold Y . (Here L ⊂ Y is a ﬁbered link, while π: Y − L → S 1 denotes the ﬁbration of the open book decomposition.) Then by presenting the circle S 1 as the union of two closed (connected) arcs S 1 = A 1 ∪ A 2 intersecting each other in two points, the open book decom- position (L, π) naturally induces a Heegaard decomposition Y = U 1 ∪ Σ U 2 of the 3-manifold Y : one only needs to verify the simple observation that U i = π −1 (A i ) ∪ L are solid handlebodies. The surface Σ along which these handlebodies are glued is simply the union of two pages π −1 (A 1 ∩ A 2 ) to- gether with the binding. This is illustrated in Figure 5.3. Σ L Σ Figure 5.3. The handlebody Ui 5.2. Contact structures and Heegaard decompositions 97 Theorem 5.2.1 (Torisu, [168]). Suppose that ξ 1 , ξ 2 are contact structures on Y satisfying: (i) ξ i [ U j (i = 1, 2; j = 1, 2) are tight, and (ii) Σ is convex in (Y, ξ i ) and L is the dividing set for both contact structures. Then ξ 1 and ξ 2 are isotopic. In addition, the set of such contact structures is nonempty. Proof. Suppose that a page F of the given open book is a genus-g surface with r boundary components. Then ∂U 1 = Σ is a closed surface of genus h = 2g + r − 1. First we would like to argue that there is at most one tight contact structure on the handlebody U 1 such that L is the dividing set on Σ. Since U 1 is a genus-h handlebody, it is clear that we can ﬁnd h homologically linearly independent curves α 1 , α 2 , . . . , α h on Σ which bound h disjoint disks in U 1 so that when we cut along these disks we get the 3-disk D 3 . The key point of our construction is that we can choose α 1 , α 2 , . . . , α h in such a way that each α k intersects the dividing set L ⊂ Σ twice for k = 1, 2, . . . , h. We depicted a choice of such curves for r = 3, g = 2 in Figure 5.4. The disk D k spanned by α k can be visualized as the disk which is swept out in U 1 (see Figure 5.3) by swinging the left-half of the curve α k until it coincides with its right-half. α α α α α α L L L 6 5 4 3 1 2 Figure 5.4. The α-curves. Now we proceed exactly as in the proof of Theorem 5.1.30. First we put the curves α 1 , α 2 , . . . , α h into Legendrian position and make the spanning disks D 1 , D 2 , . . . , D h convex. The diving set on each D k will be an arc connecting two points on the boundary, for k = 1, 2, . . . , h. Then we cut 98 5. Convex surfaces in contact 3-manifolds along these disks and round the edges to get a connected dividing set on the remaining D 3 and use Eliashberg’s theorem to show the uniqueness of a tight contact structure on U 1 with the assumed boundary condition. Clearly we can prove the same result for the handlebody U 2 . To ﬁnish this part of the proof of the theorem we need to show the existence of a tight contact structure η 1 on U 1 (and η 2 on U 2 ) which has L as its dividing set on Σ. The idea is to embed U 1 into an open book whose compatible contact structure (see Chapter 9) is Stein ﬁllable, and hence tight. Such an embedding of the genus-g handlebody into a Stein ﬁllable contact structure will be shown in Exercise 11.3.5(c). Suppose now that η j is a tight contact structure on U j whose dividing set is equal to the binding L ⊂ ∂U j for j = 1, 2. Let φ: ∂U 1 →∂U 2 be the diﬀeomorphism deﬁning the Heegaard decomposition Y = U 1 ∪ Σ U 2 . The tight contact structure η j on U j induces a foliation T j on Σ. Now φ(T 1 ) (as well as T 2 ) is a singular foliation on ∂U 2 divided by L (since φ is the identity on L). Then by Giroux’s ﬂexibility Theorem 5.1.16 we can isotope Σ in Y so that φ(T 1 ) and T 2 agree by this isotopy and hence we can glue the tight contact structure η 1 and η 2 to get a contact structure ξ on Y . The uniqueness of such a contact structure ξ on Y follows from the uniqueness of η 1 and η 2 . Notice that the tightness of η 1 and η 2 does not imply tightness for the glued up contact structure ξ. Example 5.2.2. Consider the open book decomposition of S 3 induced by the positive (resp. negative) Hopf link. The associated contact structure ξ is tight (resp. overtwisted). Let (Y i , ξ i ) be a contact 3-manifold for i = 1, 2. To deﬁne the contact connected sum (Y = Y 1 #Y 2 , ξ = ξ 1 #ξ 2 ) just delete a Darboux ball D i from Y i (i = 1, 2) and glue Y 1 −int D 1 to Y 2 −int D 2 by a diﬀeomorphism f : ∂(Y 1 − int D 1 ) → ∂(Y 2 − int D 2 ) which takes the dividing set Γ 1 on ∂(Y 1 − int D 1 ) to the dividing set Γ 2 on ∂(Y 2 − int D 2 ). This operation is well deﬁned by Eliashberg’s Theorem 5.1.26. Remark 5.2.3. Torisu [168] also proves that the contact structure asso- ciated to a plumbing of two ﬁbered links (cf. Chapter 9) is the contact connected sum of the corresponding contact structures. (See [68] for an alternative proof.) Exercise 5.2.4. Show that the contact structure associated to the open book of S 3 induced by a (p, q)-torus knot is tight. (Hint: See Example 9.1.4.) 6. Spin c structures on 3- and 4-manifolds Spin c structures turn out to be very useful tools in understanding homotopic properties of contact structures. In addition, gauge theoretic invariants — such as Seiberg–Witten and Ozsv´ath–Szab´o invariants — are deﬁned for spin c 3- and 4-manifolds. This chapter is devoted to the review of spin c structures — with a special emphasis on the 3- and 4-dimensional case. Throughout this chapter we will assume that the reader is familiar with the basics of the theory of characteristic classes. (For an excellent reference see [116].) For a more complete treatment of spin c structures the reader is advised to turn to [113]. 6.1. Generalities on spin and spin c structures We begin our discussion by recalling the related and much more standard subject of spin structures. Spin structures By deﬁnition the n-dimensional (n ≥ 3) spin group Spin(n) is the universal (double) cover of SO(n). In other words, Spin(n) is a simply connected Lie group with a map ρ: Spin(n) → SO(n) which is the principal Z 2 - bundle of the unique nontrivial real line bundle on SO(n) (n ≥ 3). (Recall that π 1 _ SO(n) _ = H 1 _ SO(n); Z _ = Z 2 .) Let X be a given oriented Riemannian n-manifold and let p: P SO(n) →X denote the principal SO(n)- bundle of orthonormal frames in TX. A spin structure on X is a principal Spin(n)-bundle s: P Spin(n) → X with a map π: P Spin(n) → P SO(n) such that p◦π = s and ﬁberwise π is just the double cover ρ: Spin(n) →SO(n). 100 6. Spin c structures on 3- and 4-manifolds In other words, the associated principal SO(n)-bundle P Spin(n) ρ SO(n) is isomorphic to P SO(n) . Two spin structures P 1 and P 2 are equivalent if there is a bundle isomorphism ϕ: P 1 →P 2 such that s 1 = s 2 ◦ϕ where s i : P i →X are the bundle maps of the principal Spin(n)-bundles for i = 1, 2. The set of equivalence classes of spin structures on X will be denoted by Spin(X). Remark 6.1.1. More generally, for any principal SO(n)-bundle E → X a similar deﬁnition provides spin structures on E. Theorem 6.1.2. An oriented Riemannian n-manifold X admits a spin structure if and only if w 2 (X) = 0. In that case the number of inequivalent spin structures is equal to ¸ ¸ H 1 (X; Z 2 ) ¸ ¸ . In fact, H 1 (X; Z 2 ) admits a free and transitive action on Spin(X). In a similar fashion, it can be shown that a principal SO(n)-bundle E →X admits a spin structure if and only if w 2 (E) = 0, and the number of inequivalent spin structures is again equal to ¸ ¸ H 1 (X; Z 2 ) ¸ ¸ . Spin c structures The group Spin c (n) is deﬁned as S 1 Spin(n)/Z 2 where Z 2 = _ ±(1, 1) ∈ S 1 Spin c (n) _ , and ¦±1¦ is deﬁned as ker ρ ⊂ Spin(n). It follows that Spin c (n) admits an S 1 -ﬁbration over SO(n) (n ≥ 3); this map ρ: Spin c (n) → SO(n) can be characterized as the principal S 1 -bundle of the unique nontrivial complex line bundle on SO(n) (n ≥ 3). Notice that H 2 (SO(n); Z) = Z 2 for n ≥ 3. Again, a spin c structure on an n-dimensional manifold X is a principal Spin c (n)-bundle s: P Spin c (n) → X with a map π: P Spin c (n) →P SO(n) such that s = p ◦ π and ﬁberwise π is just ρ; equiva- lently, in bundle theoretic terms P Spin c (n) ρ SO(n) ∼ = P SO(n) . As in the spin case, spin c structures can be deﬁned for any principal SO(n)-bundle. The map α: Spin c (n) →S 1 we get by the formula _ ±(z, A) _ →z 2 enables us to associate a line bundle — the determinant line bundle — L = P Spin c (n) α C to a given spin c structure P Spin c (n) . In an equivalent way, a spin c structure on X can be regarded as an element u ∈ H 2 (P SO(n) ; Z) whose restriction to every ﬁber of P SO(n) →X is the unique nontrivial element of H 2 (SO(n); Z): by considering the S 1 -bundle s u : L →P SO(n) corresponding to u, the com- position s u ◦ p provides a principal Spin c (n)-bundle structure on L and hence a spin c structure in the above sense. Similarly to the spin case, we say that spin c structures P 1 , P 2 → X are equivalent if there is a bundle 6.1. Generalities on spin and spin c structures 101 isomorphism h: P 1 → P 2 satisfying s 1 = s 2 ◦ h, where s i : P i → X are the bundle projections. The set of equivalence classes of spin c structures on a ﬁxed manifold X will be denoted by Spin c (X). As the above reformulation shows, we can regard Spin c (X) as subset of H 2 (P SO(n) ; Z). Theorem 6.1.3. Let P Spin c (n) be a given spin c structure with determinant line bundle det P Spin c (n) = L. Then c 1 (L) ≡ w 2 (X) (mod 2). In addition, if c ∈ H 2 (X; Z) satisﬁes c ≡ w 2 (X) (mod 2) then there is a spin c structure with determinant line bundle L satisfying c 1 (L) = c. Proof (sketch). The natural map α ρ: Spin c (n) →SO(2) SO(n) can be shown to be the unique double cover of SO(2) SO(n) which extends to a double cover of SO(n + 2), hence TX admits a spin c structure if and only if there is a line bundle L such that TX ⊕ L admits a spin structure. This latter is equivalent to w 2 (TX ⊕ L) = w 2 (TX) + c 1 (L) ¸ ¸ 2 = 0, proving the claim. The group H 2 (X; Z) admits a free and transitive action on Spin c (X) (if the latter is nonempty) as follows: for s ∈ Spin c (X) ⊂ H 2 (P SO(n) ; Z) and a ∈ H 2 (X; Z) the action of a on s is given by s →s +p ∗ (a) where p: P SO(n) → X is the bundle map of the frame bundle. The natural group homomorphism Spin(n) → Spin c (n) shows that a spin structure induces a spin c structure. It follows that such an induced spin c structure has trivial determinant line bundle. Conversely, if det (P Spin c (n) ) is trivial for a spin c structure then it can be induced by a spin structure, since the triviality of the determinant line bundle shows that the cocycle structure of P Spin c (n) can be homotoped into the kernel ker α = Spin(n). The collar neighborhood theorem for the embedding ∂X ֒→X provides a splitting of TX[ ∂ = T(∂X)⊕R near the boundary, implying in particular, that a spin (or spin c ) structure on X naturally induces a similar structure on ∂X. (As always, R denotes the trivial real line bundle.) After having dispensed with the above general discussion, in the rest of this chapter we focus on the 3- and 4-dimensional case and relate spin c structures to other geometric objects on such manifolds. 102 6. Spin c structures on 3- and 4-manifolds 6.2. Spin c structures and oriented 2-plane fields We start with the 3-dimensional case. It is fairly easy to see that Spin(3) = SU(2) ∼ = S 3 and Spin c (3) = U(2). Notice also that from the theory of characteristic classes it follows that for a 3-manifold X we have w 2 (X) = w 2 1 (X), therefore any oriented 3-manifold admits a spin (and so spin c ) structure. A spin c structure on an oriented 3-dimensional Euclidean vector space V can be given by specifying a complex hermitian plane W with a map γ : V → Hom C (W, W) satisfying γ(v) ∗ γ(v) = −[v[ 2 id W . Globally, a spin c structure on an oriented Riemannian 3-manifold is simply a continuous family of spin c structures on the tangent spaces, i.e., a pair (W, ρ) where W →Y is a hermitian C 2 -bundle (a U(2)-bundle) on the 3-manifold Y and ρ: T C Y →Hom C (W, W) is a bundle homomorphism satisfying ρ(v) ∗ ρ(v) = −[v[ 2 id W . The equivalence with the deﬁnition given in Section 6.1 is clear: P Spin c (3) = P U(2) corresponds to the principal U(2)-bundle of W while ρ: T C Y →Hom C (W, W) and the map π: P Spin c (3) →P SO(3) determine each other. Next we discuss a more geometric presentation of spin c structures on 3-manifolds; this presentation will be more suitable for our purposes in our subsequent discussions. Let Ξ(Y ) denote the space of oriented 2-plane ﬁelds on Y , while V ect(Y ) stands for the set of vector ﬁelds of length 1. Notice that by considering the oriented unit normal of an oriented 2-plane ﬁeld we get a bijection Ξ(Y ) →V ect(Y ). Deﬁnition 6.2.1. Two nowhere vanishing vector ﬁelds v 1 and v 2 are said to be homologous if v 1 is homotopic to v 2 outside a disk D 3 ⊂ Y (through nowhere vanishing vector ﬁelds). This equivalence relation — together with the identiﬁcation given above — induces an equivalence relation on Ξ(Y ) and hence on π 0 _ Ξ(Y ) _ . For the proof of the following statement see [169]. Proposition 6.2.2 (Turaev, [169]). Spin c (Y ) can be identiﬁed with the set of equivalence classes of elements of π 0 _ Ξ(Y ) _ under the equivalence relation given by homology. Exercises 6.2.3. (a) Show that an oriented 2-plane ﬁeld reduces the struc- ture group of TY →Y from SO(3) to U(1), and this latter group admits a natural lift to U(2). 6.2. Spin c structures and oriented 2-plane ﬁelds 103 (b) Verify that a C 2 -bundle W → Y admits a nowhere vanishing section. Show that a spin c structure (W, ρ) induces a nowhere vanishing section of TY →Y . (c) Using the solutions of the above exercises prove Proposition 6.2.2. In fact, the above correspondence can be reﬁned as follows: Lemma 6.2.4 (Kronheimer–Mrowka, [86]). Let us ﬁx a closed, oriented 3- manifold Y . There is a one-to-one correspondence between the space Ξ(Y ) of oriented 2-plane ﬁelds on Y and isomorphism classes of pairs (t, φ) where t ∈ Spin c (Y ) and φ ∈ Γ(W) is of unit length. An oriented 2-plane ﬁeld, or more speciﬁcally a contact structure ξ naturally induces a spin c structure which will be denoted by t ξ . Let p: π 0 _ Ξ(Y ) _ →Spin c (Y ) denote the map associating t ξ to ξ. In the rest of this section we brieﬂy recall the classiﬁcation of oriented 2-plane ﬁelds (up to homotopy) on Y , cf. also Chapter 11 of [66]. Trivializing TY and consid- ering the oriented normal of an oriented 2-plane ﬁeld, a map Y →S 2 can be associated to ξ ∈ Ξ(Y ). In particular, on Y = S 3 the oriented 2-plane ﬁelds are in one-to-one correspondence with elements of [S 3 , S 2 ] = π 3 (S 2 ) ∼ = Z. Using the Pontrjagin-Thom construction, the space [Y, S 2 ] can be identiﬁed with the framed cobordism classes of framed 1-manifolds in Y . Homotopies outside of a disk (i.e., spin c structures) can be parameterized by the 1- manifolds in Y up to cobordism, i.e., with H 1 (Y ; Z) ∼ = H 2 (Y ; Z). A ﬁber p −1 (t) for a spin c structure t admits an [S 3 , S 2 ] ∼ = Z-action: for any n we can “twist” the given framing of the framed link corresponding to the oriented 2-plane ﬁeld by n. Viewing this action from another point of view, oriented 2-plane ﬁelds (or, equivalently, the orthogonal vector ﬁelds corresponding to them) inducing a ﬁxed spin c structure t ∈ Spin c (Y ) can be assumed to be identical outside of a disk D 3 ⊂ Y . Then Z acts on p −1 (t) by connect summing a given (Y, v) (where [v] ∈ p −1 (t)) with the elements of _ (S 3 , w) [w is a nowhere zero vector ﬁeld on S 3 _ . By pulling back the generator of H 2 (S 2 ; Z) with the map f ξ : Y → S 2 associated to ξ ∈ Ξ(Y ) we get a second cohomology class Γ ξ ∈ H 2 (Y ; Z). This class will depend on the chosen trivialization of TY , but for ξ 1 , ξ 2 ∈ Ξ(Y ) the diﬀerence Γ ξ 1 − Γ ξ 2 is independent of this choice, since it can be identiﬁed with the obstruction of f ξ 1 being homotopic to f ξ 2 on Y − D 3 . This observation again shows the existence of a natural free and transitive H 2 (Y ; Z)-action on Spin c (Y ). It is not hard to see that c 1 (ξ) ∈ H 2 (Y ; Z) 104 6. Spin c structures on 3- and 4-manifolds (where we regard ξ as an oriented R 2 -, hence a complex line bundle) is equal to 2Γ ξ : by deﬁnition Γ ξ is the pull-back of [S 2 ] = PD[point] ∈ H 2 (S 2 ; Z) while ξ is the pull-back of the tangent bundle TS 2 , hence c 1 (ξ) = f ∗ ξ _ c 1 (TS 2 ) _ = f ∗ ξ (PD[two points]). Consequently, if H 2 (Y ; Z) ∼ = H 1 (Y ; Z) has no 2-torsion, then c 1 (ξ) deter- mines the spin c structure t ξ induced by ξ. Notice that c 1 (ξ) = c 1 (t ξ ) for the induced spin c structure t ξ , since a second cohomology class uniquely ex- tends from Y −D 3 to Y on a 3-dimensional manifold. Recall that Z admits a transitive action on p −1 (t) for any spin c structure t. In the statement below, Z 0 is understood to be equal to Z. Proposition 6.2.5 (Gompf, [64]). Let t ∈ Spin c (Y ) be a given spin c structure. The ﬁber p −1 (t) can be identiﬁed with Z d(t) where d(t) denotes the divisibility of c 1 (t), and is zero if c 1 (t ξ ) is a torsion class. In conclusion, the homotopy type of a 2-plane ﬁeld ξ is uniquely speciﬁed by the induced spin c structure t ξ and the framing of the corresponding 1- manifold in Y . This latter invariant is an element of Z d(t) in general, and it is hard to work with, except in the case of torsion ﬁrst Chern class c 1 (t ξ ). In this case the set of framings (an aﬃne set for Z) can be lifted to a subset of Q as follows: Suppose that ξ ∈ Ξ(Y ) has torsion ﬁrst Chern class c 1 (ξ) and suppose furthermore that (X, J) is an almost-complex 4-manifold with ∂X = Y and ξ is homotopic (as an oriented 2-plane ﬁeld) to the complex tangencies along ∂X, i.e., to TY ∩ JTY . Lemma 6.2.6 (Gompf, [64]). The expression d 3 (ξ) = 1 4 _ c 2 1 (X, J) −3σ(X) −2χ(X) _ ∈ Q deﬁnes an invariant of ξ. The proof is a standard exercise relying on the following Theorem 6.2.7 (Hirzebruch signature theorem for 4-manifolds). For a closed almost-complex 4-manifold (X, J) we have c 2 1 (X, J) = 3σ(X) + 2χ(X). Exercises 6.2.8. (a) Verify that if c 1 (ξ) is torsion then c 2 1 (X, J) ∈ Q is well-deﬁned. (Hint: Cf. Remark 3.1.11.) (b) Show that _ d 3 (ξ) [ ξ ∈ Ξ(S 3 ) _ ⊂ Z + 1 2 . (Hint: Use the fact that for a unimodular form Q and characteristic vector c we have Q(c, c) ≡ σ(Q) (mod 8).) In fact, the above two sets are equal. 6.3. Spin c structures and almost-complex structures 105 It is not hard to see (cf. [64]) that for any oriented 2-plane ﬁeld ξ ∈ Ξ(Y ) there is an almost-complex 4-manifold (X, J) such that ∂X = Y and ξ is homotopic to the oriented 2-plane ﬁeld of complex tangencies along ∂X. The rational number d 3 (ξ) deﬁned for those ξ ∈ Ξ(Y ) which have torsion c 1 (ξ) is called the 3-dimensional invariant of the oriented 2-plane ﬁeld ξ. Proposition 6.2.5 now specializes to Theorem 6.2.9 (Gompf, [64]). Suppose that the oriented 2-plane ﬁelds ξ 1 , ξ 2 induce the same spin c structure t and that c 1 (t) is torsion. Then [ξ 1 ] = [ξ 2 ] if and only if d 3 (ξ 1 ) = d 3 (ξ 2 ). Later we will show explicit computations for d 3 (ξ) of some contact struc- tures ξ. 6.3. Spin c structures and almost-complex structures Next we consider the geometric interpretation of spin c structures on 4- manifolds. Recall that Spin(4) = SU(2) SU(2) and so Spin c (4) = S 1 SU(2) SU(2)/ ±(1, id, id); alternatively, Spin c (4) = _ (A, B) ∈ U(2) U(2) [ det A = det B _ . The isomorphism between the above groups is given by the map (A, B) → _ α, A _ α −1 0 0 α −1 _ , B _ α −1 0 0 α −1 __ with α 2 = det A = det B. Spin and spin c structures in dimension 4 can also be deﬁned as follows. First let V be a 4-dimensional oriented Euclidean vector space. A spin structure on V is a pair (V + , V − ) of 1-dimensional quaternionic vector spaces with hermitian metrics together with an isomor- phism γ : V → Hom H (V + , V − ) compatible with the metrics. (Note that the group of symmetries of V is SO(4), while for the spin structure (V ± , γ) the group of symmetries is Spin(4).) Globally, for an oriented, Rieman- nian 4-manifold X a spin structure is a triple (S + , S − , ρ) where S ± → X are quaternionic line bundles with hermitian metrics (i.e., SU(2)-bundles) and ρ: TX →Hom H (S + , S − ) is a bundle isomorphism compatible with the chosen metrics. Using the cocycle structures of the bundles S ± → X it is fairly easy to see that this deﬁnition coincides with the general one given in Section 6.1. 106 6. Spin c structures on 3- and 4-manifolds A spin c structure on a vector space V is a pair (V + , V − ) of com- plex planes with hermitian metrics such that det C V + ∼ = det C V − to- gether with an isomorphism γ : V ⊗ C → Hom C (V + , V − ) which satisﬁes γ(v) ∗ γ(v) = −[v[ 2 id V +. (It is not hard to verify that the symmetry group of (V ± , γ) is isomorphic to Spin c (4).) Once again, by globalizing the above construction, we can deﬁne a spin c structure on X by a triple (W + , W − , ρ) where W ± → X are hermitian C 2 -bundles with det W + ∼ = det W − and ρ: TX ⊗ C → Hom C (W + , W − ) is a bundle isomorphism which satisﬁes ρ(v) ∗ ρ(v) = −[v[ 2 id W +. The proof of equivalence is again an easy exercise. As a simple homological argument shows (see e.g. [66]) the set of spin c structures on an oriented 4-manifold X is always nonempty, and hence it is (noncanonically) isomorphic to H 2 (X; Z). An almost-complex structure J naturally induces a spin c structure s J : the almost-complex structure re- duces the structure group of TX from SO(4) to U(2), and the map A → __ det A 0 0 1 _ , A _ ∈ U(2) U(2) provides the desired lift from U(2) to Spin c (4). Alternatively, since J gives rise to the bundles Λ p,q J (X), we can take W + to be equal to Λ 0,0 J (X)⊕Λ 0,2 J (X) and W − = Λ 0,1 J (X) together with ρ deﬁned as ρ(x)(α, β) = √ 2((x +iJx)α −∗ _ (x +iJx) ∧ ∗β _ ). If J is deﬁned away from ﬁnitely many points on X, we still get an induced spin c structure s J ∈ Spin c (X) since the above construction provides a spin c structure on X − ¦x 1 , . . . , x n ¦ where J is deﬁned and (since both S 3 and D 4 admit unique spin c structures) it extends uniquely to X. It can be shown that J 1 , J 2 induces the same spin c structure if J 1 is homotopic to J 2 outside of a 1-dimensional submanifold (containing all points where J i are undeﬁned). Hence Proposition 6.3.1. The set of spin c structures Spin c (X) on X can be identiﬁed with ¦J almost-complex structure on X −¦x 1 , . . . , x n ¦ for some x 1 , . . . , x n ∈ X¦/ ∼, where ∼ is the equivalence relation described above. 6.3. Spin c structures and almost-complex structures 107 If X is a compact 4-manifold with J as above, then the complex tangencies along ∂X provide an oriented 2-plane ﬁeld, giving a geometric interpretation of the restriction map Spin c (X) →Spin c (∂X) of spin c structures. Exercises 6.3.2. (a) Show that s = (W ± , γ) ∈ Spin c (X) is induced by an almost-complex structure if and only if c 2 (W + ) = 0. (b) Verify that for a closed 4-manifold X the identity c 2 (W + ) = 1 4 _ c 2 1 (W + ) −3σ(X) −2χ(X) _ holds. If the second cohomology group contains torsion elements (for example, if Y is a rational homology 3-sphere) it is quite complicated to work with spin c structures directly. In such cases c 1 might not determine the spin c structure, and we cannot work with torsion second cohomology classes through their values on embedded surfaces. The underlying smooth 3-manifold can always be presented as the oriented boundary of a smooth 4-manifold built from a 0-handle and some 2-handles only. Studying spin c structures on Y through their extensions to simply connected 4-manifolds (i.e., to manifolds where we do not have torsion (co)homologies) turns out to be very useful in numerous situations. Exercise 6.3.3. Show that if X is simply connected then the restriction map Spin c (X) → Spin c (∂X) is onto. (Hint: Apply the long exact cohomology sequence of the pair (X, ∂X) and use the fact that H 1 (X; Z) = 0.) Therefore, instead of studying t ∈ Spin c (∂X) we can focus on some s ∈ Spin c (X) with s[ ∂X = t. Since π 1 (X) = 1, the spin c structure s ∈ Spin c (X) is uniquely determined by c 1 (s), and this class is speciﬁed by its values on the second homologies of X. So suppose that X is a compact 4-manifold with boundary, given by a Kirby diagram involving a unique 0-handle and t 2-handles, attached along the knots K i (i = 1, . . . , t) with framings n i (i = 1, . . . , t). The corresponding basis of H 2 (X; Z) is denoted by α 1 , . . . , α t . Suppose furthermore that J is an almost-complex structure on X with ﬁrst Chern class c 1 (X, J) satisfying ¸ c 1 (X, J), α i _ = m i for i = 1, . . . , t. Recall that c 1 (X, J) is a characteristic cohomology element, that is, ¸ c 1 (X, J), α i _ ≡ Q X (α i , α i ) (mod 2). 108 6. Spin c structures on 3- and 4-manifolds Denote the induced oriented 2-plane ﬁeld of complex tangencies on Y = ∂X by ξ. Using the notation of Section 2.3 (see text preceding Exercises 2.3.4) we get that the Poincar´e dual of c 1 (X, J) is equal to t i=1 m i [D i ]. This element maps to t i=1 m i µ i ∈ H 1 (Y ; Z). From the relations among the homology classes µ i we get a presentation of H 1 (Y ; Z), and we can easily identify c 1 (ξ) and decide whether it is torsion or not. If c 1 (ξ) is torsion then for some n ∈ N the class nc 1 (ξ) = 0, implying that nPD _ c 1 (X, J) _ maps to zero under the map H 2 (X, Y ; Z) ϕ 2 −→H 1 (Y ; Z), hence it is in the image of ϕ 1 : H 2 (X; Z) → H 2 (X, Y ; Z). Since ϕ 1 is explicitly described in Section 2.3, it is a simple matter of solving a linear system of equations to ﬁnd c ∈ H 2 (X; Z) with the property that ϕ 1 (c) = nPD _ c 1 (X, J) _ . The linking matrix of the Kirby diagram deﬁning X enables us to determine cc = Q X (c, c), leading to a computation of c 2 1 (X, J) since this latter term is equal to 1 n 2 c c ∈ 1 n Z. Having this quantity at hand now it is an easy exercise to determine d 3 (ξ) since the linking matrix of the Kirby diagram provides χ(X) and σ(X). Of course, in general it is rather hard to ﬁnd an appropriate (X, J) for a given (Y, ξ). As we will see, for a contact 3-manifold (Y, ξ) given by a contact surgery diagram such (X, J) can be described quite easily. The above discussion naturally extends to cobordisms as well. Suppose that (Y, t) is a given spin c 3-manifold and the cobordism is deﬁned by attaching a (4-dimensional) 2-handle along K ⊂ Y . Fix a 4-manifold X with ∂X = Y which admits a handle decomposition with 0- and 2-handles only, and let s ∈ Spin c (X) be chosen in such a way that s[ ∂X = t. Note that since X is simply connected, the spin c structure s is determined by the values of c 1 (s) on a generating system of the second homology group H 2 (X; Z). A spin c structure s 1 on X ∪ W extending s ∈ Spin c (X), that is, a spin c cobordism (W, s 1 ) from (Y, t) can be speciﬁed now by the value m of c 1 (s 1 ) on the 2-homology deﬁned by the 2-handle giving rise to W. Since H 2 (W, ∂W; Z) ∼ = Z¸g) with g a generator, the value ¸ c 1 (s 1 ), g _ = n speciﬁes the extension s 1 of t. The computation of the self-intersection of g and so of c 2 1 (s 1 ) follows the same line as it is discussed in Section 2.3. Exercises 6.3.4. (a) Suppose that [K] = 0 in H 1 (Y ; Z). Determine the number of possible extensions of a given spin c structure t to the cobor- dism given by the handle attachment along the knot K ⊂ Y with surgery coeﬃcient being equal to 0 (with respect to the Seifert framing). 6.3. Spin c structures and almost-complex structures 109 (b) Suppose that (Y, t) is a spin 3-manifold. Show that s 1 ∈ Spin c (W) is a spin extension if and only if, with the above notations, ¸ c 1 (s 1 ), g _ = 0. (c) Find an example of a spin c cobordism (W, s) such that ¸ c 1 (s 1 ), g _ = 0 but s 1 ∈ Spin c (W) is not spin. (Hint: Start with a nonspin structure t ∈ Spin c (Y ) and extend it.) (d) Consider the cobordism W given by Figure 2.12. Let the spin c structure s ∈ Spin c (W) satisfy ¸ c 1 (s), g _ = n. Determine c 2 1 (s) and 1 4 _ c 2 1 (s) − 3σ(W) −2χ(W) _ ∈ Q. 7. Symplectic surgery After these preparatory chapters now we are ready to describe the surgery scheme in the symplectic category. First we will deal with the general cut- and-paste operation and then examine the handle attachment procedure in detail. The chapter concludes with the description of a version of surgery which will be useful in the contact setting, see Chapter 11. 7.1. Symplectic cut-and-paste Deﬁnition 7.1.1. A vector ﬁeld v on a symplectic manifold (X, ω) is a symplectic dilation or Liouville vector ﬁeld if L v ω = ω. Notice that since dω = 0 we have that L v ω = dι v ω + ι v dω = dι v ω, therefore the above equation translates to dι v ω = ω. A codimension-1 submanifold Y ⊂ (X, ω) is of contact type if there is a vector ﬁeld v deﬁned on some neighborhood νY of Y which is a symplectic dilation and is transverse to Y . Remark 7.1.2. Notice the similarity with the deﬁnition of convex surfaces in contact manifolds. The important diﬀerence is that now a symplectic dilation v has a direction: −v is not a dilation anymore, since L −v ω = −ω. This orientation property is also reﬂected in the following deﬁnition: Deﬁnition 7.1.3. A codimension-0 submanifold U ⊂ X in (X, ω) is ω- convex (ω-concave) if ∂U is of contact type and the vector ﬁeld v points out of (into) U. Let L Y = TY ⊥ = _ v ∈ TX [ ω(v, x) = 0 for all x ∈ TY _ . Since ω is antisymmetric, we have that TY ⊥ ⊂ TY ; here ⊥ is taken with respect of ω. From the nondegeneracy of ω it follows that L Y is a line ﬁeld on Y . Consider the special case when Y is given as H −1 (a) for a function H: X → R and 112 7. Symplectic surgery regular value a ∈ R. Then L Y is “equal to” the vector ﬁeld v H , where v H is speciﬁed by the equation dH = ι v H ω — that is, the vector ﬁeld v H is in the line ﬁeld L Y . Recall that since ω is nondegenerate, the formula dH = ι v H ω uniquely determines v H . Therefore the fact that v H ∈ L Y easily follows from the fact that for any u ∈ TY we have ω(u, v H ) = dH(u) = 0 since H does not change in the Y direction. Theorem 7.1.4 (Weinstein, [173]). The codimension-1 submanifold i : Y → X is of contact type if and only if there is a 1-form α on Y such that dα = i ∗ ω and α is nonzero on L Y . Proof. We prove the theorem in the special case when Y = H −1 (a) for some function H: X →R and regular value a. Suppose that Y is of contact type with vector ﬁeld v. Consider α ′ = ι v ω = ω(., v) and take α = i ∗ α ′ . Now ω = L v ω = dι v ω = dα ′ implies i ∗ ω = dα. In order to evaluate α on L Y notice that α(v H ) = (ι v ω)(v H ) = ω(v H , v) = −dH(v) ,= 0 since v is transverse to Y = H −1 (a). Since v H spans L Y , the second property follows for α. For the converse direction extend the given α to α ′ deﬁned on the neighborhood νY in such a way that dα ′ = ω. (This can be done since νY retracts to Y .) If v is deﬁned by the equation ι v ω = α ′ we get that L v ω = dι v ω = dα ′ = ω, moreover −dH(v) = ω(v H , v) = ι v ω(v H ) = α ′ (v H ) ,= 0 implies that v is transverse to the level set Y = H −1 (a). Remark 7.1.5. The assumption that Y = H −1 (a) for some function H is not very restrictive. For a codimension-1 submanifold Y ⊂ X we can always ﬁnd H such that Y ⊂ H −1 (a) for some regular value a — and this description is enough for our purposes. Equality cannot always be achieved, since the complement of Y might be connected, preventing the existence of an appropriate H — this is the case, for example, if X = T 2 and Y is a homologically essential circle on it. Proposition 7.1.6. If Y ⊂ (X, ω) is of contact type then the 1-form α provided by Theorem 7.1.4 is a contact form on Y . Proof. We need to examine dα on ker α. Notice that dα = i ∗ ω and ker α ∼ = TY/L Y , on which i ∗ ω is obviously a symplectic form, proving that α is a contact form. (For ker α ∼ = TY/L Y notice that ker α ∩L Y = 0, hence the map sending u ∈ ker α ⊂ TY to [u] ∈ TY/L Y is injective, and so an isomorphism by dimension reasons.) Notice that the contact structure ξ on Y induced by α is cooriented by α. 7.1. Symplectic cut-and-paste 113 Remark 7.1.7. Alternatively, using α = ι v ω we can compute α ∧ dα = ι v ω ∧ d(ι v ω) = ι v ω ∧ L v ω = ι v ω ∧ ω = 1 2 ι v (ω ∧ ω), so α ∧ dα is nowhere zero on a hypersurface Y transverse to v, therefore ξ = ker α is a contact structure on Y (after using the appropriate orientation). Informally, the Liouville vector ﬁeld v in the deﬁnition of a hypersurface of contact type helps us to determine the symplectic structure near Y . This means that if we “know” ω on Y (through, for example, the induced contact form α) then we “know” ω near Y . To make this picture more rigorous, we prove the following Proposition 7.1.8. Suppose that Y ⊂ (X, ω) is a hypersurface of contact type (with vector ﬁeld v). Then Y admits a neighborhood νY symplecto- morphic to a neighborhood ν _ α(Y ) _ of α(Y ) ⊂ Symp(Y, ξ), where α = ι v ω and ξ = ker _ α[ Y _ . Proof. Let us denote the symplectic form d(tα) on Symp(Y, ξ) by ω ′ . According to the Tubular Neighborhood Theorem there are neighborhoods νY ⊂ X and ν _ α(Y ) _ ⊂ Symp(Y, ξ) which are diﬀeomorphic through a diﬀeomorphism sending the ﬂow of v to the ﬂow of ∂ ∂t . Notice that ω ′ [ α(Y ) = dα = ω[ Y ; furthermore we can arrange ω ′ [ T(Symp)|α(Y ) = ω[ TX| Y ; so ω ′ = ω also holds in the normal direction. Using Moser’s method now the diﬀeomorphism can be isotoped to a symplectomorphism. Now we are in the position to prove the theorem which allows us to perform symplectic cut-and-paste. Theorem 7.1.9 ([38]). Suppose that U i ⊂ X i are codimension-0 submani- folds and ω i symplectic forms on X i (i = 1, 2) such that U i are ω i -convex and the boundaries Y i = ∂U i with the induced contact structures are contacto- morphic. Then the surgered manifold (X 1 − U 1 ) ∪ U 2 admits a symplectic structure. Proof. Consider the contact forms α i = ι v i ω i (i = 1, 2) and take the symplectization Symp(Y, ξ) with contact structure ξ = ker α 1 induced by α 1 . Now α 1 (Y ) = ¦1¦ Y ⊂ Symp(Y, ξ) = (0, ∞) Y . For the con- tactomorphism Ψ: (Y, ker α 1 ) → (Y, ker α 2 ) there is a function f : Y → R such that Ψ ∗ α 2 = fα 1 ; the graph of f : Y → R in Symp(Y, ξ) will be de- noted by α 2 (Y ) ⊂ Symp(Y, ξ = ker α 1 ). Fix neighborhoods N i ⊂ X i and N ′ i ⊂ Symp(Y, ξ) of Y and α i (Y ), respectively, which are pairwise sym- plectomorphic (such neighborhoods are provided by Proposition 7.1.8). By 114 7. Symplectic surgery rescaling ω 2 we can achieve that f < 1, hence N ′ 1 and N ′ 2 can be chosen to be disjoint. Considering V ⊂ Symp(Y, ξ) bounded by N ′ 1 and N ′ 2 we can form [X 1 −(U 1 −N 1 )] ∪V ∪(U 2 ∪N 2 ). (Notice that topologically V is triv- ial, it serves as an interpolation between the symplectic structures on the two pieces.) By applying the above symplectomorphisms on the overlapping regions we can glue the symplectic forms together, producing a symplectic structure on the smooth 4-manifold (X −U 1 ) ∪ U 2 . As a special case of the above construction we outline a proof of a theorem of Gompf: Theorem 7.1.10 (Gompf, [64]). Suppose that for i = 1, 2 the closed sym- plectic 4-manifolds (X i , ω i ) contain closed symplectic 2-dimensional sub- manifolds Σ i ⊂ X i satisfying g(Σ 1 ) = g(Σ 2 ) and [Σ 1 ] 2 + [Σ 2 ] 2 = 0. Fix an identiﬁcation f : Σ 1 → Σ 2 and consider an orientation reversing lift F : ∂νΣ 1 → ∂νΣ 2 of f. Then the normal connected sum X 1 # F X 2 = (X 1 −νΣ 1 ) ∪ F (X 2 −νΣ 2 ) admits a symplectic structure. Proof (sketch). We assume ﬁrst that [Σ 1 ] 2 < 0. In that case Σ 1 admits an ω 1 -convex, and Σ 2 (with [Σ 2 ] 2 > 0) an ω 2 -concave neighborhood — as their local models show. Then all we need to do is to show that the contact structures on the boundaries are contactomorphic. Let α 1 be the contact form on ∂νΣ 1 and α 2 the pull-back of the contact form of ∂νΣ 2 by F. Let α t = tα 1 +(1 −t)α 2 _ t ∈ [0, 1] _ be a path connecting them. Since dα 1 and dα 2 are both positive multiples of π ∗ ω 1 [ Σ 1 for π: ∂νΣ 1 → Σ 1 , we conclude that the α t are all contact forms on ∂νΣ 1 : notice that dα t = t dα 1 + (1 − t) dα 2 is also a positive multiple of π ∗ ω[ Σ 1 , and ker α t is always transverse to the ﬁbers of ∂νΣ 1 → Σ 1 , hence dα t is nondegenerate on ker α t . This shows that α 1 and α 2 are isotopic, therefore Gray’s Theorem 4.1.16 shows that they are contactomorphic, hence the previous construction proves the theorem. If [Σ 1 ] 2 = [Σ 2 ] 2 = 0 then just use a function which ﬁberwise turns the punctured unit disk in R 2 symplectically inside out. Remark 7.1.11. The original proof of Gompf for the above theorem rests on the symplectic neighborhood theorem, for details see [64]. Notice that this construction — applied for CP 1 ⊂ CP 2 as it is described in Lemma 3.3.3 — veriﬁes the existence of a minimal model of a symplectic 4-manifold. In general it is quite a delicate question whether ∂X of a symplectic mani- fold X is of contact type or not (i.e., whether an appropriate vector ﬁeld exists). In addition, to apply the above gluing procedure we have to relate 7.2. Weinstein handles 115 induced contact structures on hypersurfaces of contact type. In some spe- cial cases the vector ﬁeld comes with the construction (for example, for a Stein manifold), and contactomorphism can be proved by relying on some form of classiﬁcation results of contact structures on ∂X. 7.2. Weinstein handles In the following we work out a special case (ﬁrst described by Weinstein) of the above gluing procedure — when we glue a 4-dimensional 2-handle to a symplectic 4-manifold (X, ω) with ω-convex boundary. For a more general discussion of gluing handles see [173]. Let us take the standard 2-handle H as the closure of the component of R 4 − __ x 2 1 +x 2 2 − 1 2 (y 2 1 +y 2 2 ) = −1 _ ∪ _ x 2 1 +x 2 2 − ε 6 (y 2 1 +y 2 2 ) = ε 2 _ _ which contains the origin, see the shaded region in Figure 7.1, cf. [38]. It inherits the symplectic structure ω 0 = dx 1 ∧dy 1 +dx 2 ∧dy 2 from the standard structure on R 4 . Consider the vector ﬁeld v = 2x 1 ∂ ∂x 1 −y 1 ∂ ∂y 1 + 2x 2 ∂ ∂x 2 −y 2 ∂ ∂y 2 . Exercises 7.2.1. (a) Show that v is equal to ∇f for the function f = x 2 1 +x 2 2 − 1 2 y 2 1 − 1 2 y 2 2 with respect to the standard Euclidean metric. (b) Check that α = ι v (ω 0 ) = 2x 1 dy 1 +y 1 dx 1 + 2x 2 dy 2 +y 2 dx 2 . (c) Show that L v ω 0 = ω 0 and that v is transverse to the boundary of the standard 2-handle. (Hint: Use the fact that L v ω 0 = dι v ω 0 . For transversality compute df(v) and show that it is equal to 4x 2 1 +4x 2 2 +y 2 1 +y 2 2 .) (d) Show that the attaching circle S = ¦x 1 = x 2 = 0, y 2 1 +y 2 2 = 2¦ ⊂ ∂H is Legendrian with respect to the contact structure ξ = ker α generated by v on the boundary of the 2-handle. Similarly, the belt circle B = ¦y 1 = y 2 = 0, x 2 1 + x 2 2 = ε 2 ¦ ⊂ ∂H is Legendrian. Notice that in this part of ∂H the vector ﬁeld points out of H. The orientation of ∂H near S given by v is opposite to the orientation ∂H inherits from H, while the two orientations coincide near B. Now applying the previous construction we get 116 7. Symplectic surgery , y y 1 2 , x x 1 2 B B S S H Figure 7.1. The standard 4-dimensional 2-handle H Theorem 7.2.2. Suppose that (X, ω) is a symplectic 4-manifold with ω- convex boundary ∂X and L ⊂ ∂X is a Legendrian curve (with respect to the induced contact structure). Then a 2-handle H can be attached to X along L in such a way that ω extends to X ∪ H as ω ′ and ∂(X ∪ H) is ω ′ -convex. Proof. According to the Legendrian neighborhood theorem, L ⊂ ∂X and S ⊂ ∂H admit contactomorphic neighborhoods. Choose ε in the deﬁnition of H so small that the attaching region of H becomes a subset of this neighborhood. The contactomorphism between the neighborhoods of L ⊂ ∂X and S ⊂ ∂H will provide a suitable gluing map. Notice that since this map is dictated by Theorem 4.1.15, the framing of the handle attachment is also given. The last statement follows from patching the vector ﬁelds together. Remark 7.2.3. The same gluing scheme has been developed for any 2n- dimensional index k-handle (with k ≤ n) in [173]. For example, the 4- manifold # m S 1 D 3 admits a symplectic structure with ω-convex boundary 7.2. Weinstein handles 117 — just repeat the handle attachment for m 1-handles starting with the standard 4-disk _ D 4 , ω st [ D 4 _ and vector ﬁeld v = x ∂ ∂x +y ∂ ∂y +z ∂ ∂z +t ∂ ∂t . In order to have a complete picture about the topology of X ∪ H we need to identify the framing of the 2-handle H we have to use in the above construction. Recall that L admits a canonical framing (as a Legendrian knot), hence we need to understand the framing of the gluing relative to this canonical one. We think of a framing as a vector ﬁeld in the tangent space along the knot transverse to its tangent vector ﬁeld. Therefore we need to identify two vector ﬁelds along L: v 1 is the vector ﬁeld in ξ which is transverse to the tangent of L (providing the contact framing), while v 2 is the image of the direction we push-oﬀ the attaching circle when measuring framings. Notice that for this computation we can work in the standard handle H: the vector ﬁeld v 2 is deﬁned as an image of a vector ﬁeld in H, while v 1 is the image of the corresponding vector ﬁeld along the Legendrian knot S ⊂ H, since the gluing map is a contactomorphism. For this computation, ﬁx a parametrization of the attaching circle S = ¦x 1 = x 2 = 0, y 2 1 + y 2 2 = 2¦ as (0, 0, √ 2 cos t, √ 2 sin t). Then the unit tangent vectors along S are given by s ′ (t) = (0, 0, −sin t, cos t). Restricting the contact form α to the tangent to S we get √ 2 cos t dx 1 + √ 2 sint dx 2 , therefore the contact framing (i.e., the vector ﬁeld along S(t) which is orthogonal to s ′ (t) and is in the kernel of α) can be represented by the vector ﬁeld V (t) = (sin t, −cos t, 0, 0). Now the framing of the gluing is measured by pushing oﬀ S in the (x 1 , x 2 )-direction in the handle. The corresponding vector ﬁeld can be chosen, for example as (1, 0, 0, 0). Since the two unit length vector ﬁelds intersect each other once, the diﬀerence of the two framings can be clearly represented by a meridian of the knot. Taking the orientations into account, we see that the contact framing makes one positive full turn around the origin, therefore the framing we get by pushing the knot slightly in the (x 1 , x 2 )-direction is (−1) when compared to the contact framing. In conclusion Theorem 7.2.4 (Weinstein). Suppose that (X, ω), ∂X and L are as in Proposition 7.2.2. If we attach a (4-dimensional) 2-handle H with framing −1 with respect to its canonical contact framing to ∂X along L then ω extends to X ∪ H as in Proposition 7.2.2. Corollary 7.2.5. Let L ⊂ (S 3 , ξ st ) be a given Legendrian link. Then L equips the 4-manifold X deﬁned by handle attachment along the smooth link underlying L (with framings tb(L i ) −1) with a symplectic structure ω such that ∂X is ω-convex. 118 7. Symplectic surgery Proof. Apply the above theorem for D 4 with the symplectic structure it inherits from (R 4 , ω 0 ) and for the outward pointing radial vector ﬁeld v = x ∂ ∂x +y ∂ ∂y +z ∂ ∂z +t ∂ ∂t . Exercises 7.2.6. (a) Let the 3-manifold Y be given as 0-surgery on the right-handed trefoil knot. Present Y as the ω-convex boundary of a sym- plectic 4-manifold. (Hint: Take the Legendrian knot of Figure 1.4 and compare framings.) (b) Find a symplectic 4-manifold with ω-convex boundary diﬀeomorphic to the 3-manifold given by the surgery diagram of Figure 7.2. (Hint: Convert the 0-framed circle into a 1-handle, see Figure 7.3.) 0 4 Figure 7.2. Stein ﬁllable 3-manifold 0 4 4 4 Figure 7.3. Convert appropriate 2-handle into 1-handle In the construction above we always assumed that X has ω-convex boun- dary, that is, a symplectic dilating vector ﬁeld transverse to the boundary exists. In the gluing construction discussed above, however, we only need the existence of this vectorﬁeld near the Legendrian knot L ⊂ ∂X. As it truns out, the necessary vector ﬁeld exists near the given knot under much weaker assumptions: it exists if (X, ω) is only a weak ﬁlling of the contact 3-manifold (Y = ∂X, ξ), see Section 12.1. 7.3. Another handle attachment 119 7.3. Another handle attachment A similar scheme would work by gluing the handle to X along the Leg- endrian knot B of Figure 7.1, i.e., along the belt circle of H. This time, however, the symplectic structures of X and H do not match up, since the vector ﬁeld on the handle points in the wrong direction. Therefore the re- sulting 4-manifold carries no natural symplectic structure. Notice also that the framing coeﬃcient of this latter operation is +1 with respect to the contact framing of the knot in ∂X. This operation will have interesting in- terpretation in the realm of contact surgery, see Chapter 11. Viewing this latter construction from another point of view, we see that a symplectic 2-handle H can be glued along B (with framing (+1) relative to the con- tact framing) to a symplectic 4-manifold X along a Legendrian knot lying in an ω-concave part of ∂X. In this case the symplectic structure will ex- tend to the handle attachment. Return now to the picture when gluing the handle along ω-convex boundary with framing (+1) (relative to the contact framing). Lemma 7.3.1. Take a Legendrian curve L ⊂ (∂X, ξ) and push it oﬀ along its contact framing to get another Legendrian knot L ′ . If we attach a 2- handle along L with framing (−1) (with respect to the contact framing) and another 2-handle along L ′ with framing (+1) (again, measured with respect to the contact framing) then the resulting 4-manifold will have boundary diﬀeomorphic to ∂X. Proof. This fact is quite obvious from the smooth point of view, since by sliding L ′ over L we will get a 0-framed circle which is just the boundary of a small normal disk to L. Surgering out the corresponding sphere of self- intersection 0, we end up with L passing through a 1-handle, i.e., we get a cancelling pair of handles. This means that we can erase them without changing the 4-manifold. Since the surgery along the sphere does not change the boundary, we conclude that the boundary after the handle attachments is diﬀeomorphic to ∂X. As we will see, the new 3-manifold we get after the handle attachment with framing (+1) carries a natural contact structure. In Section 11.2 we will sharpen the above lemma to prove contactomorphism for the resulting structure after a (−1)- and a (+1)-surgery on L and its contact push oﬀ L ′ . In another context we will see that although the symplectic structure does not extend through the handle when glued along the belt circle B, an 120 7. Symplectic surgery appropriate almost-complex structure does extend to H −¦pt.¦, providing, for example, a spin c structure on the manifold X ∪ H. We will discuss this aspect of the gluing theorem in the next section. 8. Stein manifolds In this chapter we interpret the Weinstein handle attachment in the Stein category, leading us to Eliashberg’s celebrated theorem. To put this result in the right perspective, we ﬁrst recall rudiments of Stein manifold theory. The chapter concludes with a discussion about surfaces in Stein manifolds. For a more detailed treatment of this topic the reader is advised to turn to [70]. 8.1. Recollections and definitions Let X be a complex manifold. The holomorphic convex hull of K ⊂ X is ´ K = _ x ∈ X ¸ ¸ ¸ ¸ ¸ f(x) ¸ ¸ ≤ sup y∈K ¸ ¸ f(y) ¸ ¸ for all f holomorphic on X _ . The manifold X is holomorphically convex if for all K ⊂ X compact the holomorphic convex hull ´ K is compact. This property is equivalent to the requirement that for any inﬁnite discrete set D ⊂ X there is a holomorphic function f : X → C which is unbounded on D. Yet another equivalent property for a domain Ω ⊂ C n is the existence of a holomorphic function f : Ω →C which cannot be extended holomorphically to any larger domain. The traditional deﬁnition of Stein manifolds requires holomorph convexity (which, as we remarked above, resembles to being a domain of holomorphy) and the existence of many holomorphic functions. More precisely: Deﬁnition 8.1.1. A complex manifold X is a Stein manifold if it is holo- morphically convex, for each x ,= y ∈ X there is a holomorphic function f : X → C such that f(x) ,= f(y) and for every x ∈ X there are holomor- phic functions f 1 , . . . , f n and a neighborhood U of x such that z i = f i [ U (i = 1, . . . , n) give local coordinates on U. 122 8. Stein manifolds Exercise 8.1.2. Show that if X is Stein then it is noncompact. For example, C n and every closed analytic submanifold of C n is Stein. The next theorem asserts that the converse also holds: Theorem 8.1.3 (Narasimhan, Bishop, Remmert, Eliashberg–Gromov). Let q be an integer greater than or equal to 3n 2 +1. Then an n-dimensional Stein manifold biholomorphically and properly embeds into C q . Therefore the following deﬁnition (in the dimension of our interest) is equiv- alent to the one given above: Deﬁnition 8.1.4. The 2-dimensional complex manifold S is a Stein surface if it admits a proper biholomorphic embedding S ֒→ C 4 . That is, S is a smooth aﬃne 2-dimensional complex analytic submanifold in C 4 . Another, technically more involved equivalent way of deﬁning Stein man- ifolds is to require the vanishing of the sheaf cohomology groups H q (X, o) for q ≥ 1 and any coherent sheaf o (see [70], for example). For our purposes yet another, more topological deﬁnition will be the most suitable. First a few related deﬁnitions are in place: Deﬁnition 8.1.5. A smooth function ϕ: X → R on a complex manifold X is (strictly) plurisubharmonic if ϕ is (strictly) subharmonic on every holomorphic curve C ⊂ X. Recall that ϕ is subharmonic if for r small enough ϕ(x 0 ) ≤ 1 2πr _ B(x 0 ,r) ϕ(x) dx; or alternatively ∆ϕ ≥ 0 for the Laplace operator ∆. A function ϕ: X → R is an exhausting function if _ x ∈ X [ ϕ(x) < c _ is relatively compact in X for all c ∈ R. Recall that a map ϕ: X → Y is proper if the inverse image of a compact set is compact. (Hence a proper function ϕ: X →[0, ∞) is exhausting.) For example, the function z → [z[ 2 = n i=1 z i z i is a plurisubharmonic exhausting function on C n . Moreover, if f is holomorphic (and not identi- cally 0 on any component) then log [f[ is plurisubharmonic; e.g. log [z[ 2 is plurisubharmonic. Let now Y ⊂ X be a codimension-1 submanifold of X. The complex tangencies along Y form a complex hyperplane distribution in TY , which can be (at least locally) given as ker α for some 1-form α. The Levi form L Y (x, y) is deﬁned as L Y (x, y) = dα(x, Jy) (where J is multiplication by √ −1). Taking the orientation of Y into account, L Y is deﬁned up to multiplication by a positive function. If Y is given as ϕ −1 (a) for some 8.1. Recollections and deﬁnitions 123 smooth function ϕ: X → R (which can be assumed, at least, locally) then its Levi form can be given as L Y (x, y) = n i,j=1 ∂ 2 ϕ ∂z i ∂z j x i y j . Deﬁnition 8.1.6. The hypersurface Y ⊂ X is strictly pseudoconvex (or J-convex) if its Levi form is positive deﬁnite. In this case the complex hyperplane distribution can be proved to give a contact structure, since the deﬁnition requires dα to be nondegenerate on ker α. Remark 8.1.7. It can be proved that if Y is strictly pseudoconvex then it cannot be touched by a holomorphic curve from inside. More precisely, if Y = ϕ −1 (0) is pseudoconvex and Y is oriented as the boundary of X = ϕ −1 _ (−∞, 0] _ then any holomorphic curve C ⊂ X with boundary is transverse to Y , in particular int C ∩ Y = ∅. This property explains “convexity” in the deﬁnition. A function ϕ: X → R turns out to be strictly plurisubharmonic if the associated Levi form ( ∂ 2 ϕ ∂z i ∂z j ) is positive deﬁnite. More precisely, if ϕ −1 (a) is J-convex for all a (oriented as the boundary of the sublevel set ¦ϕ ≤ a¦) then there is a diﬀeomorphism h: R →R such that the function ˜ ϕ = h◦ϕ is plurisubharmonic. A more invariant reformulation of the above fact can be given as follows. Suppose that a smooth function ϕ: X →R is given on the complex manifold X. Consider the associated 2-form ω ϕ = −dJ ∗ dϕ. (Here J ∗ : T ∗ X → T ∗ X is the dual of J. The operator J ∗ d is frequently denoted by d C , hence ω ϕ = −dd C ϕ.) This 2-form gives rise to a symmetric tensor g ϕ (x, y) = ω ϕ (x, Jy). Proposition 8.1.8. The smooth function ϕ: X → R on the complex manifold X is strictly plurisubharmonic if and only if g ϕ is a Riemannian metric. In particular, this property implies that the exact 2-form ω ϕ is nondegenerate, hence is an exact symplectic form, while g ϕ is a K¨ ahler metric on X. Note that if X is complex 2- (hence real 4-) dimensional then the deﬁnite- ness of L Y on the (real) 2-dimensional distribution on a 3-manifold Y ⊂ X is in fact equivalent to requiring that the distribution is a contact structure. Therefore in this dimension ϕ: X → R is strictly plurisubharmonic if and only if the complex tangencies provide a contact structure on the smooth points of the level sets ϕ −1 (a). The gradient vector ﬁeld ∇ϕ (with respect to the Riemannian metric g ϕ ) is a symplectic dilation since L ∇ϕ ω ϕ = d(ι ∇ϕ ω ϕ ) 124 8. Stein manifolds and (ι ∇ϕ ω ϕ )(v) = ω ϕ (∇ϕ, v) = g ϕ (∇ϕ, Jv) = −dϕ(Jv) = −J ∗ dϕ(v), hence d(ι ∇ϕ ω ϕ ) = −dJ ∗ dϕ = ω ϕ . Recall that on C n there are many plurisubhar- monic functions, and so Stein manifolds (being complex submanifolds of C n for some n) admit many plurisubharmonic functions as well. In fact, the converse of this statement also holds: Theorem 8.1.9 (Grauert, [69]). If a complex manifold X admits a proper plurisubharmonic function ϕ: X →[0, ∞) then X is Stein. For X n ⊂ C m the distance function f(z) = [z −p[ 2 from a generic point p ∈ C m deﬁnes a proper Morse function f : X n →[0, ∞) with critical points of index at most n [114]. In conclusion Theorem 8.1.10 (Grauert, [69]). The complex surface S is Stein if and only if it admits a proper Morse function f : S →[0, ∞) such that away from the critical points f −1 (t) is a contact 3-manifold (with complex tangent lines as ξ) for all t. Remarks 8.1.11. (a) It is not very hard to see that a plurisubharmonic function satisﬁes the maximum principle, i.e., on a connected compact com- plex space it is constant. In other words, if C is a holomorphic curve with boundary then for a plurisubharmonic function ϕ the restriction ϕ[ C has no local maximum in the interior int C. This property is closely related to the alternative reformulation of pseudoconvexity described in Remark 8.1.7. (b) According to a result of Eliashberg and Gromov, two plurisubharmonic functions ϕ and ψ with complete gradient ﬂows deﬁne symplectomorphic symplectic structures (X, ω ϕ ) and (X, ω ψ ) on a Stein manifold X. A compact manifold W with boundary will be called a Stein domain if there is a Stein manifold X with plurisubharmonic function ϕ: X →[0, ∞) such that W = ϕ −1 _ [0, a] _ for some regular value a. So a compact manifold with boundary (and complex structure on its interior) is a Stein domain if it admits a proper plurisubharmonic function which is constant on the boundary. More generally, a cobordism W (with boundary −Y 1 ∪ Y 2 ) is a Stein cobordism if W is a complex cobordism with a plurisubharmonic function f : W →R such that f −1 (t i ) = Y i , t 1 < t 2 . 8.2. Handle attachment to Stein manifolds 125 8.2. Handle attachment to Stein manifolds In this section we show a way to adapt the handle attachment scheme given in the previous chapter to the case of Stein surfaces. Recall that in this setting we assume that ∂W is the level set of a plurisubharmonic function and we consider the contact structure on ∂W provided by the distribution of complex tangencies. The main theorem (due to Eliashberg) is the following Theorem 8.2.1 (Eliashberg, [25]). Suppose that W is a (complex) 2- dimensional Stein domain and L ⊂ ∂W is a Legendrian knot. By attaching a Weinstein handle H to W along L, the Stein structure can be extended to W ∪ H. Proof (sketch). The idea of the proof is the following: ﬁrst we approximate the Legendrian knot L with a C ∞ -close real algebraic Legendrian knot. In this way the attaching map can be chosen to be complex analytic, providing us a complex structure on W ∪ H. Notice that by the framing assumption complex lines will match up under the gluing. Therefore the proof reduces to extending the plurisubharmonic function ϕ which already exists on W. Now suppose that we glue the 2-handle to the Stein domain ϕ −1 _ [0, c +ε] _ , hence the plurisubharmonic function ϕ is already deﬁned on some parts (containing the attaching circle) of the 2-handle. The extension of ϕ to the 2-handle now proceeds by turning it into a standard model and then extending. For details see [25]; for an explicit description of the shape of the 2-handle see [50]. Corollary 8.2.2. A Legendrian link L ⊂ (S 3 , ξ st ) determines a Stein mani- fold X L . Topologically X L is given by 2-handle attachments along the link L with framings tb(L i ) −1 on the individual components. Remarks 8.2.3. (a) Similar (simpler) result holds for attachment of 1- handles: after attaching a 1-handle to a Stein domain the Stein structure always extends. (b) The product Y I of a contact 3-manifold (Y, ξ) and I = [0, 1] can be equipped with a Stein structure. In addition, if (Y, ξ) is overtwisted then the framing condition yields no restriction in the gluing. Therefore a cobordism (involving only 2-handles) on an overtwisted 3-manifold always admits Stein structure. (For the deﬁnition of overtwisted structures see Section 4.) (c) The result generalizes to arbitrary dimension n > 2, with the simpliﬁ- cation of dropping the framing condition. 126 8. Stein manifolds The theory of Stein manifolds from the point of view of handle calculus was carefully developed by Gompf in [65], see also Chapter 11 of [66]. Here we highlight only one result of [65], which will be important in our later considerations: the identiﬁcation of the ﬁrst Chern class of a Stein structure given by handle attachments. Suppose that the Stein surface (X, J) is given by attaching Weinstein 2-handles to (D 4 , J st ) along the Legendrian link L = (L 1 , . . . , L t ) ⊂ (S 3 , ξ st ), and suppose that Σ L i ⊂ X denotes the surface corresponding to the knot L i (see Section 2.3). The value of c 1 (X, J) of the resulting complex structure on the homology deﬁned by the knot is given by the following Proposition 8.2.4 (Gompf, [65]). The ﬁrst Chern class of the resulting complex structure is given by ¸ c 1 (X, J), [Σ L i ] _ = rot(L i ). Proof. By deﬁnition, the value of the ﬁrst Chern class c 1 (X, J) is equal to the obstruction of extending a complex trivialization of TD 4 to the complex 4-manifold X we get after the handle attachment. To determine this obstruction, we ﬁx trivializations on D 4 and on the handle and compute the obstruction for splicing them together. Along the boundary S 3 we can take the vector ﬁeld ∂ ∂x (spanning the standard contact structure ξ st , regarded as ker(dz +xdy) on the ﬁnite part of S 3 ), and an inward pointing normal v — these two vector ﬁelds span TD 4 over C —, and extend them over the 4-disk. In the handle consider the tangent vector ﬁeld τ and the outward pointing normal w along the attaching circle. These two vector ﬁelds extend to a complex trivialization of the tangent bundle of the handle where the spanning disk of the attaching circle is viewed as part of iR 2 ⊂ C 2 . Now under the handle attachment we map w to v and τ into ξ. The obstruction for extending the trivialization given on D 4 is now simply the rotation number of τ with respect to the chosen vector ﬁeld ∂ ∂x , which is by deﬁnition the rotation number of the Legendrian knot K ⊂ (S 3 , ξ st ). Remark 8.2.5. Similar statement holds when we glue Stein 1-handles ﬁrst to D 4 and then add the 2-handles; for details see [65]. Exercises 8.2.6. (a) Equip the handlebodies given by Figure 2.14 with Stein structures. (b) Equip RP 3 with a contact structure. (c) Find contact structures on lens spaces. 8.3. Stein neighborhoods of surfaces 127 8.3. Stein neighborhoods of surfaces In this section we describe a way to ﬁnd Stein neighborhoods of certain embedded surfaces in complex 4-manifolds. Let us ﬁx a complex manifold (X 4 , J) and an embedded oriented surface Σ 2 ⊂ X 4 . As always, J : TX → TX denotes the almost-complex structure induced by the complex structure of X; all dimensions are understood to be real. The theory presented below resembles to the discussion about Bennequin’s inequality given in Section 4.3. Deﬁnition 8.3.1. A point p ∈ Σ is a complex point if T p Σ = JT p Σ, i.e., the tangent space of Σ at p is a complex line. The noncomplex points of the embedding are called real points. For a generic embedding complex points are isolated. Saying the above condition in another way, p ∈ Σ is an isolated complex point if there are complex coordinates (z 1 , z 2 ) in X such that Σ locally can be given by _ z 2 = f(z 1 ) _ with p = (0, 0) and ∂f ∂z (0, 0) = 0. This can be seen by noting that the vectors X(z 1 ) = (1, ∂f ∂x (z 1 )) and Y (z 1 ) = (1, ∂f ∂y (z 1 )) (with z 1 = x + iy) form a real basis of T (z 1 ,f(z 1 )) Σ and a point is complex if and only if these vectors are complex scalar multiples of each other, i.e., the determinant _ 1 ∂f ∂x (z 1 ) 1 ∂f ∂y (z 1 ) _ = −2i ∂f ∂z (z 1 ) = 0. Yet another way to see the picture is the following: If Gr 2 (4) denotes the Grassmannian of oriented 2-planes in R 4 then consider the associated bundle Gr 2 (X) = P X Gl 2 (R) Gr 2 (4) for the principal frame bundle P X → X. By taking the tangent planes of Σ we get a lift of the embedding Σ ֒→ X to F : Σ → Gr 2 (X). The complex tangent lines deﬁne a subset CGr(X) ⊂ Gr 2 (X) and for the projection π: Gr 2 → X we have that π _ CGr(X)∩F(Σ) _ ⊂ Σ is precisely the set of complex tangencies. The fact that complex tangencies are isolated for a generic embedding now follows from a general transversality result of Thom. Deﬁnition 8.3.2. The index I p ∈ Z of an isolated complex point p ∈ Σ is deﬁned as the winding number of ∂f ∂z around a small circle around p. 128 8. Stein manifolds To have a well-deﬁned notion, one has to check that this quantity is inde- pendent of f, see [49]. In fact, by choosing appropriate coordinates f can be written either as z 2 = z k 1 z 1 with k ≥ 0 or as z 2 = z −k+1 with k < 0, and so I p = k. The index of the embedding Σ ⊂ X is deﬁned as I(Σ) = p∈Σ I p . Deﬁnition 8.3.3. The complex point p ∈ Σ is elliptic if I p = 1 and hyperbolic if I p = −1. In the Grassmannian picture the index I p can be interpreted as an intersection multiplicity: if π(P) = p for P ∈ CGr(X) ∩ F(Σ) ⊂ Gr 2 (X) then I p is the multiplicity of the intersection of CGr(X) with F(Σ) at P. For a generic embedding Σ ⊂ X all complex points are isolated and either elliptic or hyperbolic. Now the orientation of Σ makes us able to assign a sign to each complex point p: this sign is positive if the complex orientation of T p Σ coincides with the orientation of it inherited from Σ and it is negative otherwise. Said another way, CGr(X) falls into the disjoint union CGr (X) + ∪CGr (X) − according to whether the orientation of the 2- plane is the complex one or its opposite. The complex point p ∈ Σ is positive if p = π(P) for P ∈ CGr (X) + ∩F(Σ) and negative if P ∈ CGr (X) − ∩F(Σ). We deﬁne I + (Σ) (I − (Σ)) as the sum of I p for all positive (resp. negative) complex points of Σ ⊂ X. Obviously I + (Σ) + I − (Σ) = I(Σ). It turns out that I ± (Σ) are topological invariants, more precisely Theorem 8.3.4 (Lai, [90]). With the above conventions I(Σ) = I + (Σ) + I − (Σ) = χ(Σ) + [Σ] 2 and I + (Σ) − I − (Σ) = ¸ c 1 (X), [Σ] _ . In conclusion I ± (Σ) = 1 2 (χ(Σ) + [Σ] 2 ± ¸ c 1 (X), [Σ] _ ). Proof (sketch). By choosing a metric on X ﬁx a projection of TX[ Σ to the normal bundle νΣ. For proving the ﬁrst identity, choose a vector ﬁeld v in the tangent bundle TΣ. Applying J to it and projecting the result to νΣ we get a section σ of the normal bundle νΣ. The projection provides a zero of σ at p ∈ Σ if and only if Jv p is in the tangent plane T p Σ, which happens if and only if either p is a complex point or v p = 0. Checking the signs of the zeros the ﬁrst formula follows. For the second formula consider the section s = ω(v 1 , v 2 ) −1 v 1 ∧ C v 2 of the complex line bundle Λ 2 C (TX) ¸ ¸ Σ , where ω is a symplectic form on Σ and ¦v 1 , v 2 ¦ is a local frame for TΣ. Now the zeros of s correspond to those points of Σ where v 1 and v 2 are not independent over C, i.e., in the complex points of Σ. A careful checking of the signs expresses the (signed) sum of zeros of s as I + (Σ) − I − (Σ). On the other hand the sum of zeros of s is equal to c 1 (Λ C (TX) _ [Σ] _ = ¸ c 1 (X), [Σ] _ . The expressions for I ± (Σ) now 8.3. Stein neighborhoods of surfaces 129 follow by adding and subtracting the above formulae. Suppose now that either [Σ] ,= 0 or Σ is not a sphere. Theorem 8.3.5. If there is an open subset U ⊂ X such that Σ ⊂ U and U admits a Stein structure then I ± (Σ) ≤ 0. Proof. By the adjunction inequality of Theorem 1.2.1 we have [Σ] 2 +[ ¸ c 1 (U), [Σ] _ [ ≤ −χ(Σ) once U is Stein (and Σ is not a null-homologous sphere). Since c 1 (U) = c 1 (X), the formulae for I ± (Σ) imply the result. According to a beautiful result of Forstneri´c, the converse of the above theorem also holds: Theorem 8.3.6 (Forstneri´c, [49]). If I ± (Σ) ≤ 0 for a generic embedding Σ ⊂ X then there is a Stein domain U ⊂ X containing an isotopic copy of Σ. Proof (sketch). The proof of this theorem involves two major steps. First we use a cancellation theorem due to Eliashberg and Kharlamov: if p, q ∈ Σ are complex points with equal sign and I p +I q = 0 then Σ can be isotoped to cancel p and q without changing the other complex points or introducing new complex tangencies. Therefore the assumption guarantees that we can isotope Σ to have only hyperbolic complex point. Then a local construc- tion near real and hyperbolic points together with a patching argument provides a neighborhood U of Σ with a plurisubharmonic function show- ing its Stein property: In a complex chart around a complex hyperbolic point p j take the nonnegative function ρ j (z 1 , z 2 ) = [z 2 −z 2 1 [. Now deﬁne ρ 0 on a tubular neighborhood of Σ as h(v) where we implicitly identiﬁed the tubular neighborhood with the normal bundle of Σ and h is the quadratic form of a Riemannian metric on the normal bundle νΣ. For the complex points ¦p 1 , . . . , p m ¦ choose χ j smooth cut-oﬀ functions (j = 1, . . . , m) sup- ported by the (disjoint) complex coordinate neighborhoods which are con- stant 1 near p j . The function ρ = m j=1 χ j ρ j + (1 −χ j )ρ 0 can be shown to be plurisubharmonic, proving the fact that Σ admits a neighborhood with Stein structure. In fact, this argument provides a Stein neighborhood basis for Σ, that is, Stein neighborhoods ¦U α ¦ α∈(0,1) with U β = ∪ α<β U α and U β = ∩ α>β U α and all U α retract to Σ. It is known that the presence of an elliptic point on 130 8. Stein manifolds Σ obstructs the existence of such a basis: the existing holomorphic Bishop disks [27] around the elliptic point (which cannot ﬁt into all U α ) would provide holomorphic extensions of functions. Notice the similarity between these ideas and the ones involved in the proof of Bennequin’s inequality (Section 4.3). 9. Open books and contact structures Recently Giroux [63] proved a central result about the topology of contact 3-manifolds. He showed that there is a one-to-one correspondence between contact structures (up to isotopy) and open book decompositions (up to positive stabilization/destabilization) on a closed oriented 3-manifold. This chapter is devoted to the introduction of relevant notions and also some parts of the proof of this beautiful correspondence. 9.1. Open book decompositions of 3-manifolds Deﬁnition 9.1.1. Suppose that for a link L in a 3-manifold Y the com- plement Y − L ﬁbers as π: Y − L → S 1 such that the ﬁbers are interiors of Seifert surfaces of L. Then (L, π) is an open book decomposition of Y . Traditionally the Seifert surface F = π −1 (t) is called a page, while L the binding of the open book decomposition. The monodromy of the ﬁbration π is called the monodromy of the open book decomposition. Any locally trivial bundle with ﬁber F over an oriented circle is canonically isomorphic to the ﬁbration I F/(1, x) ∼ _ 0, h(x) _ → I/∂I ≈ S 1 for some self-diﬀeomorphism h of F. In fact, the map h is determined by the ﬁbration up to isotopy and conjugation by an orientation preserving self-diﬀeomorphism of F. The isotopy class represented by the map h is called the monodromy of the ﬁbration. Conversely given a compact oriented surface F with nonempty boundary and h ∈ Γ F (the mapping class group of F) we can form the mapping torus F(h) = I F/(1, x) ∼ _ 0, h(x) _ . Since h is the identity on ∂F, the boundary ∂F(h) of the mapping torus F(h) can be canonically identiﬁed with r copies of T 2 = S 1 S 1 , where the ﬁrst S 1 factor is identiﬁed with I/∂I and the second one comes from a component 132 9. Open books and contact structures of ∂F. Hence by gluing in r copies of D 2 S 1 to F(h) so that ∂D 2 is identiﬁed with S 1 = I/∂I and the S 1 factor in D 2 S 1 is identiﬁed with a boundary component of ∂F, F(h) can be completed to a closed 3-manifold Y equipped with an open book decomposition. In conclusion, an element h ∈ Γ F determines a 3-manifold together with an “abstract” open book decomposition on it. Notice that by conjugating the monodromy h of an open book on a 3-manifold Y by an element in Γ F we get an equivalent open book on a 3-manifold Y ′ which is diﬀeomorphic to Y . In Example 9.1.4(b) we illustrate a method to convert an abstract open book to an open book concretely embedded into an ambient 3-manifold. See also Section 15.2. Remark 9.1.2. We deﬁne a ﬁbered link as a link L ⊂ Y whose complement Y − L ﬁbers over S 1 , in such a way that each ﬁber intersects a tubular neighborhood of L in a curve isotopic to L. It is clear that the binding of an open book decomposition is a ﬁbered link and conversely a ﬁbered link naturally induces an open book decomposition with our deﬁnition. Theorem 9.1.3 (Alexander, [9]). Every closed and oriented 3-manifold admits an open book decomposition. Proof (sketch). There are several diﬀerent proofs of this classical theorem of Alexander. We ﬁrst outline a proof using branched covers. Every closed and oriented 3-manifold Y is a 3-fold branched cover Y → S 3 . The proof of this fact rests on the following. Choose a Heegaard decomposition of Y as U 1 ∪ f U 2 (here U 1 and U 2 are solid genus-g handlebodies and f ∈ Γ g is a mapping class) and represent U 1 , U 2 as triple branched covers of B 3 . Gluing the two B 3 together we can realize any mapping class for the gluing of the two handlebodies [75, 117], hence the result follows. We can assume that the branch locus of this cover is transverse to the pages of an open book decomposition of S 3 . Notice that S 3 admits several open book decomposi- tions (see examples below). Thus we get an open book decomposition of Y by lifting an open book decomposition of S 3 to the cover. Next we describe a proof given in [148]. It is well-known that every closed and oriented 3-manifold is obtained from S 3 by a (±1)-surgery along a link L. Moreover we can assume that there is an unknot K ⊂ S 3 such that each component L i of L links K exactly once. Consider the trivial open book of S 3 whose binding is K and whose pages are the spanning disks for K with trivial monodromy (cf. Example 9.1.4(a)). It is clear that when we remove a neighborhood V i of L i to perform surgery along L i we puncture once every page of this open book on S 3 . Observe that the boundary of 9.1. Open book decompositions of 3-manifolds 133 a puncture (which is a meridional curve to L i in S 3 ) becomes longitudinal by a (±1)-surgery along L i . By performing surgery along L i we glue in an annulus (bounded by L i and this longitudinal curve on ∂V i ) to a puncture on each page of the trivial open book of S 3 to obtain a page of an open book on Y . Notice that a page of this open book on Y is planar, i.e., it is a disk with holes since to each puncture we glue in an annulus. The binding is given by K ∪ L and the monodromy is given by an (appropriate) Dehn twist along a curve parallel to each boundary component L i and identity near K. (Yet another proof of this theorem using Lefschetz ﬁbrations was given by Harer, see Section 10.2.) Examples 9.1.4. (a) Taking L to be the z-axis and considering the half-planes with boundary L we get an open book decomposition on R 3 . Alternatively, take π: R 3 − _ (0, 0, z) _ →S 1 ⊂ R 2 given by π(x, y, z) = 1 _ x 2 +y 2 (x, y). This open book decomposition extends to the one point compactiﬁcation S 3 as a genus-0 open book decomposition with binding the unknot and monodromy equal the identity id D 2 of the disk. The resulting open book decomposition is called the standard open book decomposition on S 3 . This picture is another manifestation of the fact that S 3 is the union of two solid tori, one is the neighborhood of the binding and the other one is the union of the pages. (b) Let h be the right-handed Dehn twist along the middle circle S 1 ¦ 1 2 ¦ in S 1 [0, 1]. Using h as monodromy, it deﬁnes a 3-manifold together with an (abstract) open book decomposition. We denote the corresponding open book decomposition by ob + . Using methods we will discuss in the next chapter, we can see that the 3-manifold given by h is the 3-sphere S 3 : Just consider the Lefschetz ﬁbration given by h and realize that it can be built using a single 1-handle and a 2-handle cancelling it; see Figure 9.1 for a Kirby diagram of this 4-manifold. In fact, the binding of the resulting open book decomposition can be identiﬁed with the positive Hopf link and the pages are just the obvious Seifert surfaces. To see this, slide the circles representing ∂ _ S 1 [0, 1] _ over the (−1)-framed 2-handle and cancel the 1-handle/2-handle pair. Then the circles a = S 1 0 and b = S 1 1 will be linked once in the new S 3 . This is depicted by the ﬁne lines in Figure 9.1. Taking h −1 corresponds to reversing the orientation on the Lefschetz ﬁbration and hence on its boundary S 3 . Therefore the resulting 134 9. Open books and contact structures −1 a a b b Figure 9.1. The circles a and b after handle cancellation open book decomposition has the negative Hopf link as its binding — this open book decomposition will be denoted by ob − . An alternative way to give ob ± is by considering H ± = ¦r 1 r 2 = 0¦ ⊂ S 3 equipped with polar coordinates (r 1 , θ 1 , r 2 , θ 2 ) coming from S 3 ⊂ C 2 and π ± (r 1 , θ 1 , r 2 , θ 2 ) = θ 1 ±θ 2 . The standard open book decomposition becomes L = ¦r 2 = 0¦ and π(r 1 , θ 1 , r 2 , θ 2 ) = θ 2 in these coordinates. (c) Let p and q be relatively prime integers such that p, q ≥ 2. It is well-known that a (p, q) torus knot T (p,q) is a ﬁbered knot. This gives an open book decomposition of S 3 , where the ﬁber is a surface of genus 1 2 (p − 1)(q − 1) with one boundary component and the monodromy is a product of (p − 1)(q − 1) right-handed Dehn twists along nonseparating (i.e., homologically essential) curves. Next we will brieﬂy describe the plumbing operation (which is a special case of the Murasugi sum) and explain how to construct the ﬁbered surface of a torus knot (i.e., a page of the open book of S 3 induced by the torus knot) by plumbing positive Hopf bands (cf. [7, 74]) depicted in Figure 9.2. Recall that the monodromy of a positive (resp. negative) Hopf link is a right- handed (resp. left-handed) Dehn twist along the core circle of the Hopf band. Warning 9.1.5. Our convention for monodromy diﬀers from Harer’s [74]. We glue the end (1, x) to _ 0, h(x) _ in the mapping torus to calculate the monodromy h as opposed to gluing (0, x) to _ 1, h(x) _ as Harer does. Let H + denote the positive Hopf link and F + its ﬁbered surface (the positive Hopf band) in S 3 as it is shown in Figure 9.2. Suppose that (L, F) is another ﬁbered link with its ﬁbered surface in an arbitrary 3-manifold Y . Choose an arc α in F connecting two points on the boundary L = ∂F. Take a neighborhood να of α in F and thicken this into a 3-ball. Now apply the 9.1. Open book decompositions of 3-manifolds 135 β (a) (b) Figure 9.2. (a) positive and (b) negative Hopf bands same for the curve β depicted in Figure 9.2 on the Hopf band F + and take a connected sum of Y with S 3 along these 3-balls such that να = αI and νβ = β I are glued in a way that α is identiﬁed with I ⊂ νβ and β is identiﬁed with I ⊂ να. In fact plumbing a Hopf band is a special case of a more general operation called the Murasugi sum which is deﬁned similarly for gluing arbitrary ﬁbered links along their ﬁbered surfaces. It is proven in [151] (see also [52]) that by plumbing two ﬁbered links we get a new ﬁbered link whose monodromy is the product of the monodromies of these ﬁbered links in the following sense: First extend the monodromies of each ﬁbered surface onto the glued up surface (obtained by plumbing) by identity and then take the product of the resulting diﬀeomorphisms. For example, we can plumb two positive Hopf links to get a (2, 3) torus knot (the right-handed trefoil) with its ﬁbered surface. Simply identify a neighborhood of the arc α in one Hopf band with a neighborhood of the arc β in the other Hopf band, transversely as shown in Figure 9.3. The resulting monodromy will be the product of two right-handed Dehn twists along the curves also drawn in Figure 9.3(c). Note that the two curves (one of which is drawn thicker) intersect each other only once and they stay parallel when they go through the left twist on the surface. We can iterate this plumbing operation to express the monodromy of a (2, q) torus knot as a product of (q −1) right-handed Dehn twists. 136 9. Open books and contact structures (a) (b) (c) α β Figure 9.3. Plumbing two Hopf bands Exercise 9.1.6. Describe an abstract open book corresponding to the (2, q) torus knot. By attaching more positive Hopf bands we can construct the ﬁbered surface of a (p, q) torus knot for arbitrary p and q. We depicted the (3, 5) torus knot with its ﬁbered surface in Figure 9.4. We would like to view this ﬁgure as two rows of “gates”. First construct the row of gates in the back as described above and then plumb a Hopf band in the front row and proceed as above to obtain a second row of gates. It should be clear that we 9.1. Open book decompositions of 3-manifolds 137 can iterate this process to build as many rows of gates as we wish. Hence the monodromy of the (p, q) torus knot is a product of right-handed Dehn twists. Figure 9.4. Monodromy of the (3, 5) torus knot Exercise 9.1.7. Show that the curves depicted on the Seifert surface of the (3, 5) torus knot in Figure 9.4 are homologically essential. Deﬁnition 9.1.8. Suppose that an open book decomposition with page F is speciﬁed by h ∈ Γ F . Attach a 1-handle to the surface F connecting two points on ∂F to obtain a new surface F ′ . Let α be a closed curve in F ′ going over the new 1-handle exactly once. Deﬁne a new open book decomposition with h ◦ t α ∈ Γ F ′ , where t α denotes the right-handed Dehn twist along α. The resulting open book decomposition is called a positive stabilization of the one deﬁned by h. If we use a left-handed Dehn twist instead then we call the result a negative stabilization. The inverse of the above process is called positive (negative) destabilization. Notice that the resulting monodromy depends on the chosen curve α. We can view the stabilization/destabilization as plumbing/deplumbing Hopf bands. Since plumbing a Hopf band on the 3-manifold level is equal to the 138 9. Open books and contact structures connected sum with S 3 , by deﬁnition we do not change the topology of the underlying 3-manifold. There is another technique, called twisting, for constructing new open book decompositions of 3-manifolds. Suppose that C is a curve embedded in a page F of a given open book decomposition in S 3 . Twisting is deﬁned as performing a (±1)-surgery along C with respect to the framing C acquires by the page. It is easy to see that by this operation we add a Dehn twist along C to the monodromy of the original open book decomposition. In particular if C is unknotted in S 3 and the surgery coeﬃcient of C in S 3 turns out to be ±1 (with respect to its Seifert framing) then the resulting manifold is again S 3 and hence we get a new open book of S 3 . Theorem 9.1.9 (Harer, [73]). Every open book decomposition in S 3 is related to the trivial one by a sequence of plumbings and twistings. In fact Harer conjectured that one can entirely omit the twisting operation in the theorem above. Harer’s conjecture was recently veriﬁed by Giroux (and also independently by Goodman [68]). It was showed that any two open book decompositions of an arbitrary integral homology sphere are related by a sequence of plumbings. (See Corollary 9.2.14.) 9.2. Compatible contact structures Deﬁnition 9.2.1. An open book decomposition of a 3-manifold Y and a (cooriented) contact structure ξ on Y are called compatible if ξ can be represented by a contact form α such that the binding is a transverse link, dα is a volume form on every page and the orientation of the transverse binding induced by α agrees with the boundary orientation of the pages. This deﬁnition of compatibility is natural in the sense that the conditions that α > 0 on the binding and dα > 0 on the pages is a strengthening of the contact condition α ∧ dα > 0 in the presence of an open book on Y . Exercises 9.2.2. (a) Show that the condition that dα is a volume form on every page is equivalent to the condition that the Reeb vector ﬁeld of α is transverse to the pages. (Hint: Recall that the Reeb vector ﬁeld R α is determined as the unique direction where dα degenerates, and a volume form is nondegenerate.) 9.2. Compatible contact structures 139 (b) Show that an open book decomposition and a contact structure are compatible if and only if the Reeb vector ﬁeld of α is transverse to the pages (in their interiors) and tangent to the binding. Intuitively an open book is compatible with a contact structure if we can push the contact planes arbitrarily close to the tangents of the pages (except on the binding) of the open book. Recall that the Reeb vector ﬁeld R α for α is transverse to the contact planes. Next we would like to look at the simplest example of a compatible open book decomposition and a contact structure. Recall that the standard contact structure on R 3 can be given as the kernel of the form α 1 = dz+xdy and the pages of the standard open book decomposition on R 3 is given by the half-planes around the z-axis (see Example 9.1.4). Notice that dα 1 is degenerate when restricted to a page, and therefore clearly is not a volume form. In fact ∂ ∂z is the Reeb vector ﬁeld for α 1 and it is tangent to all the pages of the open book decomposition. However, if we multiply α 1 by the positive function f(x, y, z) = e −x 2 + e −y 2 we get a contact form which represents the same contact structure and using this form, rather than the standard one, we show that that the standard contact structure and the standard open book decomposition on R 3 are compatible. It is clear that the binding (the z-axis) is transverse to the contact planes and ∂ ∂z orients the binding. Let α = (e −x 2 +e −y 2 )(dz +xdy). Then dα = −2xe −x 2 dx ∧ dz −2ye −y 2 dy ∧ dz + _ (1 −2x 2 )e −x 2 +e −y 2 _ dx ∧ dy is a volume form on the pages since we can easily check that ∂ ∂z is not the direction that dα degenerates so the Reeb vector ﬁeld of α cannot be tangent to the pages. So we checked two conditions in the deﬁnition but we still have to verify the condition about the orientations. We want to show that the orientation on the binding induced from a page (which is oriented by dα) agrees with ∂ ∂z . This can be easily checked by evaluating dα on the basis ¦ ∂ ∂z , u¦ for any vector u in the xy-plane. Exercise 9.2.3. Consider the contact form β = (x 2 +y 2 +1)(dz +xdy) on R 3 . Show that dβ is a volume from on the pages of the standard open book decomposition but the orientation it induces on the binding is − ∂ ∂z . Example 9.2.4. Next we give an example of a compatible contact struc- ture and an open book decomposition on a closed manifold. Consider the standard tight contact structure ξ st on S 3 = _ (z 1 , z 2 ) ∈ C 2 [ [z 1 [ 2 +[z 2 [ 2 = 1 _ . 140 9. Open books and contact structures The contact structure ξ st can be given by the kernel of the 1-form α = r 2 1 dθ 1 +r 2 2 dθ 2 , where z j = r j e iθ j , for j = 1, 2. The simplest open book decomposition of S 3 is given by the binding L = ¦r 2 = 0¦ and the ﬁbration π(r 1 , θ 1 , r 2 , θ 2 ) = θ 2 . Then, for a ﬁxed θ 2 , π −1 (θ 2 ) is the interior of a disk bounded by the binding L. This is the trivial open book decomposition of S 3 where the binding is an unknot, the pages are disks and the monodromy is the identity. We can see that the trivial open book decomposition of S 3 is compatible with ξ st as follows: The tangent to the binding L is given by ∂ ∂θ 1 and the contact form is dθ 1 when restricted to ¦r 2 = 0¦. Therefore the binding is transverse to the contact structure ξ st . The contact form restricted to a page is r 2 1 dθ 1 and thus d(r 2 1 dθ 1 ) = 2r 1 dr 1 ∧ dθ 1 is a volume form. This also shows that the orientation induced on the binding as the boundary of a page coincides with the orientation induced by the contact form α. The roots of Giroux’s result invoked at the beginning of this chapter go back to the following classical result of Thurston and Winkelnkemper: Theorem 9.2.5 (Thurston–Winkelnkemper, [167]). Every open book de- composition of a closed and oriented 3-manifold Y admits a compatible contact structure. Proof. We describe the construction of Thurston and Winkelnkemper fol- lowing the expositions given in [1, 122]. Recall that if an open book decom- position of Y is given then Y is diﬀeomorphic to F(h) _ (∂F D 2 ), where F is an oriented surface with boundary, h is a self-diﬀeomorphism of F preserving ∂F pointwise and F(h) = _ F [0, 1] _ /((x, 1) ∼ _ h(x), 0 _ ) is the relative mapping torus of the element h ∈ Γ F . To simplify notation, below we assume that the boundary ∂F has only one component. To ﬁnd a contact form on Y , we ﬁnd a contact form on F [0, 1] ﬁrst, which descends to the quotient F(h) and then extend it over the solid torus S 1 D 2 ≃ ∂FD 2 . Let (t, θ) be coordinates for a collar neighborhood C of ∂F such that t ∈ ( 1 2 , 1] and ∂F = ¦t = 1¦. We claim that the set o of 1-forms η satisfying (1) dη is a volume form on F, and (2) η = tdθ near ∂F, 9.2. Compatible contact structures 141 is nonempty and convex. For proving the claim, choose a volume form Ω on F with _ F Ω = 1 and Ω[ C = dt ∧ dθ. Let η 1 be any 1-form on F which equals tdθ near ∂F. Then by Stokes’ Theorem we obtain _ F (Ω −dη 1 ) = _ F Ω − _ F dη 1 = 1 − _ ∂F η 1 = 1 − _ ∂F dθ = 0. Hence the closed 2-form Ω −dη 1 represents the trivial class in cohomology and vanishes near ∂F. By deRham’s theorem there is a 1-form γ on F with dγ = Ω −dη 1 and γ vanishes near ∂F. Deﬁne η 2 = η 1 + γ. Then dη 2 = Ω is a volume form on F and η 2 = tdθ near ∂F, showing that o , = ∅. Let ϕ 1 and ϕ 2 be two 1-forms in o. Then d _ τϕ 1 + (1 −τ)ϕ 2 _ = τdϕ 1 + (1 −τ)dϕ 2 > 0 on F and τϕ 1 + (1 −τ)ϕ 2 = tdθ near ∂F, which shows the convexity of the set o. Let η be any 1-form in o. Then h ∗ η also belongs to the set o: dh ∗ η = h ∗ dη is a volume form on F and h ∗ η = η = tdθ near ∂F. By convexity, the 1-form ˜ η (x,τ) = τη x + (1 −τ)(h ∗ η) x is in o for each τ and descends to the quotient F(h) where x is in the ﬁber and τ is in the base circle. Thus d˜ η induces a volume form when restricted to a page of our open book decomposition. Notice that when we glue the two ends of F I, the forms η and h ∗ η match up on that ﬁber. Moreover, since h — and hence h ∗ — is the identity near ∂F, we have ˜ η (x,τ) = tdθ for all (x, τ) = _ (t, θ), τ _ near ∂F(h) = ∂F S 1 . Let dτ be a volume form on S 1 . We claim that α 1 = ˜ η +κπ ∗ dτ is a contact form on F(h) for some suﬃciently large constant κ > 0, where π denotes the projection of F(h) onto the circle S 1 = [0, 1]/ ∼. To prove the claim we ﬁx a point (x, τ) ∈ F(h) and choose an oriented basis ¦u, v, w¦ of T (x,τ) F(h) such that d˜ η (x,τ) (u, v) > 0 and π ∗ (u) = π ∗ (v) = 0. This means 142 9. Open books and contact structures that the vectors u and v are tangent to the ﬁber and w is transverse to the ﬁbration. Thus we get (α 1 ∧ dα 1 ) (x,τ) (u, v, w) = (˜ η ∧ d˜ η) (x,τ) (u, v, w) +κ (x,τ) _ dτ(π ∗ w)d˜ η (x,τ) (u, v) _ . Hence we conclude that (α 1 ∧ dα 1 ) (x,τ) (u, v, w) > 0 for κ (x,τ) suﬃciently large since dτ(π ∗ w)d˜ η (x,τ) (u, v) is positive by the choice of the oriented basis (u, v, w). By compactness of F(h), there is a suﬃciently large κ > 0 such that α 1 ∧ dα 1 > 0 on F(h). Now we would like to identify a collar neighborhood of ∂F(h) with a tubular neighborhood of the binding. Let D(r) denote a disk of radius r. We use polar coordinates (r, φ) for D(1.5) (near the binding) and identify coordinates as (θ, r, φ) ≈ (θ, 2 − t, τ) where 1 ≤ r ≤ 1.5. Then the 1-form α 1 deﬁned above is given by α 1 = (2 −r)dθ +κdφ on ∂F (D(1.5) − D(1)) since ˜ η = tdθ near the boundary and π ∗ dτ is identiﬁed with dφ. We have to extend this form smoothly onto ∂F D(1.5). Note that the form (2 − r)dθ + κdφ is a positive contact form away from r = 0 but it does not extend across r = 0. Consider the 1-form α 2 = (2 −r 2 )dθ +r 2 dφ instead, which is a contact form near r = 0 since α 2 ∧dα 2 = 4rdθ ∧dr ∧dφ. Now we claim that we can “connect” α 1 and α 2 by contact 1-forms, i.e., we can ﬁnd smooth functions f 1 , f 2 : [0, 1.5] →R so that the 1-form α = f 1 (r)dθ +f 2 (r)dφ is a contact form on ∂F D(1.5) where α equals α 2 near r = 0 and equals α 1 for 1 ≤ r ≤ 1.5. Note that the necessary and suﬃcient condition for α to be a positive contact form is that α ∧ dα > 0 which is equivalent to the condition f 1 f ′ 2 −f 2 f ′ 1 > 0 (away from r = 0) as the following simple calculation shows: 9.2. Compatible contact structures 143 α ∧ dα = (f 1 dθ +f 2 dφ) ∧ (f ′ 1 dr ∧ dθ +f ′ 2 dr ∧ dφ) = (f 1 f ′ 2 −f 2 f ′ 1 ) dθ ∧ dr ∧ dφ. To guarantee the condition f 1 f ′ 2 − f 2 f ′ 1 > 0 we choose smooth functions f 1 (r) and f 2 (r) such that f 1 (r) = _ 2 −r 2 if 0 ≤ r ≤ 0.5 2 −r if 1 ≤ r ≤ 1.5 f 2 (r) = _ r 2 if 0 ≤ r ≤ 0.5 κ if 1 ≤ r ≤ 1.5 satisfying f ′ 1 (r) < 0 (0.5 ≤ r ≤ 1), and f ′ 2 (r) > 0 (0.5 ≤ r ≤ 1). It is clear that we can ﬁnd such smooth functions. We claim now that the smooth 1-form α is compatible with the given open book decomposition. Note that by construction dα = dα 1 = d˜ η is a volume form on the ﬁbers of F(h). On the other hand dα = f ′ 1 dr ∧ dθ + f ′ 2 dr ∧ dφ is a volume form on ¦φ = constant, 0 ≤ r ≤ 1.5¦ in ∂F D(1.5). Note that the orientation dt ∧dθ on the collar of a ﬁber F in F(h) and the orientation −dr ∧dθ on the surface ¦φ = constant, 0 < r ≤ 1.5¦ in the solid torus ∂F D(1.5) match up (via the identiﬁcation t = 2 − r ) to orient the pages of the given open book decomposition. So we conclude that dα induces a volume form when restricted to a page of our open book decomposition. Moreover, the tangent to the binding is given by ∂ ∂θ and it is clearly transverse to (2−r 2 )dθ+r 2 dφ. Finally notice that the orientation induced on the binding by the volume form dα agrees with ∂ ∂θ . Exercise 9.2.6. We proved above that the 1-form α 1 = ˜ η + κπ ∗ dτ is a contact 1-form for suﬃciently large κ. Show that the contact planes ker α 1 approach the tangents of the pages in the open book decomposition as κ →∞. Theorem 9.2.5 was substantially reﬁned by Giroux [63]. He proved Proposition 9.2.7. Any two contact structures compatible with a given open book decomposition are isotopic. 144 9. Open books and contact structures Proof. Suppose that ξ 0 and ξ 1 are two contact structures compatible with a given open book decomposition of a closed oriented 3-manifold Y . Then there are contact forms α 0 and α 1 such that ξ i = ker α i , where dα i is a positive volume form on the pages and α i is transverse to the binding L, for i = 0, 1. Choose coordinates (θ, r, φ) near L in which the binding and the pages are given by ¦r = 0¦ and ¦φ = const¦, respectively. Let α = f(r)dφ, where f(r) is a nondecreasing function which equals 0 for small r and which equals 1 for r ≥ r 0 . Extend α to Y as π ∗ dτ (which agrees with dφ in a tubular neighborhood of the binding), where π: Y − L → S 1 is the ﬁbration and dτ is a volume form on S 1 . Notice that in this way we deﬁne a global 1-form α on Y which vanishes near the binding. Then the 1-forms α i,t = α i + tα, t ≥ 0 are all contact. It is easy to see that α i,t is a contact form away from the binding: α i,t ∧dα i,t = α i ∧dα i +tπ ∗ dτ ∧dα i > 0 since α i is a contact form and dα i is a volume form on the pages. Moreover for t large enough, the forms α s,t = (1 − s)α 0,t + sα 1,t (0 ≤ s ≤ 1) are also contact. Again, when we consider α s,t ∧ dα s,t away from the binding, the only terms which are not necessarily positive are s(1 − s)α 1 ∧ dα 2 and s(1 − s)α 2 ∧ dα 1 . But the rest of the terms are positive and some of them are multiplied with the parameter t. This shows that for large enough t, α s,t is contact for all 0 ≤ s ≤ 1 and hence α 0,t and α 1,t are isotopic, which in turn implies that α 0 and α 1 are isotopic. For the converse direction, we get the following theorem. Theorem 9.2.8. Every closed contact 3-manifold (Y, ξ) admits a compati- ble open book decomposition. Here we outline a construction of a compatible open book described in Goodman’s thesis [68] which is a slight modiﬁcation of Giroux’s original construction. An alternative construction will be given in Chapter 11. Deﬁnition 9.2.9. A contact cell decomposition of (Y, ξ) is a CW-decom- position of Y such that (i) the 1-skeleton of Y is a Legendrian graph, (ii) each 2-cell D is convex with tw(∂D, D) = −1, i.e., the contact planes twist negatively once (along ∂D) with respect to the surface D, and (iii) the contact structure ξ is tight when restricted to the 3-cells. In fact every contact 3-manifold admits a contact cell decomposition. In order to ﬁnd such a cell decomposition ﬁrst cover (Y, ξ) by a ﬁnite number of Darboux balls. Then take a cell decomposition of Y such that each 3-cell 9.2. Compatible contact structures 145 lies in the interior of one of these Darboux balls. Isotope the 1-skeleton to be Legendrian by the Legendrian Realization Principle (cf. Lemma 5.1.18) and make each 2-cell convex (cf. Remark 5.1.7). If tw(∂D, D) < −1 then take a reﬁnement of the cell decomposition by appropriately subdividing D. Deﬁnition 9.2.10. Let G be the 1-skeleton of a contact cell decomposition of (Y, ξ). The ribbon R of G is a (smoothly embedded) surface in Y such that the surface R retracts onto G, T p R = ξ p for p ∈ G and T p R ,= ξ p for p / ∈ G. A ribbon R for the 1-skeleton of any contact cell decomposition of (Y, ξ) can be constructed such that B = ∂R is the binding of an open book on Y compatible with ξ. Notice that the ribbon R is a page of this open book. Theorem 9.2.8 can be sharpened by determining the relation between two open book decompositions compatible with a ﬁxed contact structure, in a similar fashion as Proposition 9.2.7 does in the converse direction. This result says that two open book decompositions are compatible with the same contact structure if and only if they admit common positive stabilization. We will not deal with the proof of this statement in these notes, although the proof involves similar ideas as described above for the converse direction. Summarizing the above results, together with this last missing identiﬁcation, we get Giroux’s theorem announced in the introductory section: Theorem 9.2.11 (Giroux, [63]). (a) For a given open book decomposition of Y there is a compatible contact structure ξ on Y . Contact structures compatible with a ﬁxed open book decomposition are isotopic. (b) For a contact structure ξ on Y there is a compatible open book de- composition of Y . Two open book decompositions compatible with a ﬁxed contact structure admit common positive stabilization. Remark 9.2.12. When we stabilize a compatible open book ob ξ on (Y, ξ) we take a connected sum of Y with S 3 at the topological level, so that the resulting manifold is diﬀeomorphic to Y . In the case of positive stabilization the resulting open book is obtained by plumbing a positive Hopf band F + to a page of ob ξ . Recall that the contact structure on S 3 compatible with the open book ob + (induced by the positive Hopf link H + ) is the standard tight contact structure. Hence by Remark 5.2.3, the contact structure compatible with the resulting open book is a contact connected sum ξ#ξ st , which is isotopic to ξ. In conclusion, positive stabilization does not change the (compatible) contact structure. On the other hand, negative 146 9. Open books and contact structures stabilization does change the contact structure since the contact structure on S 3 compatible with the open book ob − (induced by the negative Hopf link H − ) is an overtwisted contact structure. Consequently, the contact structure compatible with an open book obtained by negative stabilization is necessarily overtwisted. The next natural question is how to read oﬀ contact topological properties of a given contact structure from a compatible open book decomposition. Recall that an open book decomposition can be speciﬁed by giving a map- ping class in the mapping class group of the page. It seems that tightness and ﬁllability properties of the contact structure translate to factorizabil- ity of the monodromy of some compatible open book decomposition. The diﬃculty in using such characterizations lies in the fact that the open book decomposition is not uniquely deﬁned for a contact structure, rather it is given up to a complicated equivalence relation between various mapping class groups. The relation between properties of mapping classes and the corresponding contact structures is still not completely understood. By the classiﬁcation of overtwisted contact structures one can prove Corollary 9.2.13 (Giroux). A contact 3-manifold (Y, ξ) is overtwisted if and only if it admits a compatible open book decomposition which is a negative stabilization of another open book decomposition. Proof. We already showed in Remark 9.2.12 that the contact structure compatible with an open book obtained by a negative stabilization is over- twisted. It is not hard to see that negative stabilization of an open book changes the oriented 2-plane ﬁeld induced by the compatible contact struc- ture by adding one to its 3-dimensional invariant. This follows from the connected sum formula for the 3-dimensional invariant (cf. Chapter 11). The classiﬁcation of overtwisted contact structures shows that any oriented 2-plane ﬁeld v 0 is homotopic to an overtwisted contact structure ξ 0 and we know that there is an open book ob ξ 0 compatible with ξ 0 . Let (Y, ξ) be an overtwisted contact 3-manifold. Now take the oriented 2-plane ﬁeld v 0 for which the contact structure compatible with the negative stabilization of the corresponding open book ob ξ 0 is homotopic to the oriented 2-plane ﬁeld induced by ξ. Consider the open book ob which is obtained by a negative stabilization of ob ξ 0 . By construction, the contact structure ξ ob compatible with ob is homotopic (as an oriented 2-plane ﬁeld) to ξ. Now since ξ is over- twisted, then (again by the classiﬁcation of overtwisted contact structures) ξ and ξ ob are isotopic, hence ξ and ob are compatible, proving the result. 9.2. Compatible contact structures 147 Corollary 9.2.14 (Giroux). In an integral homology 3-sphere, any two open book decompositions can be related by a sequence of plumbing and deplumbing positive and negative Hopf bands. Proof (sketch). We plumb suﬃciently many negative Hopf bands to one of the given open book decompositions so that the 3-dimensional invariants of the resulting contact structures become equal. (We might have to plumb extra negative Hopf bands to guarantee that both contact structures are overtwisted (cf. Corollary 9.2.13)). Thus these overtwisted contact struc- tures are homotopic and therefore isotopic. Consequently, the resulting open book decompositions have a common positive stabilization by Theo- rem 9.2.11. Notice that by the assumption it follows that the 3-manifold supports a unique spin c structure, hence oriented 2-plane ﬁelds are classi- ﬁed by their 3-dimensional invariants d 3 ∈ Z up to homotopy. Another corollary makes use of Legendrian surgery: Corollary 9.2.15 (Giroux, Matveyev). The contact 3-manifold (Y, ξ) is Stein ﬁllable if and only if Y admits an open book decomposition compatible with ξ whose monodromy admits a factorization into right-handed Dehn twists only. A proof of this theorem is given in Theorem 10.3.4. For various notions of ﬁllability of contact structures see Chapter 12, and for factorizations of mapping classes into Dehn twists see Chapter 15. Next we will discuss a criterion given by Goodman ([68]) to detect over- twistedness of a contact structure by examining the monodromy of a com- patible open book decomposition based on a diﬀerent point of view. Notice that the contact structures compatible with a given open book decompo- sition are all tight or all overtwisted by Proposition 9.2.7. Hence we will call an open book decomposition overtwisted if a contact structure com- patible with this open book decomposition is overtwisted. Let α, β ⊂ F be properly embedded oriented arcs which intersect transversely on an oriented surface F. The algebraic intersection number i alg (α, β) is the oriented sum over interior intersections. The geometric intersection number i geom (α, β) is the unassigned count of interior intersections, minimized over all boundary ﬁxing isotopies of α and β. The boundary intersection number i ∂ (α, β) is one-half the oriented sum over intersections at the boundaries of the arcs, after the arcs have been isotoped, ﬁxing boundary, to minimize geometric intersection. See Figure 9.5 for sign conventions. In particular, given an arc α on a page F of an open book decomposition with monodromy h, we 148 9. Open books and contact structures − α β α α β β + + Figure 9.5. Sign convention for intersection numbers on a surface can consider i alg _ α, h(α) _ , i geom _ α, h(α) _ and i ∂ _ α, h(α) _ . Here h(α) is oriented by reversing the orientation on h(α) which is obtained by pushing forward the orientation of α by the monodromy h. Deﬁnition 9.2.16. A properly embedded arc α is sobering for a monodromy h if i alg _ α, h(α) _ +i geom _ α, h(α) _ +i ∂ _ α, h(α) _ ≤ 0, and α is not isotopic to h(α). Proposition 9.2.17 (Goodman, [68]). If there is a sobering arc α ⊂ F for h then the given open book decomposition is overtwisted. In order to prove this result, Goodman constructs a surface with Legendrian boundary which violates Eliasberg’s inequality given by Theorem 4.3.7. As an example, we consider the simplest case: the open book decomposition ob − of S 3 induced by the negative Hopf link H − with its ﬁbered surface F − . Exercise 9.2.18. Show that the arc α across the annulus F − in Figure 9.6 is a sobering arc for the monodromy h of the open book decomposition. Recall that h is a left-handed Dehn twist along the middle circle of the annulus. (Hint: Observe that i ∂ _ α, h(α) _ = −1, while i alg _ α, h(α) _ = i geom _ α, h(α) _ = 0.) In the light of Proposition 9.2.17 this implies that the induced open book decomposition is overtwisted. The arc α, however, is not a sobering arc 9.2. Compatible contact structures 149 α ( ) α h Figure 9.6. A sobering arc for H + . In fact, since the monodromy of the open book decomposition ob + is a right-handed Dehn twist, the compatible contact structure is Stein ﬁllable by Corollary 9.2.15 and therefore it gives the standard tight contact structure by the classiﬁcation of contact structures on S 3 . Proposition 9.2.19 (Goodman, [68]). If an arc α ⊂ F satisﬁes i alg _ α, h(α) _ +i geom _ α, h(α) _ +i ∂ _ α, h(α) _ = −1, then the open book decomposition with page F and monodromy h n is overtwisted for any n > 0. For an application of Proposition 9.2.19 consider a genus-g surface F with only one boundary component and let δ be a curve parallel to the boundary. Then the open book decomposition with page F and monodromy t −n δ is overtwisted for n > 0, where t δ denotes the right-handed Dehn twist along δ. To see this, we ﬁrst observe that t δ = (t a 0 t a 1 t a 2g−1 t a 2g ) 4g+2 where the curves a 0 , a 1 , . . . , a 2g are depicted in Figure 15.5. Now the arc α shown in Figure 9.7 will be a sobering arc for h = t −1 a 2g t −1 a 2g−1 t −1 a 1 t −1 a 0 , since i alg _ α, h(α) _ +i geom _ α, h(α) _ +i ∂ _ α, h(α) _ = −1. We depict α and h(α) in Figure 9.7. Then since t −n δ = h n(4g+2) , the open book decomposition with monodromy t −n δ is overtwisted for n > 0 by Proposition 9.2.19. On the other hand, the open book decomposition with page F and monodromy t n δ (n ≥ 0) is Stein ﬁllable and hence tight. 150 9. Open books and contact structures (α) h 2g α α a Figure 9.7. The action of the mapping class h on the arc α Corollary 9.2.20 (Goodman, [68]). An open book decomposition is over- twisted if and only if it has a common positive stabilization with an open book decomposition which has a sobering arc. Proof. If the ﬁnal open book decomposition has a sobering arc then it is overtwisted by Theorem 9.2.17, and positive stabilization/destabilization does not change the contact structure (up to isotopy). Conversely, if the contact structure is overtwisted then it has a positive stabilization which is a negative stabilization of some other open book decomposition. On the other hand, negative stabilization can be realized by plumbing a negative Hopf band. Now the solution of Exercise 9.2.18 shows the existence of a sobering arc, concluding the proof. 9.3. Branched covers and contact structures Let F denote an orientable compact surface and Y a closed, orientable 3- manifold. Deﬁnition 9.3.1. A smooth surjective map π: F → D 2 is called a simple d-fold cover if there is a ﬁnite set Q in the interior of D 2 , called the branch set and each p ∈ D 2 has a neighborhood U over which π: π −1 (U) → U behaves as follows: (1) if p / ∈ Q then π[ π −1 (U) is a trivial d-fold cover, and (2) if p ∈ Q then π −1 (U) has d − 1 components, one of which is a disk projecting to U as a double cover branched over p, i.e., can be modeled by the complex map z → z 2 around the origin, while the others are disks projecting diﬀeomorphically. 9.3. Branched covers and contact structures 151 Deﬁnition 9.3.2. A smooth map h: Y →S 3 is called a simple d-fold cover with branch set B ⊂ S 3 if it is locally diﬀeomorphic to the product of an interval with a simple d-fold cover of a disk D 2 and the branch points (multiplied by the interval) form the set B. Theorem 9.3.3 ([118]). Every open book decomposition of Y with con- nected binding is a simple 3-fold cover of S 3 branched over a closed braid. Proof. Let F be the page of a given open book decomposition on Y with connected binding. Choose a 3-fold simple branched cover π: F → D 2 . Then we can realize the monodromy of the open book decomposition as the lift of a braid (viewed as a diﬀeomorphism of the disk D 2 ﬁxing ∂D 2 and the branch set in D 2 ) under the branched covering map π. Now consider the closure of this braid (viewed as a geometric object in the usual sense) in S 3 with respect to an axis A. Note that S 3 has an open book decomposition with disks D t as pages, A = ∂D t as binding, and the identity map as monodromy. This construction gives a branched covering map h: Y → S 3 such that each page of the given open book decomposition is realized as h −1 (D t ) and the binding is simply equal to h −1 (A). Corollary 9.3.4 (Giroux). Every closed contact 3-manifold (Y, ξ) is a simple 3-fold cover of (S 3 , ξ st ) branched along a transverse link. Proof. For a given contact 3-manifold (Y, ξ) Theorem 9.2.8 provides an open book decomposition of Y (with connected binding) which is compatible with the contact structure ξ. By Theorem 9.3.3, on the other hand, this open book decomposition is a simple 3-fold cover h: Y → S 3 branched over a closed braid B. Let α be the standard contact form on S 3 . Note that h ∗ α is not a contact form on Y . We denote by C the set of points in Y where h fails to be a local diﬀeomorphism. Next we follow the discussion in [67] to construct a contact form on Y using the branched covering map h. There is a tubular neighborhood U 1 of B with coordinates (θ, x, y) where gα = dθ +xdy −y dx for some positive function g on U 1 . Extend g to a positive function on S 3 such that g = 1 outside a tube U 2 slightly larger than U 1 and then deﬁne α ′ = gα. Using polar coordinates, α ′ can be given by dθ +r 2 dφ on U 1 − B. Then we deform h by an isotopy of S 3 supported in U 1 to a nonsmooth covering map H which is a local diﬀeomorphism on Y −C. Since 152 9. Open books and contact structures H is a local diﬀeomorphism, β = H ∗ α ′ is a contact form on Y −C. Moreover, we can ﬁnd local coordinates (θ 0 , r 0 , φ 0 ) in a tubular neighborhood U of C such that the form β is given by (1 +nr 2 0 )dθ 0 +r 2 0 dφ 0 for some integer n. We can extend this form to a smooth contact form on Y , again denoted by β. To see this we write the form β in Cartesian coordinates as β = _ 1 +n(x 2 0 +y 2 0 ) _ dθ 0 +x 0 dy 0 −y 0 dx 0 and check that β ∧ dβ = 2 dθ 0 ∧ dx 0 ∧ dy 0 > 0. Let A denote the axis of the closed braid B which is the branch set of h: Y → S 3 . We can assume that both the braid B and the axis A are transverse to the standard contact structure ξ st on S 3 . Consider the trivial open book decomposition of S 3 , whose pages are the disks D t , the binding is the axis A = ∂D t and the monodromy is the identity map. The pages of the open book decomposition on Y are then given by h −1 (D t ) and the binding is equal to h −1 (A). To show that the contact form β is compatible with the open book decomposition on Y , we need to check the conditions given in Deﬁnition 9.2.1. It is easy to see that the binding h −1 (A) of our open book decomposition is transverse to the contact structure β on Y since A is transverse to α and the covering map is a local diﬀeomorphism on the points of the binding. We also need to check that dβ induces a volume form on every page of the open book decomposition on Y . Since β = H ∗ α ′ = h ∗ α away from the set C, the contact form β is simply the pull back of α by the covering map h. Now dβ = dh ∗ α = h ∗ dα is a volume form on h −1 (D t ) − (h −1 (D t ) ∩ U), because dα induces a volume form on a page D t of the trivial open book decomposition on S 3 and the covering map h is a local diﬀeomorphism. Near the set C, the contact form β is given by (1 +nr 2 0 )dθ 0 +r 2 0 dφ 0 in local coordinates. Note that dβ = 2r 0 dr 0 ∧ dφ 0 = 2 dx 0 ∧ dy 0 on a small disk obtained by ﬁxing θ 0 and clearly it is a volume form on this small disk which doubly covers a disk in S 3 with a branching point at r 0 = 0. 9.3. Branched covers and contact structures 153 This proves that dβ is a volume form on every page. The condition about the orientation is clear since we use the same orientation preserving map to pull back the contact form and to construct the branched cover. Thus, by Proposition 9.2.7 the contact structure ker β is isotopic to ξ, which ﬁnishes the proof. 10. Lefschetz fibrations on 4-manifolds In the light of recent results it turns out that both closed symplectic 4-man- ifolds and Stein surfaces admit a purely topological description in terms of Lefschetz ﬁbrations and Lefschetz pencils. In this chapter we give the necessary deﬁnitions and sketch this topological descriptions of symplectic and Stein manifolds. In the discussion we include achiral Lefschetz ﬁbra- tions as well; these more general objects are useful in viewing open book decompositions as boundaries of certain achiral Lefschetz ﬁbrations. The chapter concludes with some applications of Lefschetz ﬁbrations in various low dimensional problems. 10.1. Lefschetz pencils and fibrations Deﬁnition 10.1.1. (a) Suppose that X and Σ are given oriented 4- and 2- dimensional manifolds. The smooth map f : X 4 →Σ 2 is an achiral Lefschetz ﬁbration if df is onto with ﬁnitely many exceptions ¦p 1 , . . . , p k ¦ = C ⊂ int X (called the set of critical points), the map f is a locally trivial surface bundle over Σ−f(C) and around p i ∈ C and q i = f(p i ) ∈ f(C) there are complex charts U i , V i on which f is of the form z 2 1 + z 2 2 . We call the ﬁbers f −1 (q i ) (q i ∈ f(C)) singular, while the other ﬁbers are regular. A ﬁbration is relatively minimal if no ﬁber contains a sphere with self-intersection ±1, i.e., we cannot blow down X without destroying its ﬁbration structure. (b) An achiral Lefschetz pencil on X (with ∂X = ∅) is a nonempty ﬁnite set B ⊂ X (called the base point set) together with a map f : X−B →CP 1 such that each point b ∈ B has a coordinate chart on which f can be given by the projectivization C 2 − ¦0¦ → CP 1 and around its critical points f behaves as in (a). 156 10. Lefschetz ﬁbrations on 4-manifolds (c) An achiral Lefschetz ﬁbration/pencil is called a Lefschetz ﬁbration/pencil if the complex charts U i , V i around the critical and base points p i and q i = f(p i ) appearing in the above deﬁnition respect the given orientations of X and Σ. Remark 10.1.2. Notice that in Deﬁnition 10.1.1(a) the manifolds might have boundaries. If the typical ﬁber f −1 (t) is a closed surface then f −1 (∂Σ) = ∂X, but the deﬁnition also allows f −1 (t) to have boundary, in which case f −1 (∂Σ) forms only part of ∂X. Notice that the notion of achiral Lefschetz ﬁbrations/pencils is more general than the one without the adjective achiral ; although the terminology might suggest the contrary. We did not take the courage for changing this unfortunate phenomenon, we will rather remind the reader for this subtlety of the subject. Deﬁnition 10.1.3. Two Lefschetz ﬁbrations f : X → Σ and f ′ : X ′ → Σ ′ are equivalent if there are diﬀeomorphisms Φ: X →X ′ and φ: Σ →Σ ′ such that f ′ ◦ Φ = φ ◦ f. Next we show that a Lefschetz critical point corresponds to gluing a 4- dimensional 2-handle, and then we determine its attaching map (see also [66]). Recall that near the critical point f(z 1 , z 2 ) equals z 2 1 +z 2 2 , so a nearby regular ﬁber is given by z 2 1 + z 2 2 = t, and after multiplying f by a unit complex number we can assume t > 0. For the discussion below, assume that the chart does respect the orientation of X around the critical point. If we intersect the ﬁber with R 2 ⊂ C 2 , we obtain the circle x 2 1 + x 2 2 = t in R 2 (where z j = x j + iy j and R 2 is spanned by x 1 and x 2 ). This cir- cle bounds a disk D t ⊂ R 2 and as t → 0 the disk D t shrinks to a point in R 2 . By deﬁnition ∂D t = F t ∩ R 2 is the vanishing cycle of the critical point, and we explicitly see the singular ﬁber F 0 being created from F t by the collapse of ∂D t . A regular neighborhood νF 0 of the singular ﬁber is obtained from the neighborhood νF t by adding a regular neighborhood of D t . This latter neighborhood is clearly a 2-handle H attached to νF t , with core disk equal to D t . (In fact, a corresponding Morse function can be given locally by g = −Re f, or g(z 1 , z 2 ) = y 2 1 + y 2 2 − x 2 1 − x 2 2 . This Morse function provides a handlebody decomposition of the relative handlebody built on ∂νF t for a regular ﬁber F t .) Suppose that ∂νF t contains a ﬁber F s , 0 < s < t. Then the core of the 2-handle H is D s and the attach- ing circle is the vanishing cycle ∂D s ⊂ F s . We describe the framing of the handle attachment by comparing it with the framing on ∂D s ⊂ ∂νF t deter- mined by the surface F s . At a point ( √ s cos θ, √ s sin θ, 0, 0) ∈ ∂D s ⊂ R 2 ⊂ R 2 iR 2 ∼ = C 2 the vector (−sin θ, cos θ, 0, 0) is tangent to ∂D s . 10.1. Lefschetz pencils and ﬁbrations 157 Since ∂D s lies in F s , which is holomorphic in the given local coordi- nates, the vector ﬁeld v(θ) = (0, 0, −i sin θ, i cos θ) on ∂D s is also tan- gent to F s . This can be seen explicitly by taking, for example, the curve ( √ s −t 2 cos θ, √ s −t 2 sin θ, −it sinθ, it cos θ) on the ﬁber and consider its tangent at the point ( √ s cos θ, √ s sin θ, 0, 0). Notice that the dot product of v(θ) with the tangent vector of the circle is zero, therefore v(θ) provides the normal to ∂D s inside the surface F s . This framing has to be compared with the one we get by considering a parallel nearby circle to the attaching cir- cle in the 2-handle. In the tangent space of the 2-handle the corresponding vector ﬁeld can be chosen to be (0, 0, 0, i), for example. This choice imme- diately shows that the two framings diﬀer by one. By taking the orientation into account, if we measure the surface framing with respect to the push- oﬀ on the 2-handle, we have to compute the winding number of the curve (−sin θ, cos θ) around the origin, and this quantity can be easily veriﬁed to be 1. On the other hand, we would like to specify the framing coeﬃcient of the 2-handle with respect to the surface framing, therefore the above argument translates to −1. For the case of a critical point admitting an orientation reversing coordinate chart the above argument passes through with an orientation reversal at the last moment, implying that the framing coeﬃcient in that case is +1 (with respect to the ﬁber framing). The above reasoning can be obviously inverted in the following sense: Suppose that f : X → Σ is an achiral Lefschetz ﬁbration with ∂Σ ,= ∅ and γ ⊂ ∂X is a given knot which lies in a ﬁber. Let X ′ be given by attaching a 2-handle to X along γ with framing ±1 relative to the surface framing of γ. Then f extends as an achiral Lefschetz ﬁbration to X ′ →Σ. Remark 10.1.4. By adding the standard shaped 2-handle, the map will not extend to X ′ as a Lefschetz ﬁbration, since we added only a small neigh- borhood of the critical point of the new singular ﬁber, but not the whole ﬁber. On the other hand, f can be extended to a manifold diﬀeomorphic to X ′ (similarly to the procedure of “smoothing corners”, encountered in Remark 2.1.1), and this is the content of the above argument. Next we determine the monodromy around a critical value. For any bundle with ﬁber F over an oriented circle, the monodromy is determined by a single diﬀeomorphism ψ representing the image of the canonical generator of π 1 (S 1 ) in Γ F . The bundle is then canonically isomorphic to the ﬁbration I F/((1, x) ∼ _ 0, ψ(x) _ → I/∂I ≈ S 1 . Given a Lefschetz ﬁbration f : X →Σ and a disk D ⊂ Σ inheriting the orientation of Σ, we can consider the monodromy of the bundle f[ ∂D provided that the oriented circle ∂D 158 10. Lefschetz ﬁbrations on 4-manifolds avoids the critical values of f. If D contains no critical values then f[ D is trivial, as is the monodromy (i.e., ψ is isotopic to id F ). If D contains a unique critical value, however, the monodromy is nontrivial provided f[ D is relatively minimal. A local computation shows Proposition 10.1.5. If f −1 (c) contains a unique critical point then the monodromy around c is a Dehn twist along a simple closed curve. If the orientation of the chart containing c ∈ C respects the orientation of X then the Dehn twist is right-handed, otherwise it is left-handed. The simple closed curve is isotopic to the vanishing cycle of the singular ﬁber under examination. (For the deﬁnition of a Dehn twist see Appendix 15.) Notice the assumption about the number of critical points in a ﬁber. It is not hard to see that any Lefschetz ﬁbration admits a perturbation such that f is injective on C. Therefore by ﬁxing a natural generating system of π 1 (Σ − f(C)) we get a word in Γ g : if g i in the generating system is deﬁned as g i = [γ i ] with either γ i = ∂D i for disks D i satisfying [f(C) ∩ D i [ = 1 and [h i ] (i = 1, . . . , 2g(Σ)) is a natural generating system of π 1 (Σ) then the ﬁbration can be encoded by the word t 1 . . . t n Π[α i , β i ] where the t i are the monodromies around γ i and α i (resp. β i ) are the monodromies around h i (resp. h i+g(Σ) ). The word uniquely determines the ﬁbration since a Dehn twist determines its deﬁning vanishing cycle up to isotopy, and from this information the ﬁbration can be recovered by adding 2-handles along the vanishing cycles with appropriate framings. In fact, if the ﬁber of the ﬁbration f : X → Σ is a manifold with r boundary components then the resulting word naturally lives in Γ g,r . We also note that by blowing up the points of the base point set B (cf. Deﬁnition 10.1.1) we can turn a Lefschetz pencil into a Lefschetz ﬁbration. In conclusion, Lefschetz ﬁbrations can be thought of being the geometric counterparts of certain special words in various mapping class groups. This relation will be discussed in more details in Section 15.2. Suppose that f : X → D 2 is an achiral Lefschetz ﬁbration, such that each singular ﬁber contains a unique critical point. We will describe an elementary handlebody decomposition of X using essentially the deﬁnition of an achiral Lefschetz ﬁbration and Proposition 10.1.5. We select a regular value q 0 of the map f in the interior of D 2 , an identiﬁcation of the ﬁber f −1 (q 0 ) ∼ = F (a compact surface with possibly nonempty boundary), and a collection of arcs s i in the interior of D 2 with each s i connecting q 0 to q i , and otherwise disjoint from each other. We also assume that the critical values are indexed so that the arcs s 1 , . . . , s m appear in order as we 10.1. Lefschetz pencils and ﬁbrations 159 travel counterclockwise in a small circle about q 0 . Let V 0 , . . . , V m denote a collection of small disjoint open disks with q i ∈ V i for each i, see Figure 10.1. Since an achiral Lefschetz ﬁbration is a locally trivial F-bundle away from 0 1 3 m s V 2 V 3 q 2 q m q 4 q V V 1 q s 1 2 s 4 V V m 3 4 s s q 0 Figure 10.1. Fibration over the disk the critical points, we have f −1 (V 0 ) ∼ = D 2 F with ∂ _ f −1 (V 0 ) _ ∼ = S 1 F. Let ν(s i ) be a regular neighborhood of the arc s i . Now the discussion following Remark 10.1.2 shows that f −1 (V 0 ∪ν(s 1 ) ∪V 1 ) is diﬀeomorphic to D 2 F with a 2-handle H 1 attached along a circle γ 1 contained in a ﬁber ¦pt.¦ F ⊂ S 1 F. Moreover, the 2-handle H 1 is attached with framing (±1) relative to the natural framing on γ 1 inherited from the the ﬁber. (The curve γ 1 was called the vanishing cycle.) In addition, ∂ _ (D 2 F) ∪ H 1 _ is diﬀeomorphic to an F-bundle over S 1 whose monodromy is equal to the Dehn twist t γ 1 along γ 1 . Continuing counterclockwise around q 0 , we add the remaining critical values to our description, yielding that X 0 ∼ = f −1 _ V 0 ∪ _ m _ i=1 ν(s i ) _ ∪ _ m _ i=1 V i __ is diﬀeomorphic to (D 2 F) ∪ ( m i=1 H i ), where each H i is a 2-handle attached along a vanishing cycle γ i in a ﬁber of S 1 F →S 1 with relative 160 10. Lefschetz ﬁbrations on 4-manifolds framing (±1). Furthermore, the part of ∂X 0 ∼ = ∂ _ (D 2 F) ∪ _ m _ i=1 H i __ which maps to ∂D is an F-bundle over S 1 , whose monodromy is the product of Dehn twists along the vanishing cycles. We will refer to this product as the global monodromy of the ﬁbration. Suppose that an achiral Lefschetz ﬁbration f : X →Σ admits k singular ﬁbers. The Euler characteristic of X can be easily computed as χ(F)χ(Σ)+k since we add k 2-handles to a surface bundle. The computation of σ(X), however, turns out be a nontrivial issue. There is a signature formula for hyperelliptic Lefschetz ﬁbrations [35] and there is an algorithm to compute the signature for Lefschetz ﬁbrations over S 2 given in [130]. After these topological preparations we begin our discussion about the relation between Lefschetz ﬁbrations and symplectic/Stein manifolds. For the rest of this chapter we assume that all Lefschetz ﬁbrations are of the type given by Deﬁnition 10.1.1(c), i.e., the complex coordinate charts respect the orientation ﬁxed on X. We start with the case of closed symplectic manifolds; Stein surfaces will be discussed in the next section. The most important result of the subject is Donaldson’s groundbreaking theorem: Theorem 10.1.6 (Donaldson, [22]). If (X, ω) is a closed symplectic 4- manifold and [ω] ∈ H 2 (X; R) is integral then X admits a Lefschetz pencil such that the generic ﬁber is a smooth symplectic submanifold. Exercise 10.1.7. Prove that every symplectic manifold (X, ω) admits a symplectic form ω ′ such that [ω ′ ] ∈ H 2 (X; R) lifts to an integral cohomology class, i.e., it is in the image of the map H 2 (X; Z) → H 2 (X; R) induced by the inclusion Z ֒→R. Remark 10.1.8. The proof of this theorem is rather involved, here we restrict ourselves merely to a quick indication of the main idea. Let L →X be the complex line bundle with c 1 (L) = h ∈ H 2 (X; Z) (where h maps to [ω] under the map H 2 (X; Z) → H 2 (X; R)). To prove Theorem 10.1.6, Donaldson showed that if k is large enough, then L ⊗k →X admits a section s such that s −1 (0) ⊂ X is a symplectic submanifold. Using the same basic idea, he also showed that for k even larger there are linearly independent sections s 0 , s 1 ∈ Γ(L ⊗k ) such that the submanifolds _ (t 0 s 0 +t 1 s 1 ) −1 (0) ⊂ X [ [t 0 : t 1 ] ∈ CP 1 _ 10.1. Lefschetz pencils and ﬁbrations 161 are symplectic and form a Lefschetz pencil on X. The proof is based on a technique of Kodaira for embedding K¨ ahler manifolds in CP N , although the analytical details are much more subtle in the symplectic case. Speciﬁcally, it was proved that the map x → _ s 0 (x) : s 1 (x) ¸ ∈ CP 1 (deﬁned on X − _ s −1 0 (0) ∩ s −1 1 (0) _ ) provides a Lefschetz ﬁbration on some blow-up of X. The proof of Donaldson’s result, in fact, shows the following: Corollary 10.1.9 (Donaldson, [22]). If X is a closed symplectic 4-manifold then it decomposes as W ∪ D where W is a Stein domain and D is a D 2 - bundle over a surface Σ g . Proof (sketch). Take a section σ ∈ Γ(L ⊗k ) as above and consider the function log [σ[ 2 away from the zero set s −1 (0). This provides a plurisubhar- monic function on W = X−s −1 (0) for some appropriate complex structure. Since νs −1 (0) is a D 2 -bundle over the surface Σ g = s −1 (0), the conclusion follows. Donaldson’s theorem admits a converse (which is considerably simpler to prove): Theorem 10.1.10 (Gompf, [66]). If the smooth, closed 4-manifold X admits a Lefschetz ﬁbration such that the homology class of the ﬁber is nonzero in H 2 (X; R) then X admits a symplectic structure with the ﬁbers being symplectic submanifolds (at their smooth points). Remark 10.1.11. The proof of the above theorem follows the idea pio- neered by Thurston [166] providing symplectic structures on surface bun- dles, cf. Theorem 3.1.13. The extra complication of having singular ﬁbers can be taken care of by implementing the existing local models around the critical points. For details see [66, Chapter 10]. The main idea in the proof is that (by splicing forms together) we get a closed 2-form which is sym- plectic along the ﬁbers and then we add a large multiple of the pull-back of a symplectic structure from the base to it. This leaves the ﬁber directions intact and takes care for the orthogonal directions. In fact, by taking even larger multiples we can arrange that ﬁnitely many (ﬁxed) sections of the ﬁbration become symplectic as well. This leads us to the following: Corollary 10.1.12 (Gompf, [66]). If a smooth, closed 4-manifold X admits a Lefschetz pencil then it carries a symplectic structure such that the generic ﬁber is a smooth symplectic submanifold. 162 10. Lefschetz ﬁbrations on 4-manifolds Proof. By blowing up X we get X#nCP 2 equipped with a Lefschetz ﬁ- bration, moreover the n exceptional curves (being sections) can be chosen to be symplectic. Now the symplectic normal sum blows them back down, providing a symplectic structure on X. Note that the above corollary is just the converse of Donaldson’s Theo- rem 10.1.6. We just remark here that the assumption in Theorem 10.1.10 about the homology class of the ﬁber is not very restrictive: if the ﬁber genus is not equal to one or the ﬁbration has at least one singular ﬁber then it is fulﬁlled, see [66]. (For torus ﬁbrations the statement does not neces- sarily hold, as the obvious torus ﬁbration S 1 S 3 → S 2 coming from the Hopf map S 3 →S 2 shows.) 10.2. Lefschetz fibrations on Stein domains In [73] Harer proved that if a smooth 4-manifold X is obtained by attaching 1- and 2-handles to D 4 then it admits an achiral Lefschetz ﬁbration over D 2 . Notice that the boundary of an achiral Lefschetz ﬁbration f : X →D 2 acquires a canonical open book decomposition induced from the ﬁbration: compose the map f with the radial projection π: D 2 − ¦0¦ → ∂D 2 to get π ◦ f : _ ∂X − f −1 (0) _ → S 1 , providing an open book decomposition on ∂X with binding ∂f −1 (0). An alternative proof of Theorem 9.1.3 follows from this fact since every closed oriented 3-manifold Y is the boundary of a smooth 4-manifold obtained by attaching 2-handles to D 4 . Loi and Piergallini [104] (and later Akbulut and the ﬁrst author [7]) showed that a Stein domain always admits a Lefschetz ﬁbration structure: Theorem 10.2.1 (Loi–Piergallini, [104]). If W is a Stein domain then it admits a Lefschetz ﬁbration structure. In addition, we can assume that the vanishing cycles in the resulting ﬁbration are homologically essential. In fact, Loi and Piergallini proved that any Stein domain can be given as an analytic branched cover of D 4 along a holomorphic curve, or of D 2 D 2 along a positive braided surface. The theorem above follows from this result. Proof. We describe the proof of this theorem given by Akbulut and the ﬁrst author [7]. The proof explicitly constructs the vanishing cycles of the Lefschetz ﬁbration, and associates to every Stein domain inﬁnitely many pairwise nonequivalent such Lefschetz ﬁbrations. We say that a Lefschetz 10.2. Lefschetz ﬁbrations on Stein domains 163 ﬁbration is allowable if and only if all its vanishing cycles are homologically nontrivial in the ﬁber F. Note that a simple closed curve on a surface with at most one boundary component is homologically trivial if and only if it separates the surface. Sometimes we will refer to a homologically trivial (resp. nontrivial) curve as a separating (resp. nonseparating) curve. A positive allowable Lefschetz ﬁbration over D 2 with bounded ﬁbers will be abbreviated as PALF. (Here the adjective “positive” just emphasizes that we are working with Lefschetz ﬁbrations, that is, all singular ﬁbers give rise to right-handed Dehn twists in the monodromy.) In the following we digress to give the details of a construction which is due to Lyon [106]. We say that a link in R 2 is in a square bridge position with respect to the plane x = 0 if the projection onto the plane is regular and each segment above the plane projects to a horizontal segment and each one below to a vertical segment. Clearly any link can be put in a square bridge position. (Notice that we require the horizontal segment to pass over the vertical; therefore in putting a projection in square bridge position we have to pay special attention to possible illegal crossings. For these see Figure 10.2, cf. also Figure 4.3.) Suppose that the horizontal and Figure 10.2. How to handle “illegal” crossings vertical segments of the projection of the link in the yz-plane are arranged by isotopy so that each horizontal segment is a subset of ¦0¦ [0, 1] ¦z i ¦ 164 10. Lefschetz ﬁbrations on 4-manifolds z y Figure 10.3. Trefoil knot in a square bridge position for some 0 < z 1 < z 2 < < z p < 1 and each vertical segment is a subset of ¦0¦ ¦y j ¦ [0, 1] for some 0 < y 1 < y 2 < < y q < 1. Now consider the 2-disk D i = [ε, 1] [0, 1] ¦z i ¦ for each i = 1, 2, . . . , p and the 2-disk E j = [−1, −ε] ¦y j ¦ [0, 1] for each j = 1, 2, . . . , q, where ε is a small positive number. Attach these disks by small bands (see Figure 10.4) corresponding to each point (0, y i , z j ) for i = 1, . . . , p and j = 1, . . . , q. If p and q are relatively prime then the result is the minimal Seifert surface F for a (p, q) torus knot K such that K ∩ L = ∅ and L ⊂ F. It is easy to see that each component of the link L is a nonseparating curve on the surface F. Moreover we can choose p and q arbitrarily large by adding more disks of either type D or type E. This concludes our digression. 10.2. Lefschetz ﬁbrations on Stein domains 165 x D i E j L z y Figure 10.4. Attaching disks Let K be a torus knot in S 3 . It is well-known that K is a ﬁbered knot and the corresponding ﬁbration induces an open book decomposition of S 3 whose monodromy is a product of nonseparating positive Dehn twists, cf. Example 9.1.4. This factorization of the monodromy deﬁnes a PALF X → D 2 such that the induced open book decomposition on S 3 = ∂X is equal to the open book decomposition given by the torus knot. Exercise 10.2.2. Verify that for any torus knot K the 4-manifold (PALF) K underlying the corresponding Lefschetz ﬁbration is diﬀeomorphic to D 4 . (Hint: Consider the handlebody decomposition of (PALF) K and use Kirby calculus; in particular, locate cancelling 1-handle/2-handle pairs.) Returning to the proof of Theorem 10.2.1 suppose ﬁrst that W is a Stein domain built by 2-handles only. This means that we attach Weinstein 2- handles to D 4 along the components L i of a Legendrian link L in S 3 = ∂D 4 with framing tb(L i ) − 1 to get the Stein domain W. Hence our starting point is a Legendrian link diagram in (R 3 , ξ st ) ⊂ (S 3 , ξ st ). First we smooth all the cusps of the diagram and rotate everything counterclockwise to put L into a square bridge position. See Figure 10.5 for an example. (Notice that clockwise rotation results in a diagram with vertical segments passing over horizontal ones, contradicting our convention for square bridge position.) Then the construction of Lyon described above allows us to ﬁnd a torus 166 10. Lefschetz ﬁbrations on 4-manifolds z y Figure 10.5. Rotation of a Legendrian knot into square bridge position knot K with its Seifert surface F such that each L i is an embedded circle on F for i = 1, 2, . . . , n. In Figure 10.6 we depicted the embedding of the right-handed trefoil knot into the Seifert surface of the (5, 6) torus knot. Let L + i be a parallel copy of L i on the surface F, and let lk(L i , L + i ) be the linking number of L i and L + i computed with parallel orientations. This linking number is called the surface framing of L i . We will denote it by sf(L i ). Then we observe that the surface framing of L i will pick up a −1 at each left corner of the link in square bridge position and will change by the amount of writhe at each under/over-crossing. To see this, imagine a parallel copy L + i of L i on the surface F then cut out and straighten the narrow band on the surface bounded by L i and L + i . Notice, however, that this is exactly the recipe how the Thurston–Bennequin invariant of L i is calculated in its Legendrian position (before rotating and smoothing its corners): −1 for each left kink plus the writhe of the knot. Thus we get tb(L i ) = sf(L i ). This simple observation turns out to be crucial for the proof of the theorem. The Stein domain W is obtained by attaching a Weinstein 2-handle H i to D 4 along L i with framing tb(L i ) − 1 = sf(L i ) − 1 for i = 1, 2, . . . , n. By our discussion of the handle decomposition of a Lefschetz ﬁbration 10.2. Lefschetz ﬁbrations on Stein domains 167 z y x Figure 10.6. Trefoil knot on the Seifert surface of the (5, 6) torus knot in Section 10.1 we can extend the Lefschetz ﬁbration structure on the 4- manifold D 4 ∼ = (PALF) K over the 2-handles to get a new PALF since L = ¦L 1 , . . . , L n ¦ is embedded in a ﬁber F of ∂(PALF) K ∼ = S 3 . Thus we showed that W ∼ = D 4 ∪ H 1 ∪ ∪ H n ∼ = (PALF) K ∪ H 1 ∪ ∪ H n admits a PALF and the global monodromy of this PALF is the monodromy of the torus knot K composed with positive Dehn twists along the L i ’s. Notice that the Dehn twists along the L i ’s commute since they are pairwise disjoint embedded curves on the surface F. Now we turn to the general case. Suppose that W is a Stein domain obtained by attaching 1- and 2-handles to D 4 . First of all, we would like to extend (PALF) K on D 4 to a PALF on D 4 union 1-handles. Recall that attaching a 1-handle to D 4 (with the dotted-circle notation) is the same as pushing the interior of the obvious disk that is spanned by the dotted circle into the interior of D 4 and scooping out a tubular neighborhood of its image fromD 4 . To reach our goal, we represent the 1-handles with dotted-circles stacked over the front projection of the Legendrian tangle which is in standard form as it is described in [65]. Then we modify the handle decomposition by twisting the strands going through each 1-handle negatively once. In the new diagram the Legendrian framing will be the blackboard framing with one left-twist added for each left cusp. This is illustrated by the second diagram in Figure 10.7. 168 10. Lefschetz ﬁbrations on 4-manifolds tangle Legendrian tangle Legendrian tangle Legendrian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 10.7. Legendrian link diagram in square bridge position Exercise 10.2.3. Verify that the twisting operation does not change the topology of the underlying 4-manifold. (Hint: See Figure 10.8. In (b) we introduce a cancelling 1-handle/2-handle pair, in order to get (c) we slide the new dotted circle over the old one, then in (d) we slide the strands over 10.2. Lefschetz ﬁbrations on Stein domains 169 the (−1)-framed 2-handle and ﬁnally cancel the 1-handle/2-handle pair.) Determine the change of the surgery coeﬃcients on the components of the link passing through the 1-handle depicted in Figure 10.8(a). (a) (b) (c) (d) (e) −1 −1 −1 −1 −1 Figure 10.8. Introduction of negative twists Next we ignore the dots on the dotted circles for a moment and consider the whole diagram as a link in S 3 . We put this link diagram in a square bridge position as in the previous case (see Figure 10.7) and ﬁnd a torus knot K such that all link components lie on the Seifert surface F of K. Before attaching the 1-handles we isotope each dotted circle in the complement of the rest of the link such that it becomes transverse to the ﬁbers of the ﬁbration S 3 − K → S 1 , meeting each ﬁber only once, see [106] for details. Now for each 1-handle we push the interior of the disk spanned by the dotted circle into D 4 and this becomes a section of (PALF) K . Thus by attaching a 1-handle to D 4 we actually remove a small 2-disk D 2 from 170 10. Lefschetz ﬁbrations on 4-manifolds each ﬁber of (PALF) K , and hence obtain a new PALF on D 4 union a 1- handle. In other words, we extend the open book decomposition on S 3 induced by the torus knot K to an open book decomposition on S 1 S 2 . The boundary of the disk we remove from the ﬁber becomes a component of the binding of the open book decomposition on S 1 S 2 . Notice that this circle becomes a longitudional curve after the surgery on the boundary (induced by attaching the 1-handle) since we swap meridian and longitude by a 0-surgery. After attaching all the 1-handles to D 4 we get an open book decomposition on the connected sum of k copies of S 1 S 2 for some k ∈ N and a new PALF on ♮ k S 1 D 3 such that the regular ﬁber is obtained by removing disjoint small disks from F. Then as in the previous case we can extend our PALF on D 4 ∪ 1-handles to a PALF on D 4 ∪ 1-handles ∪ 2- handles. The global monodromy of the constructed PALF is the product of the monodromy of the torus knot K and right-handed Dehn twists along vanishing cycles corresponding to the 2-handles. Finally, we note that the (p, q) torus knot can be constructed using arbitrarily large p and q. Therefore our construction yields inﬁnitely many pairwise nonequivalent PALF’s, since for chosen p and q the genus of the regular ﬁber will be equal to 1 2 (p −1)(q −1). In the proof of Theorem 10.2.1 we constructed an explicit Lefschetz ﬁbration on a Stein domain which is given by its handle decomposition. The boundary of this PALF has an open book decomposition induced from the ﬁbration and it also acquires a contact structure induced from the Stein domain. It turns out that the induced open book and the contact structure are compatible. In the following we will outline a proof of this fact due to Plamenevskaya [146] which is obtained by a slight modiﬁcation of the proof of Theorem 10.2.1. Recall that in the proof of Theorem 10.2.1 we smoothly isotoped the Legendrian link into a square bridge position in order to put it on the Seifert surface of a torus knot and we forgot about the contact structure. One can modify this construction as follows: For a given Legendrian link L in (R 3 , ξ st ) there exists a surface F containing L such that dα is an area form on F (where α = dz +xdy), ∂F = K is a torus knot which is transverse to ξ st and the components of L do not separate F. The construction of the surface F is identical to the one in the proof above except that we ﬁrst isotope L by a Legendrian isotopy so that in the front projection all the segments have slope (±1) away from the points where L intersects the yz-plane, see Figure 10.9. Then we use narrow strips around these segments as in the proof of Theorem 10.2.1 and connect them by small twisted bands (twisting along with the contact planes) to construct the Seifert surface F 10.2. Lefschetz ﬁbrations on Stein domains 171 of a torus knot K = ∂F. By further isotopies we can ensure that L lies in F, dα is an area form on F and ∂F = K is transverse to ξ st . Now thicken Figure 10.9. Legendrian link diagram this one page F (which carries the Legendrian link L) into a handlebody U 1 which is the union of an interval worth of pages (see Section 5.2) so that dα is an area form on every page. Now we can ﬁber the complementary handlebody U 2 in S 3 with binding K and pages diﬀeomorphic to F since K is a ﬁbered knot. So far we obtained an open book decomposition of S 3 which is expressed as a union of two “half” open books, one of which is compatible with ξ st . We would like to extend the contact structure ξ st to the ﬁbered handlebody U 2 (as some contact structure ξ) so that it is compatible with the open book decomposition on U 2 . This can be achieved (see [146]) by an explicit construction of a contact form on U 2 similar to the one we described in Theorem 9.2.5. Hence we get a contact structure ξ on S 3 which is compatible with our open book decomposition. Since the monodromy of this open book decomposition is a product of right-handed Dehn twists, ξ is Stein ﬁllable and therefore isotopic to ξ st . Moreover, by construction ξ and ξ st coincide on U 1 so that the isotopy between ξ and ξ st can be assumed to be the identity on U 1 . Now we need to show that the contact structures ξ and ξ st are isotopic on U 2 relative to ∂U 2 . Notice that ∂U 2 can be made convex and one can check that the binding K is the dividing set on ∂U 2 . Uniqueness (up to isotopy) of a tight contact structure with such boundary conditions was shown in the proof of Theorem 5.2.1. In summary we proved 172 10. Lefschetz ﬁbrations on 4-manifolds Proposition 10.2.4 (Plamenevskaya, [146]). For a given Legendrian link L in (S 3 , ξ st ) there exists an open book decomposition of S 3 satisfying the following conditions: (1) the contact structure ξ compatible with this open book decomposition is isotopic to ξ st , (2) L is contained in one of the pages and none of the components of L separate F, (3) L is Legendrian with respect to ξ, (4) there is an isotopy which ﬁxes L and takes ξ to ξ st , (5) the surface framing of L (induced by the page F) is the same as its contact framing induced by ξ (or ξ st ). In fact item (5) in the theorem above follows from (1)-(4) by Lemma 10.2.5. Let C be a Legendrian curve on a page of a compatible open book ob ξ in a contact 3-manifold (Y, ξ). Then the surface framing of C (induced by the page) is the same as its contact framing. Proof. Let α be the contact 1-form for ξ such that α > 0 on the binding and dα > 0 on the pages of ob ξ . Then the Reeb vector ﬁeld R α is transverse to the pages (by Exercise 9.2.2) as well as to the contact planes. Hence R α deﬁnes both the surface framing and the contact framing on C. The rest of Plamenevskaya’s argument (including the case with the 1- handles) is the same as the proof of Theorem 10.2.1. In summary, we have an algorithm which constructs an explicit PALF on a Stein domain X. This algorithm also yields an open book decomposition on ∂X which is compatible with the contact structure induced from the Stein domain. Exercise 10.2.6. Find a PALF on D 2 T 2 using the given algorithm. Also ﬁnd an open book decomposition of T 3 which is compatible with the contact structure induced from the Stein domain D 2 T 2 . (Hint: See Figure 12.8 for a Stein structure on D 2 T 2 .) The converse of the above theorem also holds, namely Theorem 10.2.7 (Loi–Piergallini, [104]). Every PALF admits a Stein structure. 10.3. Some applications 173 Proof. We describe the proof given in [7]. Let X be a PALF. We can assume that the boundary of a regular ﬁber is connected by plumbing Hopf bands if necessary. It is clear that X is obtained by a sequence of steps of attaching 2-handles X 0 = D 2 F → X 1 → X 2 → → X n = X, where each X i−1 is a PALF and X i is obtained from X i−1 by attaching a 2-handle to a nonseparating curve C i lying on a ﬁber F ⊂ ∂X i−1 with framing sf(C i )−1. Notice that D 2 F has a Stein structure since it is obtained from D 4 by attaching 1-handles only. Inductively, we assume that X i−1 has a Stein structure and thus ∂X i−1 has an induced compatible contact structure. In [53] it was shown that this induced contact structure agrees with Torisu’s contact structure given in Section 5.2. Let Σ denote the double of a page F of the open book (as in Section 5.2) induced on ∂X i−1 by the PALF. We can assume that Σ is a convex surface which is divided by the binding ∂F. The simple closed curve C i on the convex surface Σ is nonisolating with respect to the dividing curve ∂F since we assumed that C i is a nonseparating curve. Then we apply the Legendrian Realization Principle (cf. Lemma 5.1.18) to make C i Legendrian such that the surface framing sf(C i ) of C i is equal to its Thurston–Bennequin framing (see Exercise 5.1.14). The result follows by Eliashberg’s handle attachment Theorem 8.2.1. Remark 10.2.8. The same proof is valid for homologically trivial (i.e., separating) vanishing cycles except that one needs to apply a fold (see [76]) in that case. A fold introduces convenient additional dividing curves so that the nonisolating condition is quaranteed even for homologically trivial curve. 10.3. Some applications In this section we use ideas developed above to solve certain low dimensional problems. Theorem 10.3.1. If W is a Stein domain then we can embed it into a minimal, closed, symplectic 4-manifold X. Proof. We know that a Stein domain W admits a PALF. By plumbing Hopf bands if necessary, we may also assume that the boundary of the regular ﬁber F is connected. The ﬁbration induces an open book decomposition of ∂W with connected binding ∂F. First we enlarge W to ˆ W by attaching a 174 10. Lefschetz ﬁbrations on 4-manifolds 2-handle along the binding ∂F with framing 0 (with respect to the surface framing) to get a Lefschetz ﬁbration over D 2 with closed ﬁbers. Hence ∂ ˆ W is an ˆ F-bundle over S 1 , where ˆ F denotes the closed surface obtained by capping oﬀ the surface F by gluing a 2-disk along its boundary. Let Γ ˆ F denote the mapping class group of the closed surface ˆ F. Now we can easily extend ˆ W into a Lefschetz ﬁbration X over S 2 with regular ﬁber ˆ F. Let t c 1 t c 2 t c k be the global monodromy of the PALF on W, where c i denotes a simple closed curve on F for i = 1, 2, . . . , n. Then this product (after capping oﬀ the boundary component) can be viewed as a product in Γ ˆ F . We clearly have t c 1 t c 2 t c k t −1 c k t −1 c k−1 t −1 c 1 = 1. By Lemma 15.1.16 we can replace every left-handed Dehn twist by a product of right-handed Dehn twists to obtain a factorization of the identity into a product of right-handed Dehn twists. This factorization gives a Lefschetz ﬁbration X over S 2 (with closed ﬁbers) which admits a symplectic structure by Theorem 10.1.10. We can assume that the genus of F (and therefore of ˆ F) is at least two so that the hypothesis [ ˆ F] ,= 0 ∈ H 2 (X; R) in Theorem 10.1.10 is automatically satisﬁed. Consequently, the Stein domain W is embedded into a closed symplectic 4-manifold X. As we will see, by taking ﬁber sum if necessary we can assume that X is minimal, cf. Proposition 10.3.9. Remark 10.3.2. The embeddability of a Stein domain into a 4-manifold with some extra structure was ﬁrst noticed by Lisca and Mati´c [97]. They proved that for any Stein domain W there is a minimal surface X of gen- eral type such that W embeds into X, i.e., there is a K¨ ahler embedding f : W ֒→X. This observation was used to distinguish homotopic but non- isotopic (Stein ﬁllable) contact structures. The above embedding of a Stein domain into a minimal, closed, symplectic 4-manifold is due to Akbulut and the ﬁrst author [8]. In another direction, we will show that any symplectic ﬁlling embeds into a closed symplectic 4-manifold, see Theorem 12.1.7. The following converse of Theorem 10.3.1 easily follows from Theorem 10.2.7 and Remark 10.2.8; compare also with Corollary 10.1.9. Corollary 10.3.3. Let f : X → S 2 be a Lefschetz ﬁbration which admits a section. Then by removing a neighborhood of the section union a regular ﬁber we get a (positive) Lefschetz ﬁbration which admits a Stein structure. A slightly weaker version of the next theorem is due to Loi and Pier- gallini. This theorem provides a connection between ﬁllability properties of 10.3. Some applications 175 a contact structure and the monodromy of a compatible open book decom- position, cf. Corollary 9.2.15. For deﬁnitions of various ﬁllability notions see Section 12.1. Theorem 10.3.4 (Giroux). A contact 3-manifold (Y, ξ) is Stein ﬁllable if and only if it admits a compatible open book decomposition with mon- odromy h ∈ Γ g,r such that h = t a 1 . . . t an with t a i right-handed Dehn twists along homotopically nontrivial simple closed curves. Each Stein ﬁlling of (Y, ξ) occurs as the Lefschetz ﬁbration corresponding to such a decomposi- tion. The genus of the ﬁbration, however, might change from one ﬁlling to another. Proof. Suppose that h = t a 1 . . . t an for some right-handed Dehn twists along homotopically nontrivial simple closed curves. Then the (positive) Lefschetz ﬁbration with total monodromy h is a Stein ﬁlling of Y . Conversely, any Stein ﬁlling of (Y, ξ) admits a PALF and thus induces an open book decom- position on the boundary which is compatible with the contact structure ξ. Notice, however, that in order to encounter all Stein ﬁllings we might need to stabilize the open book decomposition. Next we give an explicit constructions of some genus-2 Lefschetz ﬁbra- tions. This construction will be used later in our study of Stein ﬁllings of certain contact 3-manifold. Furthermore, these examples show that ﬁber sums of holomorphic Lefschetz ﬁbrations do not necessarily admit complex structures. Theorem 10.3.5 ([131]). There are inﬁnitely many (pairwise nonhomeo- morphic) 4-manifolds which admit genus-2 Lefschetz ﬁbrations but do not carry complex structure with either orientation. Proof. Matsumoto [108] showed that S 2 T 2 #4CP 2 admits a genus-2 Lefschetz ﬁbration over S 2 with global monodromy (t β 1 t β 4 ) 2 , where β 1 , . . . , β 4 are the curves depicted by Figure 10.10. Let B n denote the smooth 4-manifold obtained by the twisted ﬁber sum of the Lefschetz ﬁ- bration S 2 T 2 #4CP 2 → S 2 with itself, using the diﬀeomorphism h n of the ﬁber Σ 2 , where h denotes the right-handed Dehn twist about the curve α which is depicted in Figure 10.11. Then B n admits a genus-2 Lefschetz ﬁbration over S 2 with global monodromy (t β 1 t β 4 ) 2 (t h n (β 1 ) t h n (β 4 ) ) 2 . Standard theory of Lefschetz ﬁbrations gives that π 1 (B n ) = π 1 (Σ 2 )/ ¸ β 1 , . . . , β 4 , h n (β 1 ), . . . , h n (β 4 ) _ , showing that π 1 (B n ) = Z ⊕Z n . 176 10. Lefschetz ﬁbrations on 4-manifolds β 2 β 1 β β 4 3 Figure 10.10. Vanishing cycles α Figure 10.11. The twisting curve α Exercise 10.3.6. Show that the Lefschetz ﬁbration B n → S 2 admits a section. (Hint: Verify the statement for S 2 T 2 #4CP 2 → S 2 and splice the sections together.) The deﬁnition of B n provides a handlebody decomposition for it and shows, in particular, that the Euler characteristic χ(B n ) is equal to 12. Since B n is the ﬁber sum of two copies of S 2 T 2 #4CP 2 , we get that the signature σ(B n ) = −8, consequently b 2 (B n ) = 12, b + 2 (B n ) = 2 and b − 2 (B n ) = 10. Let M n denote the n-fold cover of B n with π 1 (M n ) ∼ = Z. Easy computation shows that b + 2 (M n ) = 2n and b − 2 (M n ) = 10n. This allows us to show that B n does not admit a complex structure (see [131] for details). Next we study the problem of ﬁnding the minimal number of singular ﬁbers a Lefschetz ﬁbration can have. (If we allow achiral ﬁbrations as well, then the answer becomes trivial.) Lemma 10.3.7 ([155]). For a given Lefschetz ﬁbration f : X → Σ there are almost-complex structures J and j on X and Σ resp., such that f is pseudoholomorphic, that is, df ◦ J = j ◦ df. 10.3. Some applications 177 Using Seiberg–Witten theory, in particular Taubes’ results on Seiberg– Witten invariants of closed symplectic 4-manifolds, this observation quickly leads us to the proofs of the following two results: Proposition 10.3.8 ([155]). Suppose that f : X →Σ is a given Lefschetz ﬁbration with g(Σ) > 0. Then the ﬁbration X →Σ is relatively minimal if and only if X as a symplectic 4-manifold is minimal. Proof. One direction of the theorem is obvious: if X contains no (−1)- sphere then the Lefschetz ﬁbration f : X → Σ is necessarily relatively minimal. For the converse direction suppose that X is not minimal. Using Taubes’ result, a (−1)-sphere S can be displaced to be a J-holomorphic submanifold, hence f[ S : S → Σ is a holomorphic map. By the assumption on the genus of Σ it is therefore constant, hence S ⊂ f −1 (p) for some p ∈ Σ, contradicting relative minimality of the ﬁbration f : X →Σ. A similar (but somewhat longer) chase for (−1)-spheres proves Proposition 10.3.9 ([154]). A Lefschetz ﬁbration X → S 2 is relatively minimal if and only if X# f X is minimal. Here are a few corollaries of the above propositions: Corollary 10.3.10. If X → Σ is a relatively minimal Lefschetz ﬁbration and g(Σ) > 0 then c 2 1 (X) ≥ 0. If X → S 2 is relatively minimal then c 2 1 (X) ≥ 4 −4g. Proof. The ﬁrst statement follows from Proposition 10.3.8 and Taubes’ result 13.1.10. For the second statement notice that σ(X# f X) = 2σ(X) and χ(X# f X) = 2χ(X) + 4g − 4, therefore by Proposition 10.3.9 we have that 0 ≤ c 2 1 (X# f X) = 3σ(X# f X) + 2χ(X# f X) = 6σ(X) + 4χ(X) + 8g −8 = 2c 2 1 (X) + 8g −8, which implies the result. Corollary 10.3.11. A genus-g Lefschetz ﬁbration X →S 2 has at least 4 5 g singular ﬁbers. 178 10. Lefschetz ﬁbrations on 4-manifolds Proof (sketch). We can assume that X is relatively minimal. Let n and s denote the number of homologically nontrivial and homologically trivial vanishing cycles, respectively. Then 4 − 4g ≤ c 2 1 (X) = 3σ(X) + 2χ(X) ≤ 3(n − s) + 2(4 − 4g + n + s) = 5n − s + 8 − 8g, implying n ≥ 4 5 g. The inequality σ(X) ≤ n−s (see [130] for example) follows from the fact that a 2-handle attachment can change the signature of the 4-manifold by at most 1, and if the vanishing cycle is homologically trivial, such an attachment reduces the signature. If we allow the base space to have higher genus then the problem of ﬁnding the minimal number of singular ﬁbers in a relatively minimal Lefschetz ﬁbration is almost completely solved, see Chapter 15. 11. Contact Dehn surgery Now we are in the position to describe the contact version of the smooth surgery scheme we started our notes with. This method provides a rich and yet to be explored source of all kinds of contact 3-manifolds. The approach to 3-dimensional contact topology we outline here was initiated by Ding and Geiges [16, 17], see also [18, 19]. Using contact surgery diagrams — and applying achiral Lefschetz ﬁbrations — we will make connection to Giroux’s theory on open book decompositions, and we will also show a way to determine homotopic properties of the contact structures under examination. We begin by reviewing the classiﬁcation of tight structures on S 1 D 2 due to Honda — this is the result which allows us to deﬁne contact surgery diagrams. 11.1. Contact structures on S 1 D 2 The idea of Honda in the classiﬁcation of tight contact structures on the solid torus S 1 D 2 is roughly the following: there is a strong relationship between contact structures on the solid torus S 1 D 2 with certain boundary condition, the thickened torus T 2 [0, 1] with certain related boundary conditions, and on the lens space L(p, q) where p and q depend on the above boundary conditions. Legendrian surgery provides many tight, in fact, Stein ﬁllable contact structures on lens spaces — this gives a lower bound for the number of structures on the solid torus. Using convex surface theory then Honda gives an upper bound for that number, which matches with the lower bound given by the surgeries. This concludes the proof. Here we will show the lower bound in the general case by using Legendrian surgery, and produce a (generally much weaker) upper bound for the number of tight structures on S 1 D 2 . In one particular case, however, our two numbers 180 11. Contact Dehn surgery will match up, giving the classiﬁcation in that case — and this is the case our contact surgery construction will rely on. Let us start by stating the result of Honda. For this, let us assume that for p ≥ q ≥ 1 the rational number − p q is equal to [r 0 , . . . , r k ] where [r 0 , . . . , r k ] denotes the continued fraction expansion of the rational number − p q , i.e., − p q = r 0 − 1 r 1 − 1 r 2 −...− 1 r k with r i ≤ −2 (i = 0, . . . , k) in case p > q. (For p = q = 1 we have k = 0 and r 0 = −1.) Notice that any nonzero slope on the boundary of the solid torus can be transformed into the form − p q with p ≥ q ≥ 1 and (p, q) = 1 by a self- equivalence S 1 D 2 →S 1 D 2 . The case p = 0 needs diﬀerent treatment, see the concluding remark of this section. Now ﬁx relative primes (p, q) with p ≥ q ≥ 1. Theorem 11.1.1 (Honda, [76]). The solid torus S 1 D 2 has exactly ¸ ¸ (r 0 + 1)(r 1 + 1) (r k−1 + 1)r k ¸ ¸ nonisotopic tight contact structures with convex boundary having two dividing curves of slope − p q . Consequently, any nonzero boundary slope can be given as the boundary of a tight contact structure on S 1 D 2 . Remark 11.1.2. Recall that the dividing set on a convex torus in a tight contact structure consists of 2n parallel circles of some common slope r. The above theorem provides the classiﬁcation of tight contact structures on S 1 D 2 with convex boundary having dividing set of two components. For results regarding the general (i.e., n > 1) case see [76] — those results will not be used in this volume. Example 11.1.3. If the boundary slope is 1 n for some n ∈ Z then by a self- diﬀeomorphism we can transform it to − 1 1 = −1, hence k = 0 and r 0 = −1, consequently (up to isotopy) there is a unique tight contact structure on S 1 D 2 with boundary slope 1 n and two dividing curves, see Theorem 5.1.30. Now we turn to the proof of the lower bound of tight structures on the solid torus with ﬁxed boundary condition. As we already saw, L(p, q) can be given as − p q -surgery on the unknot K ⊂ S 3 — and this is equivalent to attaching 4-dimensional 2-handles to D 4 along a chain of (k + 1) unknots with framings r 0 , . . . , r k , cf. Exercise 2.2.7(c). In fact, the unknots can be put in Legendrian position, and since r i < −1, by adding zig-zags we can 11.1. Contact structures on S 1 ×D 2 181 arrange r i = tb(K i ) − 1 to hold. Note that there is a certain freedom in adding the zig-zags to the Legendrian unknot shown by Figure 11.1: in total the zig-zags can be positioned in ¸ ¸ (r 0 + 1) (r k + 1) ¸ ¸ diﬀerent ways. All these choices produce diﬀeomorphic Stein domains with some induced contact structures on the boundary. It is not hard to determine the spin c structures induced by these contact structures, and a direct computation easily shows that the structures are not isotopic. Recall that a ﬁxed diagram induces a Stein structure on the underlying smooth 4-manifold X with complex structure J satisfying (11.1.1) ¸ c 1 (X, J), [Σ i ] _ = rot(K i ) with [Σ i ] ∈ H 2 (X; Z) denoting the homology element corresponding to the knot K i , cf. Chapter 2. Now c 1 (X, J) determines a spin c structure on X and its restriction to ∂X is the spin c structure induced by the contact structure of the surgery diagram. Exercise 11.1.4. Show that the contact structures given by the above Legendrian surgery diagrams on L(p, q) all have diﬀerent spin c structures. There is another way, involving much deeper theory, to distinguish these structures. According to Proposition 8.2.4 (or Equation 11.1.1) the c 1 - invariants of these Stein domains are all diﬀerent, hence Theorem 11.1.5 of Lisca and Mati´c applies: −1 −3 −1 −1 Figure 11.1. Adjusting contact framing by stabilization 182 11. Contact Dehn surgery Theorem 11.1.5 (Lisca–Mati´c, [97]). Suppose that J 1 , J 2 are two Stein structures on a ﬁxed smooth 4-manifold X and ξ 1 , ξ 2 are the induced contact structures on ∂X. If c 1 (X, J 1 ) ,= c 1 (X, J 2 ) then ξ 1 and ξ 2 are not isotopic. Remark 11.1.6. The proof of this statement rests on the fact that a Stein domain can be embedded into a minimal surface of general type, and this 4-manifold has only two Seiberg–Witten basic classes. Two isotopic contact structures with diﬀerent c 1 -invariants would produce more basic classes. An alternative proof was given by Kronheimer and Mrowka [86] using Theorem 13.2.2 and Seiberg–Witten theory, and by Plamenevskaya [146] using Heegaard Floer theory. In conclusion, the ¸ ¸ (r 0 + 1) (r k + 1) ¸ ¸ tight contact structures on L(p, q) are all distinct. In fact, the above mentioned relation between contact structures on L(p, q) and on the solid torus with some ﬁxed boundary condition, together with Theorem 11.1.1 ﬁnishes the classiﬁcation of contact structures on lens spaces: Theorem 11.1.7 (Honda, [76]). Any tight contact structure on L(p, q) is isotopic to one of the structures given as Stein boundaries above. Conse- quently L(p, q) carries ¸ ¸ (r 0 +1) (r k +1) ¸ ¸ nonisotopic tight contact struc- tures — all are Stein ﬁllable. Remark 11.1.8. In fact, for some speciﬁc contact structures on L(p, q) all Stein ﬁllings can be described, see Section 12.3 and [96]. The contact structures covered by the theorem of Lisca are the ones for which all the zig-zags in the diagram are either on the left or on the right — these two structures are actually contactomorphic and universally tight. The link description of the contact structures shows that all these structures contain a Legendrian knot K, the Legendrian realization of the normal circle to, say, the left-most surgery curve in the chain, such that L(p, q) −int νK is a solid torus. We can assume that ∂νK is a convex torus with a two- component dividing set, and by examining the gluing map we can easily see that the slope of the dividing curves on this torus is − p ′ q ′ with pq ′ −qp ′ = 1 when viewed from the complementary solid torus. From this equation we get that − p ′ q ′ has continued fraction representation [r 0 , . . . , r k−1 , r k + 1] if − p q = [r 0 , . . . , r k ]. Since the neighborhood of K is standard, in this way we found ¸ ¸ (r 0 + 1) (r k + 1) ¸ ¸ isotopy classes of tight contact structures on S 1 D 2 with boundary slope − p ′ q ′ , giving the desired lower bound for arbitrary p ′ q ′ . Notice that this lower bound is equal to the number of tight 11.1. Contact structures on S 1 ×D 2 183 contact structures stated in Theorem 11.1.1 (where the statement was given for p q rather than for p ′ q ′ ), and is equal to 1 for slopes of the form 1 n . Now we turn to the derivation of the upper bound for the number of tight structures on S 1 D 2 in general. In doing so we will follow the proof of Theorem 5.1.30. To this end ﬁx a tight contact structure ξ on S 1 D 2 with convex boundary and the ﬁxed boundary slope of the dividing set Γ ∂(S 1 ×D 2 ) on it. Again, we only deal with the case when Γ ∂(S 1 ×D 2 ) has two components. Consider the meridional simple closed curve µ ⊂ ∂(S 1 D 2 ) which becomes homotopically trivial when viewed in S 1 D 2 . Put it into Legendrian position, consider the spanning disk D ⊂ S 1 D 2 with ∂D = µ and isotope this disk into convex position. Since ξ is tight, the dividing set Γ D on D contains no closed components, hence Γ D is equal to a collection of arcs with boundary on ∂D = µ. From the fact that the contact planes rotate in the same direction when travelling around µ it follows that the intersection points Γ D ∩ ∂D and Γ ∂(S 1 ×D 2 ) ∩ ∂D follow each other in an alternating manner, that is, for consecutive intersections x, y ∈ Γ D ∩ ∂D there is a unique z ∈ Γ ∂(S 1 ×D 2 ) ∩ ∂D between x and y and vice versa, cf. Figure 11.2. Since Γ ∂(S 1 ×D 2 ) , and so Γ ∂(S 1 ×D 2 ) ∩ ∂D is given by the boundary condition, the number [Γ D ∩∂D[ and so the number of arcs in Γ D is also ﬁxed. Since there is an upper bound for the possible conﬁgurations of the embedded arcs of Γ D with these boundary conditions, this argument provides an upper bound for the tight contact structures near D in terms of the boundary slope − p q . Since by Eliashberg’s Theorem there is a unique (up to isotopy) tight contact structure on S 1 D 2 − νD = D 3 , the above reasoning provides an upper bound for the number of tight structures on S 1 D 2 with the given boundary condition encoded by the slope − p q of the dividing set Γ ∂(S 1 ×D 2 ) on the boundary. This upper bound is in general far from being sharp. Isotoping the disk D in a ﬁxed contact structure ξ we might get diﬀerent conﬁgurations for Γ D , although the contact structure has not been changed. Honda’s method of manipulating the dividing sets with bypasses yields an equivalence relation among possible conﬁgurations of dividing sets on D and concludes in a sharp upper bound for the number of tight structures on the solid torus, ﬁnishing the proof of Theorem 11.1.1. Note that for p = 1 the above argument already gives 1 as an upper bound, hence veriﬁes Theorem 11.1.1 in this simple case. 184 11. Contact Dehn surgery x y z T 2 Γ D Γ Figure 11.2. Dividing sets Γ ∂(S 1 ×D 2 ) and ΓD Remark 11.1.9. Throughout the argument above we assumed that the boundary slope is diﬀerent from zero. The reason is that there is no tight contact structure on S 1 D 2 with boundary slope zero: in this case ∂ _ ¦pt.¦ D 2 _ can be isotoped to be disjoint from the dividing curves of the boundary, therefore ¦pt.¦D 2 (after being isotoped to have Legendrian boundary) provides an overtwisted disk. 11.2. Contact Dehn surgery 185 11.2. Contact Dehn surgery Now we are in the position to deﬁne a version of Dehn surgery on 3-manifolds adapted to the contact category. The discussion presented here rests on the work of Ding and Geiges [16, 17]. Suppose that K ⊂ (Y, ξ) is a Legendrian knot in a given contact 3-manifold. As we already saw, K comes with a canonical framing, hence we can perform r-surgery on (Y, ξ) along K — the surgery coeﬃcient is measured with respect to the contact framing. In order to see that the surgered manifold Y r (K) also admits a contact structure, we have to describe the surgery procedure a little more carefully. As the Legendrian neighborhood theorem shows, for some positive δ there is a contact embedding f : (N δ , ζ) →(Y, ξ) with f(C) = K where N δ = _ (φ, x, y) [ x 2 +y 2 ≤ δ _ ⊂ S 1 R 2 , ζ = cos(2πnφ) dx −sin(2πnφ) dy and C = _ (φ, x, y) [ x = y = 0 _ , see Example 5.1.23. Let N 2δ = _ (φ, x, y) [ x 2 +y 2 ≤ 2δ _ ⊂ S 1 R 2 . Now we will cut out f(N δ ) ⊂ Y and reglue N 2δ by a diﬀeomorphism g: (N 2δ −int N δ ) →(N 2δ −int N δ ) which maps boundary to boundary and on N 2δ − int N δ ∼ = T 2 I it maps the meridian µ to pµ + qλ. Such a map obviously exists on T 2 , and this can be trivially extended to N 2δ −int N δ . Considering the contact structure ζ 1 = (g ∗ ) −1 (ζ) on N 2δ −int N δ we need the following Proposition 11.2.1. For p ,= 0 the contact structure ζ 1 extends to a tight contact structure ζ ′ on N 2δ . Proof. Using the identiﬁcation given by _ p p ′ q q ′ _ with pq ′ −p ′ q = 1 to glue S 1 D 2 back in, we need to choose the slope on the solid torus to match up with the old longitude, which is isotopic to the 186 11. Contact Dehn surgery dividing curve. Recall that the slope of the boundary of the neighborhood of a Legendrian knot can be assumed to be equal to ∞ by choosing the longitude given by the contact framing. The meridian of S 1 D 2 will map to pµ + qλ. Computing the inverse of the above matrix, the inverse image of the longitude turns out to be −p ′ µ + pλ, hence the slope of the tight contact structure on S 1 D 2 we need should be equal to − p p ′ . According to Theorem 11.1.1 this boundary condition can be fulﬁlled by a tight contact structure on S 1 D 2 once p ,= 0. Now identifying w ∈ N 2δ −int N δ with f _ g(w) _ ∈ Y −f(N δ ) we glue N 2δ to Y −f(N δ ) and get a manifold Y ′ with glued up contact structure ξ ′ . From the construction it is clear that Y ′ = Y p q (K). The contact structure ξ ′ depends on the choice of the extension of ζ 1 to N 2δ . In general this extension is not unique, but — as the classiﬁcation given in the previous section shows — uniqueness holds for p = 1. (For p = 1 we can choose p ′ = 0 and q ′ = 1, hence we have to understand tight structures on S 1 D 2 with slope 1 0 = ∞, which is equivalent to 1 −1 = −1.) Notice also that even if ξ is tight, the resulting structure ξ ′ might be overtwisted. In order to have a well-deﬁned construction one needs to check that the resulting contact structure ξ ′ is (up to isotopy) independent of all the choices made throughout the above gluing process. This is the content of Theorem 11.2.2 (Ding–Geiges, [17]). If the extension ζ 1 in N 2 is ﬁxed then the resulting contact structure ξ ′ on Y ′ is uniquely deﬁned up to isotopy. In particular, if p = 1 then the contact structure ξ ′ on Y ′ is speciﬁed up to isotopy by the Legendrian knot K and q ∈ Z. The construction above allows us to prove numerous classical results in contact topology. For example Corollary 11.2.3 (Martinet, [107]). Every 3-manifold admits a contact structure. Proof. Every closed 3-manifold can be given as rational surgery on a link in S 3 . Put the link into Legendrian position in (S 3 , ξ st ) and recompute the framing coeﬃcients with respect to the contact framing. The previous procedure provides a contact structure on the desired 3-manifold. (By adding zig-zags if necessary we can always avoid contact 0-framings.) 11.2. Contact Dehn surgery 187 Remark 11.2.4. A reﬁned version of this theorem will be discussed in Proposition 11.3.15. From the 4-dimensional point of view, integral surgeries are especially important, since these correspond to 4-dimensional 2-handle attachments. In this sense, contact (±1)-surgery produces both a 4-manifold and a unique contact structure on its boundary. Recall that for (−1)-surgery the result- ing cobordism admits a Stein structure as well. We have already considered a surgery scheme producing (+1)-surgery (with respect to the contact fram- ing) in Section 7.3. It is natural to ask which contact 3-manifold can be presented as contact (±1)-surgery along a Legendrian link in (S 3 , ξ st ). Theorem 11.2.5 (Ding–Geiges, [17]). For any closed contact 3-manifold (Y, ξ) there is a Legendrian link L = L + ∪ L − ⊂ (S 3 , ξ st ) such that contact surgery on L ± with framings (±1) relative to the contact framings provides (Y, ξ). In order to give a short proof of Theorem 11.2.5 we will ﬁrst sharpen our observation of Lemma 7.3.1. Therefore suppose that (Y, ξ) is a given contact manifold, L ⊂ (Y, ξ) is a Legendrian knot and L ′ is its contact push-oﬀ. Perform contact (−1)-surgery on L and (+1)-surgery on L ′ , resulting in the contact manifold (Y ′ , ξ ′ ). Lemma 11.2.6 (The Cancellation Lemma, Ding–Geiges [16]). The contact 3-manifolds (Y, ξ) and (Y ′ , ξ ′ ) are contactomorphic. The contactomorphism can be chosen to be the identity outside of a small tubular neighborhood of the Legendrian knot L. Proof. The complete proof of this useful lemma relies on the solution of the following two exercises. Exercises 11.2.7. (a) Computing contact Ozsv´ath–Szab´o invariants verify that the result of (+1)-surgery along the Legendrian unknot of Figure 4.2(a) is tight. (Hint: For a possible solution see Lemma 14.4.10. A direct argu- ment for the same statement is given in [18]. See also Proposition 11.3.4.) (b) Show that the result of contact (+1)-surgery on the Legendrian unknot and (−1)-surgery on its Legendrian push-oﬀ gives a tight contact structure ξ on S 3 . (Hint: From (a) deduce that the result of the (+1)-surgery is Stein ﬁllable, and conclude that ξ is also Stein ﬁllable, hence tight.) Returning to the proof of Lemma 11.2.6, the idea is as follows: consider a neighborhood N of L containing L ′ . It is easy to see that the two surgeries do 188 11. Contact Dehn surgery not change the topological type (and the gluing map) of this neighborhood, so we only need to see that after surgeries the contact structure on N is tight — this shows that the two surgeries amount to a contact ∞-surgery along L, verifying the statement. By the Legendrian neighborhood theorem this tightness can be checked on a model case, for example if L is the Legendrian unknot in (S 3 , ξ st ). In this case, however, Exercise 11.2.7 shows that the result of (−1)-surgery on L and (+1)-surgery on L ′ embeds into a tight contact S 3 , hence it is tight, completing the proof. Corollary 11.2.8. Suppose that L, L ′ are Legendrian knots in a surgery diagram for (Y, ξ) such that L ′ is the contact push-oﬀ of L and there is a neighborhood of L disjoint from the rest of the diagram and containing L and L ′ only. If we do (−1)-surgery on L and (+1)-surgery on L ′ then the diagram given by the same link after deleting L and L ′ yields the same contact 3-manifold (Y, ξ). Now we can begin the proof of Theorem 11.2.5. We will prove this theorem in two steps. First we reduce the problem to the case of an overtwisted 3-sphere, and then in the next section we ﬁnish the proof by explicit diagrams for those contact 3-manifolds. Proof. (Reduction of Theorem 11.2.5 to an overtwisted S 3 .) Perform con- tact (+1)-surgery on the Legendrian knot L ⊂ (S 3 , ξ st ) pictured by Fig- ure 11.3. It is not hard to see that the result is an overtwisted structure ξ 1 +1 Figure 11.3. Overtwisted contact structure on S 3 on S 3 , see Exercise 11.2.10(a). Now consider (Y, ξ) and take the connected sum with (S 3 , ξ 1 ). The result is an overtwisted (Y, ξ 2 ) which can be given as contact (+1)-surgery along a copy of L in a Darboux chart on (Y, ξ). By Theorem 2.2.5 the 3-manifold Y can be turned into S 3 by a topologi- cal surgery along a link, and since the complement of each knot in (Y, ξ 2 ) can be assumed to be overtwisted, this link can be isotoped to a Legen- drian position with contact framing one less than the framing prescribed by 11.2. Contact Dehn surgery 189 the topological surgery. In conclusion, a sequence of contact (+1)-surgeries turns (Y, ξ) into (S 3 , ξ ′ ) with some contact structure ξ ′ . Adding one more copy of (S 3 , ξ 1 ) to the whole process, the resulting (S 3 , ξ ′ ) can be assumed to be overtwisted. Now reversing the surgeries we get that contact (−1)- surgery on a Legendrian link in some overtwisted contact 3-sphere (S 3 , ξ ′ ) yields (Y, ξ). In conclusion, once we have a surgery presentation for (S 3 , ξ ′ ), we can combine it with the above argument to yield a proof for Theo- rem 11.2.5. (Such diagrams will be given in Lemmas 11.3.10 and 11.3.11, cf. Corollary 11.3.13.) Remark 11.2.9. Combining the above proof with an argument of Etnyre and Honda we can actually assume that the Legendrian link L ⊂ (S 3 , ξ st ) producing (Y, ξ) has only one component on which (+1)-surgery is per- formed. Etnyre and Honda [46] noticed that for any contact 3-manifold (Y, ξ) and overtwisted structure (N, ζ) there is a Legendrian link in (N, ζ) along which contact (−1)-surgery provides (Y, ξ). Using this principle with (N, ζ) = (S 3 , ξ 1 ) given by (+1)-surgery along the knot L of Figure 11.3 we have the above sharpening of Theorem 11.2.5. Exercises 11.2.10. (a) Show that the contact structure ξ 1 we get by per- forming (+1)-surgery on the Legendrian knot of Figure 11.3 is overtwisted. (Hint: Consider the Legendrian knot L shown by Figure 11.4. Show that it L +1 Figure 11.4. Boundary of an overtwisted disk in the diagram bounds a disk in the surgered manifold, and compare the contact framing on L with the one induced by this disk, cf. [18].) (b) Using the same idea, verify that contact (+1)-surgery on the stabiliza- tion of a Legendrian knot results in an overtwisted contact structure. The original proof of Ding and Geiges for Theorem 11.2.5 followed slightly diﬀerent lines. In [17] they worked out a way for turning contact rational surgeries into contact (±1)-surgeries. Since this method is very useful in 190 11. Contact Dehn surgery applications, below we describe the algorithm — for the proof the reader is advised to turn to [17, 18]. Let us ﬁrst assume that we want to perform contact r-surgery on the Legendrian knot L with r < 0. In this case the surgery can be replaced by a sequence of contact (−1)-surgeries along Legendrian knots associated to L as follows: suppose that r = − p q and the continued fraction coeﬃcients of − p q are equal to [r 0 + 1, r 1 , . . . , r k ], with r i ≤ −2 (i = 0, . . . , k). Consider a Legendrian push-oﬀ of L, add [r 0 + 2[ zig-zags to it and get K 0 . Push this knot oﬀ along the contact framing and add [r 1 + 2[ zig-zags to it to get K 1 . Do contact (−1)-surgery on K 0 and repeat the process with K 1 . After (k + 1) steps we end up with a diagram involving only contact (−1)-surgeries. According to [17, 18] the result of the sequence of (−1)-surgeries is the same as the result of the original r-surgery. Remark 11.2.11. Recall that for generic r, contact r-surgery is not unique: there is a ﬁnite set of tight structures on S 1 D 2 with the correct boundary slope. This non-uniqueness is present in the sequence of (−1)-surgeries as well: we have a freedom in adding the zig-zags in each step either on the right or on the left. It is not very hard to see that there are equally many choices in both constructions. The next proposition will guide us how to turn contact r-surgery with r > 0 into a sequence of contact (±1)-surgeries. Proposition 11.2.12 (Ding–Geiges, [17]). Fixt r = p q > 0 and an integer k > 0. Then contact r-surgery on the Legendrian knot K is the same as contact 1 k -surgery on K followed by contact p q−kp -surgery on the Legendrian push-oﬀ K ′ of K. By choosing k > 0 large enough, the above proposition provides a way to reduce a contact r-surgery (with r > 0) to a 1 k - and a negative r ′ -surgery. This latter one can be turned into a sequence of (−1)-surgeries, hence the algorithm is complete once we know how to turn contact 1 k -surgery into (±1)-surgeries. Lemma 11.2.13 (Ding–Geiges, [17]). Let K 1 , . . . , K k denote k Legendrian push-oﬀs of the Legendrian knot K. Contact 1 k -surgery on K is then isotopic to performing contact (+1)-surgeries on the k Legendrian knots K 1 , . . . , K k . Exercises 11.2.14. (a) Verify that the above algorithm is correct on the topological level, that is, the algorithm provides a surgery presentation of a 3-manifold diﬀeomorphic to the result of the given r-surgery. 11.3. Invariants of contact structures given by surgery diagrams 191 (b) Notice that in applying Proposition 11.2.12 the choice of k ∈ N is not unique. Show that after applying the Cancellation Lemma 11.2.6 suﬃciently many times the resulting diagram will be independent of the choice of k. (c) Show that for any contact 3-manifold (Y, ξ) there is (Y ′ , ξ ′ ) and a Stein cobordism W between Y and Y ′ such that H 1 (Y ′ ; Z) = 0. (Hint: Start with a contact surgery diagram L of (Y, ξ) and for every knot L i in L consider a Legendrian knot K i linking L i once, not linking the other knots in L, and having tb(K i ) = 1. Adding Weinstein handles along K i we get W; check that the resulting 3-manifold Y ′ after the handle attachment is an integral homology sphere. For more details see [159].) (d) Find an open book decomposition of # k (S 1 S 2 ) compatible with the standard contact structure. 11.3. Invariants of contact structures given by surgery diagrams In this section we show how one can read oﬀ homotopic data of a contact structure given by a contact surgery diagram. Suppose that (Y, ξ) is the re- sult of contact (±1)-surgery on the Legendrian link L = L + ∪L − ⊂ (S 3 , ξ st ). Recall that integral surgery can also be regarded as (4-dimensional) 2-handle attachment to D 4 , hence the diagram represents a compact 4-manifold X with ∂X = Y . There is, however, an additional structure on X. It is fairly easy to see that the surgery diagram for (Y, ξ) gives an achiral Lef- schetz ﬁbration on the 4-manifold X: just repeat the algorithm of Akbulut and the ﬁrst author outlined in Section 10.2. (Also take the reﬁnement of Plamenevskaya [146] given in Proposition 10.2.4 into account.) Recall that an achiral Lefschetz ﬁbration on X naturally provides an open book de- composition ob L on ∂X = Y . Next we would like to show that ξ (as the result of contact (±1)-surgeries) on Y is compatible with this open book decomposition. Notice that this step will complete a portion of the proof of Giroux’s Theorem 9.2.11 about relating open book decompositions and contact structures. Let ξ L denote the contact structure (unique up to iso- topy by Part(a) of Theorem 9.2.11, see also Proposition 9.2.7) compatible with the open book decomposition ob L . Our main result is now Theorem 11.3.1. The contact structures ξ and ξ L on Y are isotopic, hence ξ is compatible with the open book decomposition ob L deﬁned above. 192 11. Contact Dehn surgery In the light of Theorem 5.2.1 of Torisu we would like to show that both ξ and ξ L admit the properties listed under (i) and (ii) of that theorem. This is obviously satisﬁed (and explicitly stated in [168]) for ξ L , hence we only need to verify them for ξ. Lemma 11.3.2 ([160]). The restrictions ξ[ U i to the handlebodies U i (i = 1, 2) of the Heegaard decomposition induced by the open book de- composition ob L are tight. The dividing set of the convex surface Σ ⊂ Y with respect to ξ is isotopic to the binding of the open book decomposition. Proof. Consider the open book decomposition found on S 3 induced by the Lefschetz ﬁbration D 4 →D 2 in the course of the algorithm presented in Section 10.2. Recall that this Lefschetz ﬁbration is given by the factorization of the monodromy of the (p, q) torus knot deﬁned by the knot in square bridge position. Since the monodromy of this open book decomposition is the product of right-handed Dehn twists only, the corresponding contact structure is isotopic to ξ st . In addition, this open book decomposition induces a Heegaard decomposition of S 3 , and the contact handlebodies of this Heegaard decomposition — since they are contained by the tight S 3 — are tight. The Heegaard decomposition S 3 = V 1 ∪ V 2 can be chosen in such a way that L is contained in V 1 . Therefore (ii) of the assumptions of Theorem 5.2.1 obviously holds, since surgery along L will not change the convex surface ∂V 1 = ∂V 2 , and the binding of the open book decomposition remains unchanged. We only need to check (i), that is, that the contact structures ξ[ U i are tight for i = 1, 2. By our choice U 2 = V 2 and ξ[ U 2 = ξ st [ V 2 , hence we only need to deal with ξ[ U 1 . Consider Legendrian push-oﬀs for all Legendrian knots in L + in such a way that these push-oﬀs are in V 2 . This can be done, since the contact framings of the knots in L coincide with the page framing they inherit from the open book decomposition. Therefore a contact push-oﬀ can be assumed to lie on a page, and this page can be chosen to be in V 2 . Doing the prescribed surgeries along the knots of L and contact (−1)-surgeries on these push-oﬀs we get a contact 3-manifold (Y ′ , ξ ′ ) which contains ξ[ U 1 . It is easy to see that (Y ′ , ξ ′ ) is tight: by the Cancellation Lemma 11.2.6 it can be given by doing (−1)-surgery along L − ⊂ (S 3 , ξ st ), therefore (Y ′ , ξ ′ ) is Stein ﬁllable, hence tight. Since ξ[ U 1 is contained by a tight 3-manifold, it is tight, concluding the proof. Proof (of Theorem 11.3.1). By [168] and Lemma 11.3.2 both ξ L and ξ satisfy conditions (i) and (ii) of Theorem 5.2.1, hence the theorem implies that ξ and ξ L are isotopic. 11.3. Invariants of contact structures given by surgery diagrams 193 Remark 11.3.3. A similar theorem was proved by Gay [53] in the case when no (+1)-surgeries are present in the picture. Now Theorem 11.3.1 allows us to ﬁnd open book decompositions for all contact structures given by contact (±1)-surgery diagrams. Notice also that we just proved that every 3-manifold admits an open book decomposition: presenting Y as the boundary of a 4-dimensional handlebody with a unique 0-handle and some 2-handles, we get a contact surgery diagram of Y with some contact structure. Turn this diagram into (±1)-surgeries and apply the above theorem to ﬁnd an open book decomposition on Y . (This operation will change the 4-dimensional handlebody, though.) As an easy application we show that Proposition 11.3.4. Contact (+1)-surgery on the Legendrian unknot pro- vides a tight structure on S 1 S 2 . Proof. After performing the algorithm given in Section 10.2 we get an achiral Lefschetz ﬁbration X → D 2 with ﬁber diﬀeomorphic to the Seifert surface of the (2, 2) torus knot, i.e., the annulus A. The 4-manifold X is built from the Lefschetz ﬁbration D 4 → D 2 by attaching a 2-handle along the central circle C of this annulus, see Figure 11.5. Since the C A Figure 11.5. The vanishing cycle C on the annulus A monodromy of the (2, 2) torus knot is equal to the right-handed Dehn twist t C along C, the total monodromy of the induced open book decomposition on ∂X = S 1 S 2 is equal to t C t −1 C = 1. The reason for the negative exponent on the second Dehn twist is that we need to do (+1)-surgery, corresponding to a left-handed Dehn twist in the monodromy. Therefore, according to Theorem 11.3.1 the contact structure we get by (+1)-surgery 194 11. Contact Dehn surgery on the Legendrian unknot is compatible with the open book decomposition deﬁned by the identity element 1 ∈ Γ A . Corollary 9.2.15 now implies that it is Stein ﬁllable, hence the proof is complete. In addition, by the classiﬁcation of tight contact structures on S 1 S 2 the above argument also shows that the surgery described above is the same as the boundary of the Stein 1- handle. Exercises 11.3.5. (a) Verify the Cancellation Lemma 11.2.6 using open book decompositions. (Hint: Notice that the curve L ′ can be given as the push-oﬀ of L on a page. Then the Dehn twists corresponding to L and L ′ cancel in the monodromy, giving the result.) (b) Prove Proposition 11.3.4 using contact Ozsv´ath–Szab´o invariants. (Hint: Use Lemma 14.4.10.) Prove tightness for the contact structure given by (+1)-surgery along the k-component Legendrian unlink. (c) Show that any solid genus-g handlebody admits a contact structure which can be embedded into a Stein ﬁllable structure on a closed 3-manifold. Our next application concerns computability of homotopic invariants of contact structures on a 3-manifold Y . Recall form Chapter 6 (cf. also [65]) that two oriented 2-plane ﬁelds ξ 1 and ξ 2 on Y are homotopic if and only if their induced spin c structures t ξ i and 3-dimensional invariants d 3 (ξ i ) are equal. If c 1 (t ξ ) is nontorsion then d 3 (ξ) does not admit a Q-lift, but for c 1 (t ξ ) torsion, this latter invariant can be lifted to Q and can be computed as 1 4 _ c 2 1 (X i , J i ) −3σ(X i ) −2χ(X i ) _ where (X i , J i ) are almost-complex 4-manifolds with ∂X i = Y such that the oriented 2-plane ﬁelds of complex tangencies of J i along ∂X i are homotopic to ξ i . The surgery picture together with Theorem 11.3.1 easily provides such a 4-manifold X: Suppose that (Y, ξ) is given by (±1)-surgery on L = L + ∪ L − ⊂ (S 3 , ξ st ), and let X 1 denote the 4-manifold deﬁned by the diagram. As explained in Section 10.2, X 1 admits an achiral Lefschetz ﬁbration structure. Consider the oriented 2-plane ﬁeld of tangents of ﬁbers away from the set C of critical points of the ﬁbration. By taking the orthogonal complement for some metric, this oriented 2-plane ﬁeld provides an almost- complex structure on X 1 − C: deﬁne J as counterclockwise 90 ◦ rotation on these planes. This almost-complex structure obviously extends through those points of C which admit orientation preserving complex charts — just use the local model. At points of C with oppositely oriented coordinate charts (corresponding to contact (+1)-surgeries) the two branches of the 11.3. Invariants of contact structures given by surgery diagrams 195 oriented singular ﬁber provide an orientation for X incompatible with the one originally ﬁxed. The obstruction for extending J through such points can be computed using a local model, as explained in [66, Lemma 8.4.12] or in [160]. In conclusion, for these points of C we need to take the connected sum of X 1 with CP 2 with its standard complex structure for extending the almost-complex structure deﬁned on X 1 −C. Consequently X = X 1 #qCP 2 with the extended almost-complex structure is a good choice of (X, J) for the given contact structure (Y, ξ). Here q denotes the number of components in L + . By repeating the proof of [65, Proposition 2.3] verbatim (see also Proposition 8.2.4) we get Theorem 11.3.6 (Gompf, [65]). The ﬁrst Chern class c 1 (X, J) ∈ H 2 (X; Z) of the resulting almost-complex structure evaluates on the 2-homology de- ﬁned by the surgery curve K as its rotation number rot(K). Since ξ is isotopic to the oriented 2-plane ﬁeld of complex tangencies along Y = ∂X, the cohomology class c 1 (ξ) is equal to the restriction of the above c 1 (X, J) to ∂X. The class c 1 (X, J) is speciﬁed by Theorem 11.3.6, and the description of H 1 (Y ; Z) in terms of a surgery diagram then provides c 1 (ξ). Note that here X is simply connected, hence the spin c structure s J induced by J is speciﬁed by c 1 (X, J). In this way the induced spin c structure t ξ is given as s J [ ∂X . If c 1 (ξ) ∈ H 2 (Y ; Z) is torsion, then for appropriate n ∈ N the class PD _ nc 1 (X, J) _ ∈ H 2 (X, ∂X; Z) is the image of a class α ∈ H 2 (X; Z), hence c 2 1 (X, J) can be computed as 1 n 2 α 2 ∈ Q as discussed in Section 6.3. Notice also that both c 1 (X 1 − C, J) and the induced spin c structure s J extend uniquely through the points of C, hence for practical purposes we can work with this extended cohomology class c ∈ H 2 (X 1 ; Z), although it is not the ﬁrst Chern class of any almost-complex structure. When computing d 3 (ξ), this fact results a correction term in the formula. In conclusion, for ξ with torsion induced spin c structure t ξ all terms in the formula for d 3 (ξ) can be easily computed once ξ is given by a surgery diagram. This leads to Theorem 11.3.7 ([160]). Suppose that the contact 3-manifold (Y, ξ) is given by contact (±1)-surgery along the link L = L + ∪ L − ⊂ (S 3 , ξ st ). Let X 1 denote the 4-manifold deﬁned by the diagram and suppose that c ∈ H 2 (X 1 ; Z) is given by c _ [Σ K ] _ = rot(K) on [Σ K ] ∈ H 2 (X 1 ; Z), where Σ K is the surface corresponding to the surgery curve K ⊂ L. If the restriction c[ ∂X 1 to the boundary is torsion and L + has q components then d 3 (ξ) = 1 4 _ c 2 −3σ(X 1 ) −2χ(X 1 ) _ +q. 196 11. Contact Dehn surgery Proof. Since σ(X 1 ) = σ _ X 1 − ¦x 1 , . . . , x k ¦ _ and χ(X 1 ) = q + χ _ X 1 − ¦x 1 , . . . , x q ¦ _ for the critical points ¦x 1 , . . . , x q ¦ of the achiral Lefschetz ﬁbration X 1 → D 2 which lie on incorrectly oriented coordinate charts, the formula easily follows. Next we show an alternative way for verifying the above formula, cf. [18]. This method works only for knots with nonzero Thurston–Bennequin in- variants, but conceptually it is simpler — for example, it makes no use of the achiral Lefschetz ﬁbration or the open book decomposition provided by the surgery diagram. As shown in Chapter 8, the complex structure of D 4 (inherited from C 2 ) extends to all the 2-handles attached with contact framing −1. We do not have such an extension for the contact (+1)-framed 2-handles, but there is no obstruction to ﬁnding an appropriate almost- complex structure on these handles away from a point. In conclusion, we have an almost-complex structure on X 1 − ¦x 1 , . . . , x q ¦ where q is the car- dinality of L + — the knots on which we do contact (+1)-surgery. Since a spin c structure (like a 2-cohomology element) extends through a point in a 4-manifold, we have a spin c structure s on X 1 extending the spin c structure t ξ ∈ Spin c (Y ) induced by ξ. We want to determine c 1 (s) on the homo- logy classes given by the Legendrian knots in L. For this computation, ﬁx L ⊂ (S 3 , ξ st ), perform contact (+1)-surgery on it and consider the resulting 4-manifold X L with spin c structure s L ∈ Spin c (X). Let k denote the value of c 1 (s L ) on a generator for H 2 (X L ; Z). (To be precise, we need to ﬁx an orientation for L, which provides a canonical generator for H 2 (X L ; Z) ∼ = Z.) Deﬁne u as the obstruction to extending the almost-complex structure from X L − ¦pt.¦ to X L , i.e., the 3-dimensional invariant of the oriented 2-plane ﬁeld induced on the boundary S 3 of the neighborhood of the point is u. Proposition 11.3.8 ([18, 99]). If tb(L) ,= 0 then k = rot(L) and u = 1 2 . Proof. Consider 2n Legendrian push-oﬀs of L and call them L 1 , . . . , L n and L ′ 1 , . . . , L ′ n . Do contact (−1)-surgeries along L i and (+1)-surgeries along L ′ i . According to the Cancellation Lemma 11.2.6 the result is (S 3 , ξ st ) again. On the other hand, simple homological computation shows that the 3-dimensional invariant of the result of the surgery is 1 4 (n _ k 2 −rot(L) _ −n 2 tb(L) _ k −rot(L) _ 2 −2) +n _ u − 1 2 _ − 1 2 . Since d 3 (S 3 , ξ st ) = − 1 2 , the above expression implies u = 1 2 and k = rot(L) provided tb(L) ,= 0. 11.3. Invariants of contact structures given by surgery diagrams 197 Remark 11.3.9. In fact, we need to use the above expression for n = 1 and n = 2 only to draw that conclusion. Note that since u can be easily shown to be independent of L, for tb(L) = 0 the above argument gives k = ±rot(L); a more detailed study of the almost-complex structure on X L −¦pt.¦ actually proves that k = rot(L) in this case as well, see also [18]. In most cases, however, the proof for the tb(L) ,= 0 case is suﬃcient. The formula of Theorem 11.3.7 above gives us a way to distinguish contact structures given by surgery diagrams on a ﬁxed 3-manifold. For example, let L be n unlinked copies of the knot given by Figure 11.3, and take (Y, ξ n ) to be (+1)-surgery on L. Simple computation veriﬁes Lemma 11.3.10. Y = S 3 and d 3 (ξ n ) = n − 1 2 . Proof. By turning the contact framing coeﬃcients to Seifert framings, we see that Y is given by (−1)-surgery on the n-component unlink, hence we can blow all surgery curves down, showing that Y = S 3 . The corresponding 4- manifold X 1 therefore has σ(X 1 ) = −n, χ(X 1 ) = n+1 and since L = L + , we have q = n. Easy computation shows that c 2 = −n; by plugging these values into the formula of Theorem 11.3.7 the proof of the lemma is complete. A similar quick calculation shows Lemma 11.3.11. n geometrically disjoint copies of the link of Figure 11.6 provide a sequence of contact structures ξ −n on S 3 with d 3 (ξ −n ) = −n − 1 2 . Proof. Figure 11.6 shows that the manifold we get after the surgery is diﬀeomorphic to S 3 . Application of the formula for the 3-dimensional invariant d 3 now implies the result. Since S 3 admits a unique tight contact structure ξ st and d 3 (ξ st ) = − 1 2 , the contact structures ξ n for n ∈ Z−¦0¦ encountered above are all overtwisted. Exercises 11.3.12. (a) By ﬁnding the overtwisted disks show directly that the contact structures (S 3 , ξ n ) _ n ∈ Z−¦0¦ _ of the above two Lemmas are overtwisted. (b) Find an overtwisted contact structure ξ 0 on S 3 homotopic to ξ st . (Hint: Take the connected sum of ξ 1 and ξ −1 .) (c) Show that the contact structures on L(3, 1) given by Figures 11.7(a) and (b) are not isotopic but contactomorphic. (Hint: Compute the spin c structures induced by the contact structures. Verify that reﬂection induces a contactomorphism.) (d) Find open books compatible with the contact structures given by Fig- ures 11.7(a) and (b). 198 11. Contact Dehn surgery +1 −1 smoothly −5 −1 −1 −1 Figure 11.6. The contact 3-manifold (S 3 , ξ−1) −1 −1 (a) (b) Figure 11.7. Contactomorphic, nonisotopic contact structures 11.3. Invariants of contact structures given by surgery diagrams 199 The map (S 3 , ξ) → d 3 (ξ) + 1 2 gives a bijection between the space of over- twisted contact structures and Z. To see this we only need to verify that for any oriented 2-plane ﬁeld ξ on S 3 the quantity d 3 (ξ) + 1 2 is an integer. Recall that d 3 (ξ) = 1 4 _ c 2 1 (X, J) −σ(X) _ − 1 2 _ σ(X) +χ(X) _ for an appropriate simply connected almost-complex 4-manifold (X, J). The expression 1 4 _ c 2 1 (X, J) − σ(X) _ is an even integer since c 1 (X, J) is a characteristic vector, and 1 2 _ σ(X) + χ(X) _ = b + 2 (X) − 1 2 . Therefore we have Corollary 11.3.13. The above lemmas together with Exercise 11.3.12(b) show surgery diagrams for all overtwisted contact structures ξ n (n ∈ Z) on the 3-sphere. Notice that this corollary concludes the proof of Theorem 11.2.5. Exercise 11.3.14. Consider a Legendrian knot L ⊂ (S 3 , ξ st ) and its Leg- endrian push-oﬀ L ′ . Stabilize L ′ twice to get L 1 and perform contact (+1)- surgery on L and L 1 . Determine the resulting 3-manifold Y and compute d 3 (ξ) for the resulting contact structure ξ. (Hint: See [19].) Following similar lines, in fact, we can produce surgery diagrams for all overtwisted contact structures on any 3-manifold presented by a surgery diagram. This presentation (given in [18]) provides a new proof of a classical result of Lutz and Martinet: Proposition 11.3.15 (Lutz–Martinet, [105]; cf. also [18]). For a given 3- manifold Y and oriented 2-plane ﬁeld ξ ∈ Ξ(Y ) there is a contact structure homotopic to ξ. The contact structure can be chosen to be overtwisted. Exercises 11.3.16. (a) Let L 0 ⊂ (S 3 , ξ st ) be the Legendrian unknot and L 1 another Legendrian unknot linking it k times (k ∈ Z). Add two zig-zags to the Legendrian push-oﬀ L ′ 2 of L 1 and get L 2 . Perform contact (+1)-surgery on L 0 , L 1 and L 2 . Prove that the resulting manifold is diﬀeomorphic to S 1 S 2 . Determine the spin c structure of the resulting contact structure ξ. (b) Using Exercise 11.3.14 and the above result verify Proposition 11.3.15 for S 1 S 2 . (c) Prove Proposition 11.3.15 in general. (Hint: See [18].) 200 11. Contact Dehn surgery Recall that according to Eliashberg’s result, isotopy classes of overtwisted contact structures and homotopy classes of oriented 2-plane ﬁelds are in one-to-one correspondence. Therefore the solution of Exercise 11.3.16(c) provides a surgery diagram for any overtwisted contact structure on a closed 3-manifold. −1 ..... ..... +1 −1 −1 −1 m −1 Figure 11.8. Contact structure ξ −(2m+1) on S 3 Exercise 11.3.17. Show that the contact surgery diagram depicted in Figure 11.8 gives a contact structure on S 3 with d 3 = 1 2 − 2(m + 1), where m ≥ 0 is the number of unknots in the ﬁgure with vanishing rotation number. (Hint: Compute d 3 and use the classiﬁcation of overtwisted contact structures.) Notice that this surgery diagram represents some overtwisted contact structures on S 3 using unknotted surgery curves and only one (+1) surgery curve. This example also illustrates (Stein) cobordisms between various contact structures. 12. Fillings of contact 3-manifolds This chapter is devoted to the study of ﬁllability properties of contact 3- manifolds. After having the necessary deﬁnitions we will see diﬀerent types of ﬁllings, and give a family of tight, nonﬁllable contact structures. The construction of these latter examples utilizes contact surgery, while tightness is proved by computing contact Ozsv´ath–Szab´o invariants (see Chapter 14). In the last section we will concentrate on topological restrictions a contact 3-manifold imposes on its Stein ﬁllings. 12.1. Fillings Deﬁnition 12.1.1. A given contact 3-manifold (Y, ξ) is weakly symplecti- cally ﬁllable (or ﬁllable) if there is a compact symplectic manifold (W, ω) such that ∂W = Y (as oriented manifolds) and with this identiﬁcation ω[ ξ does not vanish. In this case we say that (W, ω) is a symplectic ﬁlling. (W is oriented by the volume form ω ∧ ω, while the orientation of Y is the one compatible with ξ.) (Y, ξ) is strongly symplectically ﬁllable if it is the ω-convex boundary of a compact symplectic manifold (W, ω). In other words, ω is exact near the boundary and its primitive α (i.e., a 1-form with dα = ω) can be chosen in such a way that ker _ α[∂W _ = ξ. Yet another for- mulation of strong ﬁlling is to require a transverse, symplectically dilating vector ﬁeld for the boundary (deﬁned near ∂X) pointing outwards. (Y, ξ) is holomorphically ﬁllable if there is a compact complex surface (X, J) such that the contact structure on ∂X given by the complex tangencies is con- tactomorphic to (Y, ξ). Finally, (Y, ξ) is Stein ﬁllable if it is the J-convex boundary of a Stein surface. 202 12. Fillings of contact 3-manifolds Remarks 12.1.2. (a) Without imposing the compactness condition on W, the above deﬁnition of (weak or strong) symplectic ﬁllability would be satisﬁed by all closed contact 3-manifold (Y, ξ): Consider simply Y (0, 1] equipped with the symplectic structure it inherits from the symplectization of (Y, ξ). (b) According to a result of Bogomolov, the complex structure on a holo- morphic ﬁlling can be deformed such that (X, J ′ ) becomes the blow-up of a Stein ﬁlling. Therefore the two last notions of ﬁllability in Deﬁnition 12.1.1 are the same. (c) Notice that holomorphic/Stein ﬁllability implies strong ﬁllability, which in turn implies weak ﬁllability. For a related discussion on various ﬁllability notions see [30]. Notice that a symplectic 4-manifold (W, ω) is by deﬁnition a strong sym- plectic ﬁlling if its boundary ∂W is ω-convex. Recall that by results of Chapter 7 we can attach Weinstein handles to a strong symplectic ﬁlling along Legendrian knots in a way that the symplectic structure extends to the handle and the new symplectic 4-manifold strongly ﬁlls its boundary. In this gluing process, however, the symplectically dilating vector ﬁeld is used only in a neighborhood of the attaching circle. It turns out that if L ⊂ (Y, ξ) is Legendrian and (W, ω) is a weak ﬁlling of (Y, ξ) then there is always a symplectically dilating vector ﬁeld near L, implying Theorem 12.1.3 ([16]). Suppose that (Y ′ , ξ ′ ) is given by contact (−1)- surgery along L ⊂ (Y, ξ). If (Y, ξ) is weakly ﬁllable then so is (Y ′ , ξ ′ ). It is known that there are weakly ﬁllable contact structures which are not strongly ﬁllable: for example, the contact tori (T 3 , ξ n ) with n ≥ 2 all have this property [29]. (For an even larger collection of such contact 3-manifolds see [16].) It is still unknown whether strong ﬁllability implies Stein ﬁllability. Of course one can modify a Stein ﬁlling in such a way that it does not admit a Stein structure anymore, but such an operation does not aﬀect ﬁllability properties of the boundary contact 3-manifold. Example 12.1.4. The Legendrian surgery diagram of Figure 12.1 gives a strong (in fact, Stein ﬁlling) of the boundary of the nucleus N n with the inherited contact structure. Suppose that (W, ω) is a weak ﬁlling of (Y, ξ). It is obvious that if ω is not exact near ∂W = Y then (W, ω) is not a strong ﬁlling. The exactness of ω, however, enables us to modify ω near the boundary in such a way that it 12.1. Fillings 203 . . . . . . . . . . zig−zags n−k zig−zags k −1 −1 Figure 12.1. Stein structure on the nucleus Nn becomes a strong ﬁlling, see [30]. In the special case of rational homology spheres therefore we have Theorem 12.1.5 (Ohta–Ono, [127]). Suppose that b 1 (Y ) = 0. The sym- plectic structure ω on a weak symplectic ﬁlling W of any contact structure ξ on Y can be extended to W ∪ Y [0, 1] to a strong ﬁlling of (Y, ξ). In conclusion, a contact structure on a rational homology sphere Y is weakly ﬁllable if and only if it is strongly ﬁllable. According to a recent result of Eliashberg [30] a weak ﬁlling can be symplectically embedded into a closed symplectic 4-manifold. This theorem turned out to be of central importance in recent studies of contact invariants, see [87, 143]. Here we prove this theorem in two steps. Theorem 12.1.6. If (W, ω) is a strong ﬁlling of (Y, ξ) then W can be embedded into a closed symplectic 4-manifold. 204 12. Fillings of contact 3-manifolds Proof. Consider a surgery presentation L = L + ∪ L − ⊂ (S 3 , ξ st ) of (Y, ξ), and let K denote the Legendrian link we get by considering Legendrian push- oﬀs of the knots of L + . Attaching Weinstein handles to (W, ω) along the knots of K we get a strong ﬁlling (W ′ , ω ′ ) of a contact 3-manifold (Y ′ , ξ ′ ). Notice that by the Cancellation Lemma 11.2.6 this latter contact manifold can be given as Legendrian surgery along L − , consequently it is Stein ﬁllable (although (W ′ , ω ′ ) might not be a Stein ﬁlling of it). Consider a Stein ﬁlling (X, J) of (Y ′ , ξ ′ ) and embed this ﬁlling into a closed symplectic 4-manifold (Z, ω Z ) as explained in Section 10.3. Performing symplectic cut-and-paste (as in Theorem 7.1.9) along Y ′ ⊂ Z we get a symplectic structure on the closed 4-manifold U = (Z − int X) ∪ Y ′ W ′ . Since (W, ω) is a symplectic submanifold of (W ′ , ω ′ ), this provides a symplectic embedding of (W, ω) into the closed symplectic 4-manifold U. Notice that by adding more Weinstein handles we can make sure that b + 2 (W ′ −W) and so b + 2 (U) is at least 2. Surprisingly enough, from this point the embeddability of a weak symplectic ﬁlling follows by a trivial argument. Theorem 12.1.7 (Eliashberg, [30, 42]). If (W, ω) is a weak symplectic ﬁlling of (Y, ξ) then (W, ω) embeds symplectically into a closed symplectic 4-manifold (U, ω U ). Proof. According to Exercise 11.2.14(c) the weak symplectic ﬁlling embeds ﬁrst into a weak symplectic ﬁlling (W ′ , ω ′ ) such that the boundary Y ′ = ∂W ′ is an integral homology sphere. Now Theorem 12.1.5 provides a way to modify ω ′ near ∂W ′ to achieve that the new symplectic form ω 1 provides a strong symplectic ﬁlling of (Y ′ , ξ ′ ). The application of Theorem 12.1.6 now provides a symplectic embedding of (W ′ , ω 1 ) into a closed symplectic 4-manifold, and since (W, ω) is a symplectic submanifold of (W ′ , ω 1 ), the proof is complete. It is still a question of central importance in contact topology whether a given contact structure is ﬁllable or not (in any of the above sense) and which 3-manifolds support ﬁllable contact structures. The previous chapters provided a very powerful topological tool for con- structing Stein manifolds: attach 2-handles to ♮ n S 1 D 3 along a Leg- endrian link with framing −1 relative to the contact framing. (Here ∂(♮ n S 1 D 3 ) = # n S 1 S 2 is equipped with its unique tight contact struc- ture.) In fact, every Stein domain can be given in this way. This approach has been systematically studied by Gompf in [65]; he showed, for example 12.1. Fillings 205 Theorem 12.1.8 (Gompf, [65]). Every Seifert ﬁbered 3-manifold M = M(g, n; r 1 , . . . , r k ) with one of its orientations admits a Stein ﬁllable contact structure. If g ≥ 1 then M admits Stein ﬁllable contact structures with either of its orientations. According to a result of Eliashberg, Stein ﬁllability needs to be determined only for prime 3-manifolds: Proposition 12.1.9. The connected sum (Y 1 , ξ 1 )#(Y 2 , ξ 2 ) is Stein ﬁllable if and only if both (Y i , ξ i ) are Stein ﬁllable. According to a theorem of Eliashberg and Gromov, a ﬁllable contact structure (in any of the above sense) is tight. Theorem 12.1.10 (Eliashberg–Gromov). A weakly symplectically ﬁllable contact 3-manifold (Y, ξ) is tight. Proof (sketch). Let (W, ω) be a symplectic ﬁlling of (Y, ξ) and suppose that (Y, ξ) contains an overtwisted disk. Choose a disk D with Legendrian boundary and the property that tb(∂D) = 2, and attach a Weinstein handle along ∂D to the weak ﬁlling (W, ω). The resulting 4-manifold W ′ will be a weak symplectic ﬁlling of the surgered contact 3-manifold (Y ′ , ξ ′ ) containing a sphere with self-intersection (+1). Now embed (W ′ , ω ′ ) into a closed symplectic 4-manifold U with b + 2 (U) > 1. The adjunction inequality of Theorem 13.3.3 for the sphere of positive self-intersection now provides the desired contradiction. Remark 12.1.11. The ﬁrst proof of the above theorem is due to Eliashberg and Gromov [32], and used completely diﬀerent ideas and methods. The above result might lead one to expect that all tight contact struc- tures are ﬁllable in some sense. Until recently, however, it was very hard to ﬁnd counterexample to this expectation, since the only tool for proving tightness of a given (Y, ξ) was to show that it is ﬁllable. The state traversal method tightness, and this method led to the discovery of the ﬁrst tight but not ﬁllable contact structures [44]. The introduction of Ozsv´ath–Szab´o invariants then gave a very eﬀective way for examining tightness properties of contact structures on closed manifolds, leading to a plethora of examples of tight nonﬁllable contact structures. 206 12. Fillings of contact 3-manifolds 12.2. Nonfillable contact 3-manifolds Not all contact structures are ﬁllable and there are examples of 3-manifolds which do not admit any symplectic ﬁllings. Theorem 12.2.1 (Lisca, [94]). The Poincar´e homology 3-sphere with its natural orientation reversed admits no ﬁllable contact structure. Proof. The Poincar´e homology sphere is diﬀeomorphic to the Brieskorn sphere Σ(2, 3, 5), hence the oriented 3-manifold of the theorem is equal to −Σ(2, 3, 5). Suppose W is a ﬁlling of −Σ(2, 3, 5), and embed it into a closed symplectic manifold (as it is given in Theorem 10.3.1). The fact that Σ(2, 3, 5) admits a positive scalar curvature metric implies that b + 2 (W) = 0, cf. Proposition 13.1.7(5.). Now if E stands for the positive deﬁnite E 8 - plumbing given by the plumbing graph of Figure 1.5 then W ∪ (−E) is a negative deﬁnite closed 4-manifold with nonstandard intersection form, contradicting Donaldson’s famous diagonalizability result [20]. Therefore W cannot exist. Exercise 12.2.2. Show that if _ Z n , (−E n ) _ (n = 6, 7, 8) is a sublattice of a negative deﬁnite lattice (Z k , Q) then Q cannot be diagonal. (Hint: Notice that −E 6 is contained in all these lattices. For a solution see [95].) A similar argument shows that the boundary of the positive deﬁnite E 6 - and E 7 -plumbing cannot be the boundary of a Stein domain. Notice that in the light of Proposition 12.1.9 we have many 3-manifolds which are not boundaries of Stein domains — just take connected sum with one of the above mentioned nonﬁllable manifolds. For example, the 3-manifold Σ(2, 3, 5)# _ − Σ(2, 3, 5) _ is not a Stein boundary with either orientation. The result of Theorem 12.2.1 was not suﬃcient for producing a tight, non- ﬁllable contact structure, since by a result of Etnyre and Honda [45] the oriented 3-manifold −Σ(2, 3, 5) actually does not support any tight struc- ture are all. Probably the simplest tight, nonﬁllable contact 3-manifold (Y, ξ) is given by the contact surgery diagram of Figure 1.6. Notice that as a smooth 3- manifold Y is just (+2)-surgery on the right-handed trefoil (=−Σ(2, 3, 4)). In the light of Exercise 12.2.2 the proof of Theorem 12.2.1 shows that Y supports no ﬁllable contact structures, hence Figure 1.6 must deﬁne a nonﬁllable structure. In the proof of tightness we will make use of the contact Ozsv´ath–Szab´o invariants. For an overview of these invariants and 12.2. Nonﬁllable contact 3-manifolds 207 Ozsv´ath–Szab´o homology see Appendix 14; here we will freely use the result discussed there. Recall that ¯ HF(Y ) denotes the Ozsv´ath–Szab´o homology group of the closed, oriented 3-manifold Y , while c(Y, ξ) ∈ ¯ HF(−Y ) is the contact invariant of (Y, ξ). Proposition 12.2.3. The contact 3-manifold (Y, ξ) given by Figure 1.6 has nonvanishing contact Ozsv´ath–Szab´o invariants, hence is tight. Proof. Since (Y, ξ) is deﬁned as contact (+1)-surgery along a single knot, according to Theorem 14.4.5 the contact invariant c(Y, ξ) can be given as F W _ c(S 3 , ξ st ) _ , where W is the cobordism of the handle attachment with reversed orientation. Therefore injectivity of F W gives the nonvanishing of the invariant. The cobordism W can be given by a single 2-handle attachment along the left-handed trefoil knot with framing −2. Denote the left-handed trefoil by T. Then the surgery exact triangle reads as ¯ HF(S 3 ) ¯ HF _ S 3 −2 (T) _ ¯ HF _ S 3 −1 (T) _ F W Since ¯ HF _ S 3 −n (T) _ = ¯ HF _ S 3 n (T) _ , the genus of T is 1 and S 3 5 (T) is a lens space, Propostion 14.3.5 implies that dim ¯ HF(S 3 −n (T)) = n, hence the above triangle translates to Z 2 Z 2 ⊕Z 2 Z 2 F W therefore exactness implies the injectivity of F W , concluding the proof. Exercises 12.2.4. (a) Show that S 3 5 (T) is a lens space. (Hint: Use the presentation of S 3 1 (T) as plumbing on the positive deﬁnite E 8 -diagram and truncate its long leg, cf. also Exercise 2.3.5(f).) (b) Using the result of the above proposition ﬁnd a tight contact structure on the boundary of the positive deﬁnite E 6 -plumbing. (Hint: Take the dia- gram of Figure 1.6 with surgery coeﬃcient (+1) on the right-handed trefoil, consider the Legendrian push-oﬀ of it, add a zig-zag and perform contact (−1)-surgery on the resulting knot. Verify that the resulting manifold is the boundary of the positive deﬁnite E 6 -plumbing using Kirby calculus.) 208 12. Fillings of contact 3-manifolds (c) Show that if (Y K , ξ K ) is given by contact (+1)-surgery on (Y, ξ) along a Legendrian knot K and (Y, ξ) is not ﬁllable then (Y K , ξ K ) is not ﬁllable either. (Hint: Remember that contact (+1)-surgery along a Legendrian knot can be cancelled by contact (−1)-surgery along its Legendrian push- oﬀ, so (Y, ξ) can be given as (−1)-surgery along some Legendrian knot in (Y K , ξ K ), cf. Theorem 12.1.3.) This observation leads us to a family of nonﬁllable contact structures. Con- sider k Legendrian push-oﬀs of the right-handed trefoil and perform con- tact (+1)-surgery on each component, resulting in the contact 3-manifold (Y k , ξ k ). According to the above exercise these structures are all nonﬁllable. Exercise 12.2.5. Show that Y k can be given by the surgery diagram of Figure 12.2. Conclude that ¸ ¸ H 1 (Y k ; Z) ¸ ¸ = k + 1. (Hint: Convert the copies k . . . . . . k 1 +1 +1 . . . Figure 12.2. Tight, nonﬁllable contact 3-manifold (Y k , ξ k ) 12.2. Nonﬁllable contact 3-manifolds 209 Legendrian surgery diagram into a smooth diagram and slide the trefoils over each other.) Proposition 12.2.6 ([101]). The contact Ozsv´ath–Szab´o invariant of (Y k , ξ k ) is nonzero, hence it is a tight contact structure for any k ∈ N. Proof. The proof proceeds by induction; for k = 0 the contact structure is just (S 3 , ξ st ) and for k = 1 we can apply Proposition 12.2.3. Notice that (Y k+1 , ξ k+1 ) is given as contact (+1)-surgery on (Y k , ξ k ), giving rise to a map F W : ¯ HF(−Y k ) → ¯ HF(−Y k+1 ) with the property that F W _ c(Y k , ξ k ) _ = c(Y k+1 , ξ k+1 ). As the surgery diagram of Figure 12.3 shows, the third manifold in the corresponding surgery triangle is S 3 −1 (T) again. Since dim ¯ HF(−Y k ) ≥ ¸ ¸ H 1 (Y k ; Z) ¸ ¸ = k + 1 and ¯ HF(−Y 0 ) = Z 2 , the triangle ¯ HF(−Y k ) ¯ HF(−Y k+1 ) Z 2 F W shows that ¯ HF(−Y k ) = Z k+1 2 for all k ∈ N and F W is injective. Therefore by induction c(Y k+1 , ξ k+1 ) ,= 0, ﬁnishing the proof. For related results see [101]. The above results might give the impression that nonﬁllability can follow only from some strong topological properties of the underlying 3-manifold, and nonﬁllability must hold for all contact structures on a given manifold at the same time. Below we discuss a family of examples of 3-manifolds admitting both ﬁllable and tight nonﬁllable structures. Let Y n,g → Σ g denote the circle bundle with Euler number n over the genus−g surface Σ g . Honda [77] gave a complete classiﬁcation of tight contact structures on these 3-manifolds. He showed that all tight structures are ﬁllable, with the exception of one for n = 2g > 0 and two for n > 2g > 0. Using Seiberg– Witten gauge theory it has been veriﬁed that these exceptional structures are, in fact, nonﬁllable: Theorem 12.2.7 ([99]). Suppose that g > 0 and n ≥ 2g. Then the virtually overtwisted contact circle bundles Y n,g given in [77] are not sym- plectically ﬁllable. Extending the classiﬁcation results of Honda to Seifert ﬁbered 3-manifolds, Ghiggini [60] classiﬁed tight contact structures on the Seifert ﬁbered 3- manifolds of type M(1, n; r) (cf. Chapter 2 for conventions). Through a 210 12. Fillings of contact 3-manifolds −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −1 −1 −2 −2 −2 = −Y k−1 k+1 = Y −Y = k k −2 −2 −2 1 k+ Figure 12.3. Kirby calculus in the surgery triangle sequence of exercises we show a proof of Theorem 12.2.7 in the simplest possible case: when g = 1 and n = 2. Then we show some examples of tight nonﬁllable structures on the type of Seifert ﬁbered 3-manifolds for which the classiﬁcation result of Ghiggini holds. Exercises 12.2.8. (a) Show that the surgery diagram of Figure 12.4 gives a contact structure on Y 2,1 . (Hint: Turn contact surgery coeﬃcients into Seifert framings, put dots on the 0-framed unknots and compare the result with the diagram of Figure 2.11. In doing so one might need to apply the transformation of Figure 10.8.) 12.2. Nonﬁllable contact 3-manifolds 211 +1 +1 +1 Figure 12.4. Tight nonﬁllable contact circle bundle (b) Verify that ξ is nonﬁllable. (Hint: Consider the diagram without the two Legendrian unknots. Verify that it gives a contact structure on −Σ(2, 3, 4). Finally show that contact (+1)-surgery on a nonﬁllable contact structure produces a nonﬁllable structure, cf. Exercise 12.2.4(c).) (c) By computing the contact Ozsv´ ath–Szab´o invariants of the contact structure deﬁned by the surgery diagram of Figure 12.4, show that it is tight. (Hint: Use Lemma 14.4.10 and the result of Exercise 14.3.11(c).) 212 12. Fillings of contact 3-manifolds (d) Verify that Figure 12.5 gives the same contact structure as deﬁned by Figure 12.4. (Hint: Show that the neighborhood of K in Figure 12.5 containing the linking Legendrian knots K 1 and K 2 remains tight after the surgeries on K 1 and K 2 . Since it is glued with the same framing as (+1)- surgery on K, the solution follows from uniqueness of contact surgery with coeﬃcient of the form 1 k .) K 2 1 K K +1 +1 +1 +1 −1 Figure 12.5. Another surgery diagram for the same structure as in Figure 12.4 12.2. Nonﬁllable contact 3-manifolds 213 (e) Show that the contact structures ξ α (2 ≤ α integer) deﬁned by Fig- ure 12.6 on the 3-manifolds Y α are all tight. Notice that there are α many choices to turn the (−α)-surgery into a (−1)-surgery by adding zig-zags to the knot. The proof of nonﬁllability of the contact structures encountered in Exer- cise 12.2.8(e) above requires more theory, and relies on the following theo- rem: Theorem 12.2.9 ([98]). Suppose that ξ is a contact structure on Y with induced spin c structure t ξ ∈ Spin c (Y ) such that the Seiberg–Witten moduli space / Y (t ξ ) is a smooth manifold consisting of reducible solutions only. Then any weak symplectic ﬁlling W of (Y, ξ) satisﬁes b + 2 (W) = 0 and the map H 2 (W; R) →H 2 (∂W; R) induced by the inclusion ∂W ⊂ W is zero. Remark 12.2.10. The crux of the argument is that with such moduli space the Seiberg–Witten equations over the 3-manifold admit a perturbation with no solutions, and such perturbation can be extended to the symplectic ﬁlling unless the topological properties listed in the theorem hold for the ﬁlling W. But an extension would imply vanishing SW (W,ξ) -invariants for a weak symplectic ﬁlling, contradicting Theomem 13.2.2 of Kronheimer and Mrowka. Exercise 12.2.11. Determine the spin c structure induced by ξ α . By applying results of Mrowka, Ozsv´ ath and Yu [123] the solution of the above exercise can be used to verify that the assumptions of Theorem 12.2.9 do hold for the contact structures ξ α . Notice that the surgery description involves several contact (±1)-surgeries and one contact (−α)-surgery. This latter surgery, however, is not unique. By introducing zig-zags on the corresponding Legendrian unknot it can be turned into Legendrian (−1)- surgery, but there are many diﬀerent ways to put these zig-zags on the knot. Diﬀerent choices can be distinguished by the resulting rotation numbers. Exercises 12.2.12. (a) Show that the 3-manifold of Figure 12.6 is dif- feomorphic to the Seifert ﬁbered 3-manifold M(1, 2; 1 α−1 ). (Hint: Recall deﬁnitions from Section 2.3 and perform handleslides. Notice that all the surgery coeﬃcients are given with respect to the contact framing; ﬁrst con- vert those into surgery coeﬃcients with respect to the Seifert framing.) (b) Verify that for any n there exists α ∈ N such that among the contact structures of Figure 12.6 with that ﬁxed α there are at least n noncontac- tomorphic. (Hint: Determine c 1 of the resulting contact structures with 214 12. Fillings of contact 3-manifolds +1 +1 +1 +1 −1 −α Figure 12.6. Tight nonﬁllable stuctures on Seifert ﬁbered manifolds the help of the diagram and compute the order of the ﬁrst Chern class, see [100].) 12.3. Topology of Stein ﬁllings 215 (c) Show that the manifold −Y α is the boundary of a negative deﬁnite manifold with intersection form containing −E 8 . (Hint: Embed the 4- manifold given by the surgery diagram into a blown-up CP 2 and compute the intersection form of the complement.) Theorem 12.2.13. The contact structures ξ α deﬁned by Figure 12.6 are tight and nonﬁllable. Proof. The same idea as in the proof of Theorem 12.2.1 now shows that the contact structures ξ i (i = 1, . . . , α − 1) given by Figure 12.6 on Y α are nonﬁllable. Since these structures can be given as contact (−1)-surgery on the contact structure given by Figure 12.5 and this latter structure has nonvanishing contact Ozsv´ath–Szab´o invariants, tightness of (Y α , ξ i ) follows from Corollary 14.4.8. From the solution of Exercise 12.2.12(b) now follows Corollary 12.2.14 ([100]). For any n ∈ N there is a 3-manifold Y n with at least n pairwise noncontactomorphic tight contact structures, none of them weakly symplectically ﬁllable. Notice that by the work of Gompf all these manifolds admit Stein ﬁllable contact structures. For related results see [101]. 12.3. Topology of Stein fillings We switch perspective now, and instead of examining ﬁllability properties of 3-manifolds, we study topological properties of the ﬁllings. The motivating problem of this section can be summarized as: Problem 12.3.1. Fix a contact 3-manifold (Y, ξ) and describe all Stein ﬁllings of (Y, ξ). Remark 12.3.2. Similar questions for weak (or strong) ﬁllings are not expected to have nice answers in general. The reason is that a weak (strong) ﬁlling can be blown up without destroying the ﬁlling property. In addition, if the ﬁlling contains symplectic submanifolds (e.g., a symplectic torus with self-intersection 0) then by taking symplectic normal sums we can change the topology of the ﬁlling drastically. In some cases (when no such submanifolds are present) we might be able to describe the classiﬁcation of weak ﬁllings (up to blow-up), as it is given for lens spaces and links of certain surface singularities, see Remark 12.3.8 below. 216 12. Fillings of contact 3-manifolds Let us begin the study of Stein ﬁllings by a simple observation. If W is a Stein ﬁlling of Y then π 1 (Y ) → π 1 (W) is surjective since W can be built on Y [0, 1] by attaching 2-, 3- and 4-handles only; in particular b 1 (W) ≤ b 1 (Y ). In the following we will list contact 3-manifolds for which Stein ﬁllings have been determined (up to diﬀeomorphism). We start with a famous result of Eliashberg. Theorem 12.3.3 (Eliashberg). A Stein ﬁlling of S 3 with its standard contact structure ξ st is diﬀeomorphic to D 4 . Proof. Let us ﬁx a Stein ﬁlling W of S 3 . By considering a neighborhood of a point p ∈ CP 2 together with a Liouville vector ﬁeld and using the symplectic cut-and-paste operation we get that Z = W ∪ S 3 (CP 2 −D 4 ) is a symplectic 4-manifold. Notice that CP 2 −D 4 and so Z contains a symplectic sphere with square (+1). Standard gauge theory (cf. Proposition 13.1.7(5.) and (6.)) shows that b + 2 (W) = b − 2 (W) = 0. By our observation above we also get that π 1 (W) = 1, therefore Z is homotopy equivalent, hence (by a theorem of McDuﬀ) symplectomorphic to CP 2 . In CP 2 , however, two symplectic spheres representing the generator of H 2 (CP 2 ; Z) are isotopic, showing that W is diﬀeomorphic to CP 2 −CP 1 = D 4 . Using roughly the same line of reasoning as in the proof of Theorem 12.3.3, McDuﬀ showed that the lens space L(p, 1) with the contact structure ξ st it inherits from (S 3 , ξ st ) admits a unique (up to diﬀeomorphism) Stein ﬁlling for p ,= 4, which can be given as (−1)-surgery on a Legendrian unknot with tb = p−1 and rot = 2−p. For other L(p, q)’s (still with the quotient of the standard contact structure (S 3 , ξ st )) Lisca [96] gave a complete description of Stein ﬁllings — in general, however, uniqueness fails to hold. Exercises 12.3.4. (a) Verify that the boundary of the Kirby diagram of Figure 12.7 is L(4, 1). (Hint: Blow down the (−1)-framed unknot.) (b) Replace the 0-framing in Figure 12.7 by a dot and verify that the resulting 4-manifold is the complement of a quadric in CP 2 . Equip this 4-manifold with a Stein structure. (c) Show that both the disk bundle over S 2 with Euler class −4 and the complement of the quadric in CP 2 provide Stein ﬁllings of L(4, 1) for some contact structures. By determining their homotopy types, show that the two contact structures coincide. In fact, the above two distinct examples comprise a complete list of Stein ﬁllings for (L(4, 1), ξ st ) up to diﬀeomor- phism. 12.3. Topology of Stein ﬁllings 217 −1 0 Figure 12.7. 4-manifold with lens space boundary Remark 12.3.5. For the ﬁxed lens space L(p, q) consider the continued fraction expansion [b 1 , . . . , b k ] of p p−q . Elements of the set Z p,q = _ (n 1 , . . . , n k ) ∈ Z k [ [n 1 , . . . , n k ] = 0 and 0 ≤ n i ≤ b i _ give rise to Stein ﬁllings of (L(p, q), ξ st ) as follows: First consider a linear chain of k unknots framed by n i (providing W with ∂W = S 1 S 2 ) and on that do Legendrian surgery on unknots linking the circles of the chain — there are b i − n i such circles linking the circle with framing n i . Surgering out the 4-manifold W given by the linear chain, i.e., replacing W with a 0-handle and a 1-handle we get a Stein ﬁlling of L(p, q). The determination of the homotopy type of the contact structure on the boundary shows that what we constructed are ﬁllings of ξ st (cf. the classiﬁcation result of Honda in Section 11.1). Now Lisca proves that any Stein ﬁlling of this contact lens space is diﬀeomorphic to one of the manifolds constructed above. This last step is carried out by embedding a ﬁlling into a rational surface and showing that the complement is standard, similar to the argument presented in the proof of Theorem 12.3.3. For further details see [96]. Using the same main ideas as above, Ohta and Ono described Stein ﬁllings of links of simple and simple elliptic singularities (again with speciﬁc contact structures). All these ﬁllings happened to have b + 2 = 0 and could be embedded into rational or ruled surfaces. In particular: Theorem 12.3.6 (Ohta–Ono, [127]). The Poincar´e homology 3-sphere Σ(2, 3, 5) (with its contact structure inherited from S 3 ) admits a unique (up to diﬀeomorphism) Stein ﬁlling which is the negative deﬁnite E 8 -plumbing. The same uniqueness holds for the boundary of the negative deﬁnite E 6 - and E 7 -plumbings. 218 12. Fillings of contact 3-manifolds Remark 12.3.7. In [127] it was shown that simple (or ADE) singularities with the contact structure given by the link of the singularity admit a unique Stein ﬁlling (up to diﬀeomorphism). For a simple elliptic singularity L k — which is topologically a circle bundle of Euler class −k < 0 over the 2- torus T 2 — with the contact structure given by the link of the singularity it has been proved [128] that (i) L k admits a strong symplectic ﬁlling X with c 1 (X) = 0 if and only if 0 < k ≤ 9 and such X (which can be regarded as the generalization of smoothing) is unique up to diﬀeomorphism unless k = 8, when there are two possibilities, and (ii) for k ≥ 10 a minimal strong symplectic ﬁlling is unique up to diﬀeomorphism, and such a ﬁlling is diﬀeomorphic to the minimal resolution. Finally for k ≤ 9 a minimal ﬁlling is diﬀeomorphic either to the minimal resolution or to a smoothing (i.e., a ﬁlling with c 1 = 0). The proofs of the above statements given in [127, 128] use Seiberg–Witten theory. Remark 12.3.8. Above we considered only Stein ﬁllings of the given con- tact 3-manifold. This is, however, not the greatest generality for most of the cases discussed: if Y is a rational homology 3-sphere then any weak ﬁll- ing can be deformed into a strong ﬁlling by Theorem 12.1.5, and for strong ﬁllings the same cut-and-paste argument works. The key common feature of the above results is that in each case any ﬁlling can be embedded into a closed symplectic 4-manifold with κ = −∞. In particular, all the above ﬁllings have b + 2 = 0. Next we will describe some particular cases when the above approach fails (for example, because of the existence of ﬁllings with b + 2 > 1). Using ad hoc arguments of embedding Stein ﬁllings of T 3 and ±Σ(2, 3, 11) into homotopy K3-surfaces one gets strong constraints on the topology of such Stein manifolds [158]. These methods, however, seem to be insuﬃcient in greater generality. For example, the contact structures on the 3-torus T 3 have been classiﬁed by Kanda and Giroux [62, 79] by showing that any (T 3 , ξ) is contactomorphic to one of (T 3 , ξ n ) where ξ n = ker _ cos(2πnt) dx + sin(2πnt) dy _ (n ≥ 1). Using delicate results of Gromov, Eliashberg showed [29] that (T 3 , ξ n ) is not strongly ﬁllable once n ≥ 2. For n = 1, Figure 12.8 gives a Stein ﬁlling of (T 3 , ξ 1 ). Using a version of the cut-and-paste argument outlined above, one can show Proposition 12.3.9 ([158]). If W is a Stein ﬁlling of (T 3 , ξ 1 ) then W is homeomorphic to T 2 D 2 . 12.3. Topology of Stein ﬁllings 219 0 Figure 12.8. Stein structure on D 2 ×T 2 For the understanding of Stein ﬁllings of ±Σ(2, 3, 11) the Ozsv´ath–Szab´o homology groups of these manifolds seem to play a crucial role. First we discuss a slightly more general result, since by embedding Stein ﬁllings into symplectic 4-manifolds and applying product formulae for Ozsv´ath–Szab´o invariants, one can show Theorem 12.3.10. Suppose that Y is a rational homology sphere with ¯ HF(Y, t) = Z 2 for t ∈ Spin c (Y ). If W is a Stein ﬁlling of (Y, ξ) such that t ξ = t then b + 2 (W) = 0. If ¯ HF(Y, t) = Z 3 2 then for any Stein ﬁlling W with b + 2 (W) > 0 we have c 1 (W) = 0. Remark 12.3.11. The proof of this statement falls aside from the main topic of these notes, and we do not present it here. We just note that the proof rests on the embeddability of Stein ﬁllings into Lefschetz ﬁbrations. The fact ¯ HF(Y, t) = Z 2 is equivalent to HF red (Y, t) = 0, while ¯ HF(Y, t) = Z 3 2 is the same as HF red (Y, t) = Z 2 . Computation shows that ¯ HF _ ± Σ(2, 3, 11) _ ∼ = Z 3 2 , hence a Stein ﬁlling of it with b + 2 > 0 has c 1 = 0. Surgery on the K3-surface together with the homeomorphism characterization of the K3-surface using Seiberg–Witten invariants due to Morgan and Szab´ o given in Theorem 3.3.11 provides: 220 12. Fillings of contact 3-manifolds Proposition 12.3.12. If W is a Stein ﬁlling of −Σ(2, 3, 11) then b 2 (W) = 2. If W is a Stein ﬁlling of Σ(2, 3, 11) then either b + 2 (W) = 0 or b 2 (W) = 20. Proof (sketch). It is not very hard to prove that −Σ(2, 3, 11) does not bound negative deﬁnite 4-manifold: there exists a compact 4-manifold N 2 with ∂N 2 = −Σ(2, 3, 11) such that three disjoint copies of N 2 are embedded in K3 in a way that the intersection form of K3−3 N 2 is 2E 8 . (For a Kirby diagram of N 2 see Figure 2.14.) If ∂X = −Σ(2, 3, 11) and X is negative deﬁnite then the closed negative deﬁnite 4-manifold (K3 − 3 N 2 ) ∪ 3 X would contradict Donaldson’s diagonalizability theorem [20]. If W is a Stein ﬁlling of −Σ(2, 3, 11) then for Z = (K3 −N 2 ) ∪W a suitable product formula of the Seiberg–Witten invariants and Theorem 3.3.11 implies that Z is homeomorphic to K3. This concludes the proof of the ﬁrst statement. The same reasoning shows that if W is Stein with ∂W = Σ(2, 3, 11) and b + 2 (W) > 0 then the intersection form Q W is equal to 2E 8 ⊕2H. (For more details see [158].) In fact, it is reasonable to conjecture that if W is a Stein ﬁlling of −Σ(2, 3, 11) then it is diﬀeomorphic to N 2 , and a Stein ﬁlling of Σ(2, 3, 11) is diﬀeomor- phic either to the smoothing or to the resolution of the isolated singularity ¦x 2 +y 3 +z 11 = 0¦ ⊂ C 3 — similar to the case of simple and simple elliptic singularities. Notice that in the above arguments we did not make use of the particular choice of the contact structures on ±Σ(2, 3, 11). A recent re- sult of Ghiggini and Sch¨ onenberger [59] classiﬁes tight contact structures on certain Seifert ﬁbered spaces — including ±Σ(2, 3, 11). According to these results, −Σ(2, 3, 11) admits (up to isotopy) a unique tight contact structure, which can be given as the boundary of the Stein domain of Figure 12.9. The Seifert ﬁbered space Σ(2, 3, 11) admits exactly two (nonisotopic) tight con- tact structures, both Stein ﬁllable. Returning to Problem 12.3.1, we might ask what can we say about Stein ﬁllings in general. According to the next result we cannot expect a ﬁnite list as a solution of Problem 12.3.1, since Proposition 12.3.13 ([132]). For g ≥ 2 the element ∆ 2 g ∈ Γ g,1 admits inﬁnitely many decompositions into right-handed Dehn twists with the cor- responding Lefschetz ﬁbrations having distinct ﬁrst homologies. Conse- quently, the contact 3-manifold given by ∆ 2 g through the corresponding open book decomposition admits inﬁnitely many distinct Stein ﬁllings. 12.3. Topology of Stein ﬁllings 221 −1 −1 Figure 12.9. Stein structure on the nucleus N2 In the theorem ∆ g ∈ Γ g,1 denotes the right-handed Dehn twist along a simple closed curve parallel to the unique boundary component of Σ g,1 , cf. also discussion in Chapter 15. Proof. Consider the Lefschetz ﬁbration we get by taking the desingu- larization of the double branched cover of Σ h S 2 along two copies of Σ h ¦∗¦ ⊂ Σ h S 2 and two (four for even g) copies of ¦∗¦S 2 ⊂ Σ h S 2 . The ﬁbration map can be given by perturbing the composition of the branched cover map with the projection to the second factor. It is easy to see that the resulting ﬁbration has a section of square −1, hence gives a factorization of ∆ g ∈ Γ g,1 , cf. Section 15.2. Taking a twisted ﬁber sum of two copies of this ﬁbration we get factorizations of ∆ 2 g ∈ Γ g,1 . The twisting can be chosen in such a way that the resulting 4-manifolds have diﬀerent torsion in their ﬁrst homologies, cf. the proof of Theorem 10.3.5. Now taking the complement of a section and a regular ﬁber we get Lefschetz ﬁ- brations over D 2 with nonclosed ﬁbers, hence inﬁnitely many Stein ﬁllings of the contact 3-manifold given by ∆ 2 g . The ﬁllings are distinguished by the torsion of their ﬁrst homologies. We close this section with a general result concerning the topology of Stein ﬁllings: 222 12. Fillings of contact 3-manifolds Theorem 12.3.14 ([159]). For a given contact 3-manifold (Y, ξ) there exists a constant K (Y,ξ) such that if W is a Stein ﬁlling of (Y, ξ) then K (Y,ξ) ≤ 3σ(W) + 2χ(W). In other words, the number c(W) = 3σ(W) + 2χ(W) for a Stein ﬁlling W of (Y, ξ) — which resembles the c 2 1 -invariant of a closed complex surface — is bounded from below. Remark 12.3.15. The idea of the proof is roughly as follows: Suppose that W 1 , . . . , W n , . . . is the possibly inﬁnite list of Stein ﬁllings of (Y, ξ). Consider the K¨ ahler embeddings W i → X i where X i are minimal surfaces of general type. Our aim is to control the topology of W i . So ﬁx a ﬁlling W 1 and consider T 1 = X 1 − int W 1 . Now for any other Stein ﬁlling W of (Y, ξ) we can form Z = T 1 ∪ W, and according to Theorem 7.1.9 this is a symplectic 4-manifold (with b + 2 (Z) > 1). Therefore minimality of Z would imply c 2 1 (Z) ≥ 0, giving the desired lower bound for 3σ(W) + 2χ(W) in terms of invariants of the ﬁxed 4-manifold T 1 . Minimality of Z is, however, hard to prove — although it seems to be true —, so rather we have to use a larger (still ﬁnite) set of test manifolds X i −int W i to compare the Stein ﬁlling W with. Also we may relax the minimality requirement by trying to prove that the number of blow-ups contained in the symplectic 4-manifold Z is bounded by some number depending only on (Y, ξ). In the computation the mod 2 reduced version of Seiberg–Witten theory is used; for details see [159]. Notice that K (Y,ξ) ≤ 3σ(W) + 2χ(W) can be rewritten as b − 2 (W) +C (Y,ξ) ≤ 5b + 2 (W) where C (Y,ξ) is another constant depending only on the contact 3-manifold (Y, ξ). In other words if b + 2 (W) is bounded for all Stein ﬁllings of a given contact 3-manifold then all the characteristic numbers form a bounded set. For many 3-manifolds a Stein ﬁlling has to have vanishing b + 2 -invariant. Such 3-manifolds are, for example, the ones carrying positive scalar curvature, or having vanishing reduced Floer homologies, e.g. lens spaces or boundaries of certain plumbings along negative deﬁnite plumbing diagrams [124, 139, 143]. This observation leads us to the following conjecture. Conjecture 12.3.16. The set ( (Y,ξ) = _ χ(W) [ W is a Stein ﬁlling of (Y, ξ) _ is ﬁnite. 13. Appendix: Seiberg–Witten invariants In this chapter we recall basic deﬁnition, notions and results of Seiberg– Witten gauge theory. The introduction is not intended to be complete, we rather describe arguments most frequently used in the text. We also review a variant of the theory for 4-manifolds with contact type boundary, which setting turns out to be very useful in the study of contact topological problems. The last section is devoted to a discussion centering around the adjunction inequality. For a more complete discussion of the topics appearing in this chapter the reader is advised to turn to [21, 119, 126, 149]. 13.1. Seiberg–Witten invariants of closed 4-manifolds Let us assume that X is a closed (i.e., compact with ∂X = ∅), ori- ented, smooth 4-manifold. Suppose furthermore that b + 2 (X) > 1 and b + 2 (X) − b 1 (X) is odd. Below we outline the construction of a map SW X : Spin c (X) →Z, the Seiberg–Witten invariant of X, which turns out to be a diﬀeomorphism invariant, that is, for a diﬀeomorphism f : X 1 →X 2 and spin c structure s ∈ Spin c (X 2 ) we have SW X 2 (s) = ±SW X 1 (f ∗ s). The value SW X (s) counts solutions of a pair of equations for pairs of connec- tions and sections of bundles naturally associated to the spin c structure s. In the following we will assume that the reader is familiar with basic notions of diﬀerential geometry, such as connections, covariant diﬀerentiation and Levi–Civita connections. Fix a metric g on X and suppose that the spin c structure s ∈ Spin c (X) is given by the hermitian spinor bundles W ± → X with Cliﬀord multi- plication c: T ∗ X → Hom C (W + , W − ) satisfying c(v) ∗ c(v) = −[v[ 2 id W +. The ﬁxed metric induces a connection, the Levi–Civita connection on TX and on all bundles associated to it. By ﬁxing the connection A on 224 13. Appendix: Seiberg–Witten invariants L = det W + ( ∼ = det W − ) we get a coupled connection on W ± and hence a covariant diﬀerentiation ∇ A : Γ(W ± ) →Γ(W ± ⊗T ∗ X). Composing this with the Cliﬀord multiplication c: Γ(W ± ⊗T ∗ X) →Γ(W ∓ ) we get Deﬁnition 13.1.1. The operator / ∂ A : Γ(W ± ) → Γ(W ∓ ) given as / ∂ A = c ◦ ∇ A is called the Dirac operator associated to the connection A on L. Formally, for ψ ∈ Γ(W ± ) we have ∇ A (ψ) = w ⊗ ζ with w ∈ Γ(W ± ) and ζ ∈ Γ(T ∗ X), and then / ∂ A ψ = c(ζ)w ∈ Γ(W ∓ ). We recall that the metric g on X gives rise to the Hodge star operator ∗ g : Λ i (X) →Λ 4−i (X). On two forms ∗ 2 g = id Λ + (X) , and a 2-form ω ∈ Λ 2 (X) is self-dual (anti-self-dual, or ASD) if ∗ g ω = ω (resp. ∗ g ω = −ω). The self-dual part of ω, which is equal to 1 2 (ω + ∗ g ω), is denoted by ω + . The Cliﬀor multiplication naturally extends to 2-forms and provides a bundle isomorphism ρ: Λ + (X) → su(W + ). Moreover, for any section ψ ∈ Γ(W + ) we can consider the action of ψ ⊗ ψ ∗ on W + . The traceless part of this endomorphism will be denoted by q(ψ). Now we are in the position to write down the Seiberg–Witten equations. Let η ∈ Λ + (X) be a ﬁxed self-dual 2-form. For a spinor ψ ∈ Γ(W + ) and connection A on L = det W + the η-perturbed Seiberg–Witten equations read as follows: / ∂ A ψ = 0 ρ(F + A +iη) = q(ψ) where F + A is the self-dual part of the curvature of the connection A. The gauge group G = Map(X, S 1 ) acts on the space A(L) Γ(W + ) = _ U(1)- connections on L _ Γ(W + ) by g(ψ, A) = (gψ, A−2 dg g ), and it is not hard to see that this action maps Seiberg–Witten solutions to Seiberg–Witten solutions. The conﬁguration space _ A(L) Γ(W + ) _ /G will be denoted by B. Deﬁnition 13.1.2. The set of gauge equivalence classes of solutions of the η-perturbed Seiberg–Witten equations is called the Seiberg–Witten moduli space M η (s). The union ∪ η∈Λ + (X) M η (s) is the parameterized moduli space M(s). 13.1. Seiberg–Witten invariants of closed 4-manifolds 225 These spaces admit suitable topologies; for technical reasons we actually consider solutions in some Sobolev completions, but this subtlety will be ignored in the following. Deﬁnition 13.1.3. The pair (A, ψ) ∈ B is reducible if ψ ≡ 0 and irreducible otherwise. The set of irreducible elements in B is denoted by B ∗ . We deﬁne M ∗ (s) as B ∗ ∩ M(s). Recall that the parameterized moduli space admits a map π: M(s) → Λ + (X) by associating the perturbation parameter to every solution. The structure of the moduli space can be summarized in the following Theorem 13.1.4. The map π: M(s) → Λ + (X) is a smooth, proper Fred- holm map of index d(s) = 1 4 _ c 2 1 (s) −3σ(X) −2χ(X) _ . The subspace M ∗ (s) is a smooth inﬁnite dimensional manifold. By studying reducible solutions, it can be shown that for generic η and b + 2 (X) > 0 the moduli space M η (s) consists of irreducible solutions only. The Sard–Smale theorem implies that for generic η the moduli space is a compact smooth manifold of dimension d(s). After ﬁxing a homology orientation on X (that is, an orientation for H 2 + (X; R) ⊗ H 1 (X; R)) the moduli space admits a canonical orientation. Therefore for generic η the oriented, compact submanifold M η (s) gives rise to a homology class in H ∗ (B ∗ X ; Z), which provides us a way to turn it into a number. When d(s) = 0, this simply means that we (algebraically) count the number of solutions modulo gauge equivalence of the Seiberg–Witten equations. For d(s) > 0 we evaluate suitable cohomology classes of the cohomology ring H ∗ (B ∗ X ; Z) on M η (s). In this way we produce a number SW X (s) ∈ Z for which the following result holds: Theorem 13.1.5 (Seiberg–Witten, [174]). If b + 2 (X) > 1 then the value SW X (s) ∈ Z is independent of the chosen metric g and perturbation η, providing a smooth invariant of X. For manifolds with b + 2 (X) = 1 the proof of independence from the chosen metric and perturbation does not apply, since in that case a 1-parameter family of moduli spaces might contain reducible solutions. Such solutions are ﬁxed points of a nontrivial subgroup of the gauge group and therefore require special attention. For a thorough discussion the reader is advised to turn to [119, 149]. Recall that an element K ∈ H 2 (X; Z) is characteristic if for all x ∈ H 2 (X; Z) we have that K(x) ≡ Q X (x, x) (mod 2). The set of characteristic elements in H 2 (X; Z) will be denoted by ( X . 226 13. Appendix: Seiberg–Witten invariants Deﬁnition 13.1.6. A class K ∈ ( X is a basic class if SW X (s) ,= 0 for s ∈ Spin c (X) with c 1 (s) = K. The set of basic classes will be denoted by B X . The manifold X is of simple type if K ∈ B X implies that K 2 = 3σ(X) + 2χ(X). The following proposition summarizes some of the basic properties of B X and SW X . Here (for simplicity) we assume that X is of simple type. Recall that in the deﬁnition we assumed that b + 2 (X) − b 1 (X) is odd and b + 2 (X) is greater than 1. Proposition 13.1.7. 1. The set B X of basic classes is ﬁnite and K ∈ B X if and only if −K ∈ B X . In fact, SW X (−K) = (−1) 1 4 (σ(X)+χ(X)) SW X (K). 2. If B X ,= ∅ and Σ ⊂ X is an embedded surface representing the homology class [Σ] with [Σ] 2 ≥ 0 and Σ ,= S 2 , then 2g(Σ) −2 ≥ [Σ] 2 +[K _ [Σ] _ [ for all K ∈ B X . If B X ,= ∅ and Σ ⊂ X is an embedded sphere then [Σ] 2 < 0. The above inequality is usually called the adjunction inequality, since it generalizes the formula of Theorem 3.1.9. 3. If X is a symplectic manifold then ±c 1 (X, ω) ∈ B X . For a minimal surface of general type B X = _ ± c 1 (X) _ . Moreover, in both cases SW X _ ±c 1 (X) _ = ±1. 4. If X admits a positive scalar curvature metric, or decomposes as X = X 1 #X 2 with b + 2 (X 1 ), b + 2 (X 2 ) > 0 then B X = ∅. 5. More generally, if X = X 1 ∪ N X 2 with b + 2 (X 1 ), b + 2 (X 2 ) > 0 and N admits a metric of positive scalar curvature then B X = ∅. Saying this property in a diﬀerent way, if X = X 1 ∪ N X 2 , N admits positive scalar curvature metric and B X ,= ∅ then either b + 2 (X 1 ) = 0 or b + 2 (X 2 ) = 0. 6. If X = Y #CP 2 then B X = ¦L ± E [ L ∈ B Y ¦, where H 2 (X; Z) is identiﬁed with H 2 (Y ; Z) ⊕ H 2 (CP 2 ; Z) and E is the generator of H 2 (CP 2 ; Z). According to the following theorem, the assumption on the simple type property of X is not too restrictive for our purposes, since 13.1. Seiberg–Witten invariants of closed 4-manifolds 227 Theorem 13.1.8 (Taubes). If X is a symplectic 4-manifold then X is of simple type. If there is an embedded surface Σ ⊂ X such that 2g(Σ) −2 = [Σ] 2 ≥ 0 then X is of simple type. Exercise 13.1.9. Show that if Σ 1 , Σ 2 ⊂ X are two embedded surfaces with genera g(Σ 1 ) and g(Σ 2 ) in the symplectic 4-manifold X with [Σ 1 ] = [Σ 2 ], [Σ] 2 ≥ 0 and Σ 1 is a symplectic submanifold then g(Σ 2 ) ≥ g(Σ 1 ). (This inequality is usually referred to as the “Symplectic Thom Conjecture”. For the history of this problem see [133], cf. also Theorem 13.3.8.) The next theorem describes a relation between Seiberg–Witten invari- ants and J-holomorphic submanifolds in symplectic 4-manifolds. In order to state the result, let us assume that (X, ω) is a given symplectic 4-manifold and J is a compatible almost-complex structure. Suppose furthermore that b + 2 (X) > 1. Theorem 13.1.10 (Taubes, [162], [163]; see also [84]). Suppose that (X, ω) is a symplectic 4-manifold with b + 2 (X) > 1 and SW X (K) ,= 0 for a given cohomology class K ∈ H 2 (X; Z). Assume furthermore that the class c = 1 2 _ K − c 1 (X, ω) _ is nonzero in H 2 (X; Z). Then for a generic compatible almost-complex structure J on X the class PD(c) ∈ H 2 (X; Z) can be represented by a pseudo-holomorphic submanifold. In fact, Taubes proved much more. By deﬁning a rather delicate way of counting pseudo-holomorphic submanifolds representing a ﬁxed homo- logy class PD(c) ∈ H 2 (X; Z), he proved that this number and the value SW X (c 1 (X, ω) +2c) are equal. In many applications only the direction that a nonvanishing Seiberg–Witten invariant implies the existence of pseudo- holomorphic curves is used. Note that the curve Σ representing PD(c) is not given to be connected. This observation becomes important if one wants to apply the adjunction formula to compute the genus of Σ. By Proposi- tion 13.1.7 we have that −c 1 (X, ω) ∈ B X , consequently Theorem 13.1.10 implies, in particular, that the Poincar´e dual of −c 1 (X, ω) can be represen- ted by a pseudo-holomorphic submanifold C (assuming it is nonzero). Since a pseudo-holomorphic submanifold is always symplectic, the above reason- ing shows that −c 1 (X, ω) [ω] = _ C ω > 0 for manifolds with b + 2 (X) > 1 and c 1 (X, ω) nonzero. Furthermore, it can be shown that if b + 2 (X) > 1, then a class e ∈ H 2 (X; Z) with e 2 = −1, c 1 (X, ω) PD(e) = 1 and SW X _ c 1 (X, ω) + 2PD(e) _ ,= 0 can be represented by a symplectic sphere; consequently X is nonminimal. (The fact c 1 (X, ω) + 2PD(e) ∈ B X guar- antees the existence of a pseudo-holomorphic representative for e. The two 228 13. Appendix: Seiberg–Witten invariants other assumptions — together with the adjunction formula — ensure that this representative is a sphere, see the proof of Corollary 13.1.13.) As a further application of Theorem 13.1.10, one can show that a symplectic 4- manifold with b + 2 > 1 has Seiberg–Witten simple type, cf. Theorem 13.1.8 and [84]. Theorem 13.1.10 also proves the inequality in Theorem 3.1.12: If K is a basic class, then c = 1 2 _ K−c 1 (K, ω) _ can be represented by a pseudo- holomorphic (in particular symplectic) submanifold (unless c = 0), hence c [ω] ≥ 0. Reversing the sign of K if necessary, we can assume K [ω] ≤ 0, so c 1 (X, ω) [ω] ≤ K [ω] ≤ 0, which proves the inequality. Note that equal- ity implies c [ω] = 0, hence c = 0, and consequently, K = c 1 (X, ω) (or K = −c 1 (X, ω)). Remark 13.1.11. Above we only dealt with the case of b + 2 (X) > 1; recall that for manifolds with b + 2 (X) = 1 the Seiberg–Witten invariants depend on the chosen metric and perturbation. After the appropriate modiﬁcations, the theorems and properties discussed above extend to the case of b + 2 (X) = 1. For the sake of brevity, however, here we will omit the discussion of these extensions; see [145] for a nice review of the b + 2 (X) = 1 case. Corollary 13.1.12. Suppose that the symplectic 4-manifold X satisfying b + 2 (X) > 1 is minimal. Then c 2 1 (X) ≥ 0. Proof. According to Theorem 13.1.10 the Poincar´e dual of the class −c 1 (X, ω) can be represented by an embedded J-holomorphic submanifold C = ∪ n i=1 C i ; here C i are the connected components of C. Now the adjunc- tion formula for C i reads as 2g(C i ) −2 = [C i ] 2 −c 1 (X, ω)[C i ] = 2[C i ] 2 . Now [C i ] 2 ≥ 0 holds, since [C i ] 2 < 0 implies g(C i ) = 0 and [C i ] 2 = −1 contradict- ing minimality. Therefore c 2 1 (X, ω) = [C i ] 2 ≥ 0, concluding the proof. Corollary 13.1.13. If the symplectic 4-manifold X smoothly decomposes as Y #CP 2 then it contains a symplectic (−1)-sphere, i.e., it is not minimal as a symplectic 4-manifold. Proof. According to Proposition 13.1.7(6.) we know that B X = ¦L ± E [ L ∈ B Y ¦. Therefore ±c 1 (X) = ±(L − E) for some L ∈ B Y ; now apply Theorem 13.1.10 for K = L + E. We get that 1 2 _ K − c 1 (X) _ = E can be represented by a J-holomorphic (hence symplectic) submanifold, furthermore by the adjunction formula E 2 = −1 and c 1 (X) E = 1 give 2g(E) −2 = E 2 −c 1 (X) E = −2, so g(E) = 0, therefore the representative is a sphere. 13.2. Seiberg–Witten invariants of 4-manifolds with contact boundary 229 Remark 13.1.14. In fact, we only need the existence of a basic class K ∈ B X with the property that _ K − c 1 (X) _ 2 = −4 in order to deduce that the symplectic 4-manifold (X, ω) is not minimal. By studying the Seiberg–Witten equations on 4-manifolds of the form Y 3 R, Seiberg–Witten Floer homologies can be deﬁned for closed oriented 3-manifolds. This theory has been developed in [88], see also [89]. 13.2. Seiberg–Witten invariants of 4-manifolds with contact boundary In the study of ﬁllings of contact 3-manifolds a variant of the original Seiberg–Witten equations — developed by Kronheimer and Mrowka [86] — turns out to be extremely useful. Here we restrict ourselves to a quick review of the invariants, for a more complete discussion see [86, 89]. Let X be a given compact 4-manifold with nonempty boundary and ﬁx a contact structure ξ on ∂X. Deﬁne Spin c (X, ξ) to be all spin c structures on X which restrict to the spin c structure t ξ induced by ξ. Remark 13.2.1. Recall that the set of spin c structures on X forms a principal H 2 (X; Z)-space and for a 4-manifold it is never empty. As it follows from the long exact sequence of cohomologies of the pair (X, ∂X), the above deﬁned set Spin c (X, ξ) is a principal H 2 (X, ∂X; Z)-space. The invariant SW (X,ξ) will map from Spin c (X, ξ) to Z and is roughly de- ﬁned as follows. Consider the symplectization of (∂X, ξ) and glue it to X along ∂X ¦1¦ to get X + . By choosing an almost-complex structure for ξ, by the symplectic form on Symp(∂X, ξ) we get a metric on X + − X; extend it to a metric g deﬁned on X + . On X + − X the canonical spin c structure deﬁnes a spinor Ψ 0 and a spin connection A 0 , for a spin c struc- ture s ∈ Spin c (X, ξ) extend these over X + . Take the space of pairs (A, Ψ) — spin connections and spinors for the ﬁxed spin c structure s — which solve the usual (perturbed) Seiberg–Witten equations on the noncompact Riemannian manifold (X + , g) and are close to (A 0 , Ψ 0 ) in an appropriate L 2 -sense. After dividing with the appropriate gauge group G we get the moduli space M X + ,g (s) of Seiberg–Witten solutions. The rest of the deﬁ- nition is fairly standard now: one needs to show compactness, smoothness, orientability of the (appropriately perturbed) moduli space, and SW (X,ξ) is deﬁned by counting the number of elements (with sign) in M X + ,g (s). To get 230 13. Appendix: Seiberg–Witten invariants a well-deﬁned invariant, we need to show independence of the choices (met- ric, almost-complex structure, extensions, perturbation) made throughout the deﬁnition. This argument follows the usual cobordism method applied in the closed 4-manifold case. The two notable diﬀerences from the closed case are: • There are no reducible solutions (i.e., points in the moduli space with vanishing spinor component) since Ψ 0 is nonzero on X + −X and for any element (A, Ψ) in the moduli space Ψ is close to Ψ 0 . Therefore the gauge group acts freely, the index formula provides the actual dimension of a (smoothly cut out) moduli space and there is no need to assume anything about b + 2 (X). • In the case dimM X + ,g (s) > 0 the invariant SW (X,ξ) (s) is deﬁned to be zero, since there is no reasonable constraint with which one could cut down the dimension. (The cohomology class used in the closed case vanishes for (X, ξ).) The main result of [86] concerning SW (X,ξ) is the the generalization of Theorem 13.1.10 of Taubes to the manifold-with-boundary case. Theorem 13.2.2 (Kronheimer-Mrowka, [86]). If (X, ω) is a weak sym- plectic ﬁlling of (∂X, ξ) and s ω is the spin c structure induced by an almost-complex structure compatible with the symplectic form ω then SW (X,ξ) (s ω ) = 1. Moreover, if (X, ω) is as above and SW (X,ξ) (s) ,= 0 then [ω] ∪ (s −s ω ) ≥ 0 with equality only if s = s ω . Notice that the last assertion implies that if ω is exact (for example, if (X, ω) is a Stein ﬁlling of (∂X, ξ)) then s ω is the only spin c structure with nonzero SW (X,ξ) -invariant. In addition, these invariants can be used to prove the adjunction inequalities of the type of Proposition 13.1.7(2.) for weak symplectic ﬁllings. Let us take a contact 3-manifold (Y, ξ) and consider the noncompact 4-manifold X = Y (−∞, 0] with contact type boundary. The ideas outlined above now produce a contact invariant c SW (Y, ξ) of the contact manifold in the appropriate Seiberg–Witten Floer cohomology of Y . This invariant can be shown to share many properties with the contact invariant c(Y, ξ) ∈ ¯ HF(−Y ) to be discussed in Section 14.4. In this volume we will restrict our attention to the Heegaard Floer theoretic contact invariants, for the exact deﬁnition and some basic properties of c SW see [89]. 13.3. The adjunction inequality 231 13.3. The adjunction inequality Recall the adjunction equality from complex geometry: Theorem 13.3.1. If C is a smooth, complex curve in the complex 4- manifold X then −χ(C) = [C] 2 −c 1 (X)[C]. Remark 13.3.2. In complex algebraic geometry it is customary to use the canonical bundle K X instead of c 1 (X). Since in H 2 (X; Z) we have c 1 (X) = −K X (one originates from the tangent, while the other from the cotangent bundle), the formula reads as −χ(C) = C 2 +K X C. It is not hard to see that the above formula holds for a J-holomorphic submanifold of an almost complex 4-manifold (X, J). In particular, −χ(Σ) = [Σ] 2 −c 1 (X, ω)[Σ] holds for a symplectic submanifold of a symplectic 4-manifold (X, ω). This equality admits the following generalization for smoothly embedded sub- manifolds in symplectic 4-manifolds: Theorem 13.3.3. Suppose that Σ is a smoothly embedded, closed, ori- ented 2-dimensional submanifold in the symplectic 4-manifold (X, ω) with b + 2 (X) > 1. If g(Σ) > 0 then [Σ] 2 + ¸ ¸ c 1 (X, ω)[Σ] ¸ ¸ ≤ −χ(Σ). If g(Σ) = 0 and [Σ] is nontrivial in homology then [Σ] 2 ≤ −1. Corollary 13.3.4. If (W, ω) is a weak symplectic ﬁlling of the contact 3-manifold (Y, ξ) and Σ ⊂ W is a homologically nontrivial surface with g(Σ) > 0 then [Σ] 2 + ¸ ¸ c 1 (W, ω)[Σ] ¸ ¸ ≤ −χ(Σ). Proof. Embed the symplectic ﬁlling into a closed symplectic 4-manifold X with b + 2 (X) > 1 and apply Theorem 13.3.3. Theorem 13.3.3 follows from the fact that c 1 (X, ω) of a symplectic 4- manifold is a basic class and 232 13. Appendix: Seiberg–Witten invariants Theorem 13.3.5 (The adjunction inequality). Suppose that X is a smooth, closed 4-manifold. If K ∈ H 2 (X; Z) is a basic class of the 4-manifold X with b + 2 (X) > 1 and g(Σ) > 0 then [Σ] 2 +[K _ [Σ] _ [ ≤ −χ(Σ). The theorem was proved in the case of [Σ] 2 ≥ 0 by Kronheimer–Mrowka [85] and by Morgan–Szab´ o–Taubes [121] in their proof for the Thom conjecture. A more involved argument allows [Σ] 2 to be negative in the above formula. This extension rests on the following result. Theorem 13.3.6 (Ozsv´ath–Szab´o, [133]). Suppose that Σ is a smooth, embedded, closed 2-dimensional submanifold in the smooth 4-manifold X and for a basic class K we have χ(Σ) − [Σ] 2 − K _ [Σ] _ = 2n < 0. Let ε denote the sign of K _ [Σ] _ . Then the cohomology class K + 2εPD _ [Σ] _ is also a basic class. The most spectacular application of these results is the proof of the Sym- plectic Thom Conjecture due to Ozsv´ath and Szab´ o, which improves the result of Exercise 13.1.9 by dropping the assumption on the self-intesection of the surface. Theorem 13.3.7 (Ozsv´ath–Szab´o, [133]). If Σ 1 , Σ 2 ⊂ X are two 2- dimensional connected submanifolds of the symplectic 4-manifold (X, ω), the homology classes [Σ i ] are equal and Σ 1 is a symplectic submanifold, then the genus of Σ 2 is not smaller than the genus of Σ 1 . In addition, the form of the adjunction inequality given in Theorem 13.3.5 implies an improved version of Corollary 13.3.4, already encountered in the introduction: Theorem 13.3.8. If Σ is a smoothly embedded closed, oriented 2-dimen- sional submanifold in the Stein surface S then [Σ] 2 −c 1 (S)[Σ] ≤ −χ(Σ) unless Σ is a nullhomologous sphere. Proof. Recall that a Stein surface can always be symplectically embedded into a symplectic 4-manifold X, therefore for [Σ] 2 ≥ 0 the statement follows from the usual adjunction inequality (together with the fact that c 1 (X) of 13.3. The adjunction inequality 233 a symplectic 4-manifold is a basic class). In the case of negative [Σ] 2 we use the embedding of S into a minimal surface X of general type. Assuming g(Σ) > 0 the relation of Theorem 13.3.6 implies that either the inequality is satisﬁed or c 1 (X) ±2PD([Σ]) is a basic class. (The sign here is determined by the sign of c 1 (X) _ [Σ] _ .) But for a minimal surface of general type there are only two basic classes, which are ±c 1 (X). Therefore we have either [Σ] = 0 or the diﬀerence of the two basic classes c 1 (X) and −c 1 (X) (which is 2c 1 (X)) is equal to 2PD _ [Σ] _ . This latter case, however, provides a contradiction since it implies that c 2 1 (X) = [Σ] 2 is negative, which cannot hold for a minimal surface of general type. Finally if g(Σ) = 0 then the above principle provides [Σ] 2 ≤ −2 since a sphere with self-intersection −1 would violate minimality of X. 14. Appendix: Heegaard Floer theory The topological description of contact structures as open book decom- positions provides the possibility of deﬁning contact invariants which (at least partially) can be computed from surgery diagrams. In this appendix we outline the construction of such invariants — for a complete discus- sion the reader is referred to the original papers of Ozsv´ath and Szab´ o [135, 136, 137, 138]. To set up the stage, ﬁrst we discuss Ozsv´ath–Szab´o homology groups of oriented, closed 3-manifolds (together with maps in- duced by oriented cobordisms). The deﬁnition of the group ¯ HF(Y ) for a 3-manifold Y will rely on some standard constructions in Floer homology. After presenting the surgery triangles for this theory, we outline the deﬁni- tion of the contact Ozsv´ath–Szab´o invariants and verify some of the basic properties of this very sensitive invariant. A few model computations are also given. 14.1. Topological preliminaries Recall that a closed, oriented 3-manifold Y can be decomposed as a union of two solid genus-g handlebodies Y = U 0 ∪ Σg U 1 : consider a Morse function on Y and deﬁne U 0 as the union of the 0- and 1-handles while U 1 = 2-handles ∪ 3-handle. In fact, the 1-handles can be recorded on the genus-g surface Σ g by their cocores, while the 2-handles by their attaching circles. Hence the handlebody decomposition can be presented on Σ g by two g-tuples of embedded simple closed curves ¦α 1 , . . . , α g ¦ and ¦β 1 , . . . , β g ¦ which satisfy that the α’s (and the β’s) are disjoint among themselves and form a linearly independent system in H 1 (Σ g ; Z). Of course, the α-curves might intersect the β-curves. In conclusion, a 3-manifold can be described by a Heegaard 236 14. Appendix: Heegaard Floer theory diagram _ Σ g , ¦α i ¦ g i=1 , ¦β i ¦ g i=1 _ with the and β-curves satisfying the above conditions. It is not hard to ﬁnd a Heegaard diagram of a 3-manifold given by a surgery diagram. As we saw, any rational surgery can be transformed into a sequence of integral surgeries; in the following we will describe an algorithm (given in [135]) for ﬁnding a Heegaard diagram of a 3-manifold given by integral surgery on a knot. (The general case of surgery on a link follows similar ideas.) Suppose that Y is given by an integral surgery on K ⊂ S 3 and consider the following Heegaard diagram of S 3 −νK: Fix a projection of K to some plane. Consider a tubular neighborhood of K in R 3 and add vertical tubes for every crossing of the given projection, as it is shown by the upper diagrams of Figure 14.1. By an isotopy, the resulting subset U K ⊂ R 3 can be regarded as an ε-neighborhood of the knot projection, cf. the lower diagrams in Figure 14.1. In fact, U K is a genus-g handlebody (cf. β β β β Figure 14.1. The β-curves of the Heegaard decomposition of the knot complement Figure 14.2 for the case of the trefoil knot) with the complement in S 3 also a genus-g handlebody. This last statement can be easily veriﬁed by adding g 3-dimensional 2-handles along the curves α i encircling the “holes” of U K 14.1. Topological preliminaries 237 (see Figure 14.2 again) turning U K into a solid ball. Notice that the α- curves are the cocores of the 1-handles of the complementary handlebody S 3 − U K . This speciﬁes the α-curves of the diagram for S 3 − νK. On 1 α 2 α α 3 α 4 Figure 14.2. The α-curves of a Heegaard decomposition of the complement for the trefoil knot the other hand the meridians of the vertical tubes that we attach (as it is shown by Figure 14.1) give rise to the β-curves since we can think of them as the attaching circles of the 2-handles in S 3 − νK. By attaching handles along the α-curves we ﬁll the complement of U K (minus a point), while by attaching 2-handles along the β-curves we ﬁll the vertical tubes inside U K . Therefore the Heegaard diagram _ Σ g , ¦α i ¦ g i=1 , ¦β j ¦ g−1 j=1 _ provides the knot complement S 3 −νK. Now taking a simple closed curve deﬁning any (integral) surgery along K as β g we get a Heegaard diagram for the surgered manifold. For example, if we choose the meridian of K as β g then this choice corresponds to a trivial surgery along K so that we get a Heegaard diagram of S 3 . For another example see Figure 14.3. Exercise 14.1.1. Determine the knot and compute the surgery coeﬃcient of the surgery corresponding to the Heegaard diagram of Figure 14.3. It is natural to wonder when do two Heegaard diagrams represent the same 3-manifold. It is fairly easy to list moves which do not change the resulting 3-manifold: isotoping the α- and the β-curves (by keeping the dis- jointness property), or sliding α-curves over α-curves (and β-curves over 238 14. Appendix: Heegaard Floer theory β 1 β 2 β 3 β 4 Figure 14.3. The β-curves of a Heegaard diagram of a surgery on the trefoil knot β-curves) obviously changes only the handle decomposition, not the 3- manifold. Similarly, by stabilizing the Heegaard decomposition by tak- ing the connected sum of the original diagram with the 2-torus T 2 and α, β as shown by Figure 14.4 does not change the 3-manifold. In fact, β α Figure 14.4. A cancelling pair of α- and β-curves it can be shown that these moves are all, more precisely if two diagrams represent diﬀeomorphic 3-manifolds then one diagram can be transformed into the other by a ﬁnite sequence of isotopies, handle slides and stabiliza- tions/destabilizations [135]. This observation can be used to show that a quantity deﬁned for a Heegaard diagram is, in fact, an invariant of the cor- responding 3-manifold: one only has to check that it does not change under the moves listed above. See also Remark 14.2.3. It is a little more complicated to present 4-manifolds in a similar fashion. First of all notice that a 4-dimensional cobordism W from Y 1 to Y 2 can be decomposed as a sequence of attaching 1-, 2- and 3-handles. By assuming orientability of W, the gluing of 1-handles (and so of 3-handles) is essentially unique, and so we only need to deal with 2-handle attachments, where all 14.2. Heegaard Floer theory for 3- and 4-manifolds 239 the interesting topology happens. Suppose that K ⊂ Y 1 is a framed knot, W is the cobordism given by the 2-handle attachment along K with the given framing and consider a Heegaard diagram _ Σ g , ¦α i ¦ g i=1 , ¦β j ¦ g−1 j=1 _ for Y 1 − νK. (This can be given by implementing the algorithm described above.) Let γ j = β j for j = 1, . . . , g − 1, β g = meridian of K and γ g =the curve representing the framing of K ﬁxed before. Then the Heegaard diagrams _ Σ g , ¦α i ¦ g i=1 , ¦β j ¦ g j=1 _ , _ Σ g , ¦α i ¦ g i=1 , ¦γ j ¦ g j=1 _ represent Y 1 and Y 2 . Exercise 14.1.2. Verify that the Heegaard diagram _ Σ g , ¦β i ¦ g i=1 , ¦γ j ¦ g j=1 _ deﬁnes # g−1 (S 1 S 2 ). (Hint: Displace γ j by an isotopy to make it disjoint from β j (j = 1, . . . , g −1). Destabilize (β g , γ g ) and use induction.) Therefore, the 4-manifold X we get by attaching [0, 1] U α , [0, 1] U β and [0, 1] U γ to Σ g ¦solid triangle ⊂ C¦ along the sides I Σ g , has three boundary components: Y 1 , Y 2 and # g−1 (S 1 S 2 ). (Here U α , U β and U γ stand for the handlebodies deﬁned by the corresponding sets of curves.) Now ﬁlling the boundary component # g−1 (S 1 S 2 ) with ♮ g−1 (S 1 D 3 ) we get a cobordism from Y 1 to Y 2 , which can be proved to be diﬀeomorphic to the given cobordism W we started with. Therefore the cobordism W built on Y by attaching a 2-handle along K ⊂ Y can be represented by the Heegaard triple _ Σ g , ¦α i ¦ g i=1 , ¦β j ¦ g j=1 , ¦γ k ¦ g k=1 _ , where _ Σ g , ¦α i ¦ g i=1 , ¦β j ¦ g−1 j=1 _ is a Heegaard diagram for Y −νK and ¦γ k ¦ g k=1 is given from ¦β k ¦ g k=1 , the surgery curve K and the framing as described above. 14.2. Heegaard Floer theory for 3- and 4-manifolds Let Y be a given closed, oriented 3-manifold and ﬁx a Heegaard diagram _ Σ g , ¦α i ¦ g i=1 , ¦β j ¦ g j=1 _ for Y . Without loss of generality we can assume that each α i intersects each β j transversely. Let us consider the tori T α = α 1 . . . α g , T β = β 1 . . . β g in the g-fold symmetric power Sym g (Σ g ). This symmetric power (which is a smooth manifold of dimension 2g) can be equipped with a symplectic 240 14. Appendix: Heegaard Floer theory structure ω and the Floer homology group ¯ HF(Y ) is supposed to measure how the above two (totally real) tori intersect each other “in the symplectic sense”. More precisely, deﬁne ¯ CF(Y ) as the free Abelian group generated by the intersection points T α ∩T β . (It is easy to see that since the curves α i and β j intersect transversely, so do the tori T α and T β .) We consider two intersection points to be “removable” if there is a “holomorphic Whitney disk” showing how to get rid of them. More formally, ﬁx an ω-tame almost- complex structure J on Sym g (Σ g ) and deﬁne a diﬀerential ∂ : ¯ CF(Y ) → ¯ CF(Y ) as follows: for x, y ∈ ¯ CF(Y ) the matrix element ¸∂x, y) counts the J-holomorphic maps u: ∆ 2 →Sym g (Σ g ) from the unit disk ∆ 2 ⊂ C (up to reparametrization) with u(i) = x, u(−i) = y, u(z) ∈ T α if z ∈ ∂∆ 2 and ℜe z < 0, u(z) ∈ T β if z ∈ ∂∆ 2 and ℜe z > 0. In order to get a sensitive invariant, we need to choose a base point z 0 ∈ Σ g − ( ∪ i α i ∪ j β j ) and require u(∆ 2 ) ∩ ¦z 0 ¦ Sym g−1 (Σ g ) = ∅, that is, the holomorphic disk should avoid the divisor ¦z 0 ¦ Sym g−1 (Σ g ) deﬁned by the base point. If the space of these maps (up to reparametrization) is not 0–dimensional, we deﬁne ¸∂x, y) to be zero, otherwise ¸∂x, y) = # _ holomorphic disks from x to y disjoint from ¦z 0 ¦ Sym g−1 (Σ g ) _ . Remark 14.2.1. Using a delicate construction (and ﬁxing some auxiliary data) a sign can be attached to any map of the above type in a 0–dimensional space, and in the deﬁnition of ¸∂x, y) we count the holomorphic maps with those signs. Alternatively, we can use Z 2 -coeﬃcients, which turns out to be suﬃcient for our present purposes, therefore we will always restrict our attention to this special case. The complex _ ¯ CF(Y ), ∂ _ splits as a sum ⊕ t∈Spin c (Y ) _ ¯ CF(Y, t), ∂ _ of sub- complexes: an intersection point x ∈ T α ∩ T β and the ﬁxed base point z 0 ∈ Σ g naturally determines a spin c structure s z 0 (x) ∈ Spin c (Y ) in the following way: Suppose that the Heegaard diagram is induced by a Morse function f : Y → R and ﬁx a Riemannian metric g 0 on Y . Then an in- tersection point x ∈ T α ∩ T β can be regarded as a choice of gradient lines 14.2. Heegaard Floer theory for 3- and 4-manifolds 241 for f (with respect to g 0 ) connecting index-2 and index-1 critical points of f: choose those gradient ﬂow lines which pass through the coordinates of x = (x 1 , . . . , x g ) ∈ T α ∩ T β in Σ g . The base point z 0 speciﬁes a gradient line connecting the minimum and maximum of f, therefore on the comple- ment of the neighborhood of these paths the gradient ∇f deﬁnes a nowhere vanishing vector ﬁeld. Since along any of these paths the indices of the critical points have opposite parity, the resulting vector ﬁeld extends to Y , giving rise to a well-deﬁned spin c structure on the 3-manifold. It is not very hard to verify that there is a topological Whitney disk connecting x and y if and only if s z 0 (x) = s z 0 (y). Using Gromov’s compactness theorem it can be shown that for generic choices ∂ 2 = 0, hence the Floer homology ¯ HF(Y, t) = H ∗ _ ¯ CF(Y, t), ∂ _ can be deﬁned for all t ∈ Spin c (Y ). Theorem 14.2.2 (Ozsv´ath–Szab´o, [136]). Let Y be a given closed oriented 3-manifold equipped with a spin c structure t ∈ Spin c (Y ). The Ozsv´ath– Szab´ o homology group ¯ HF(Y, t) is a topological invariant of the spin c 3- manifold (Y, t). Remarks 14.2.3. (a) In the proof of the above theorem one needs to show that ¯ HF(Y, t) is independent of the chosen Heegaard decomposition, almost-complex structure J on Sym g (Σ g ) and base point z 0 ∈ Σ g . The in- dependence from the chosen almost-complex structure is essentially built in the deﬁnition: it is a general feature of Floer homology groups associated to intersecting Lagrangian submanifolds in symplectic manifolds. (Although T α and T β are not Lagrangian in Sym g (Σ), the general theory still ap- plies because of special features of this particular case.) The independence from the Heegaard decomposition requires to show that the groups do not change under isotopies, handle slides and stabilization. The independence of isotopies is again a consequence of some general facts regarding Floer ho- mologies: any isotopy can be decomposed into a Hamiltonian isotopy and another one which can be represented by the change of the almost-complex structure on Σ g . By a good choice of the base point, independence from stabilization is a fairly easy exercise, while handle slide invariance requires to work out a special case and a way to implement this special case under general circumstances. Finally, the change of base point can be reduced to a sequence of handle slides. For the details of the arguments indicated above, the reader is advised to turn to the original papers [135, 136]. (b) In the case b 1 (Y ) > 0 one also has to assume a certain admissibility of the Heegaard diagram, which can always be achieved by appropriate isotopies of the α- and the β-curves. This condition is needed for having 242 14. Appendix: Heegaard Floer theory ﬁnite sums in the deﬁnition of the boundary operator ∂ and in the proof of independence of choices. For details see [135]. Proposition 14.2.4. The set _ t ∈ Spin c (Y ) [ ¯ HF(Y, t) ,= 0 _ is ﬁnite for any 3-manifold Y . In particular, the vector space ¯ HF(Y ) = ⊕ t∈Spin c (Y ) ¯ HF(Y, t) is ﬁnite dimensional. Proof. After ﬁxing an admissible Heegaard diagram, there are only ﬁnitely many intersection points in T α ∩ T β , hence the chain complex ¯ CF(Y ) is ﬁnite dimensional, implying the result. Examples 14.2.5. (a) Consider the lens space L(p, q). It admits a genus- 1 Heegaard decomposition with α 1 and β 1 intersecting each other in p points. These points all correspond to diﬀerent spin c structures, therefore the boundary map ∂ vanishes for any t ∈ Spin c _ L(p, q) _ , and so we have that ¯ HF(L(p, q), t) = Z 2 . In particular, ¯ HF(S 3 ) = Z 2 . (b) It is not hard to see that ¯ HF(Y, t) ∼ = ¯ HF(−Y, t). If (Y, t) decomposes as a connected sum (Y 1 , t 1 )#(Y 2 , t 2 ) then ¯ HF(Y, t) = ¯ HF(Y 1 , t 2 ) ⊗ Z 2 ¯ HF(Y 2 , t 2 ). (c) It can be shown that if t ∈ Spin c (Y ) is torsion, that is, c 1 (t) ∈ H 2 (Y ; Z) is a torsion element, then ¯ HF(Y, t) is nontrivial. In particular, if Y is a rational homology sphere (i.e., b 1 (Y ) = 0) then ¯ HF(Y, t) is nonzero for all t ∈ Spin c (Y ). Since any 3-manifold admits torsion spin c structure, the above nontriviality statement implies that ¯ HF(Y ) ,= 0 for any 3-manifold Y . (d) The 3-manifold S 1 S 2 admits a genus-1 Heegaard decomposition with two parallel circles as α- and β-curves. This Heegaard decomposition, however, is not admissible. The diagram of Figure 14.5 gives an admissible Heegaard diagram for S 1 S 2 . By analyzing the possible holomorphic disks we get that for the spin c structure t 0 with vanishing ﬁrst Chern class ¯ HF(S 1 S 2 , t 0 ) ∼ = Z 2 ⊕ Z 2 holds, while for all other spin c structures the Ozsv´ath–Szab´o homology group vanishes. Consequently ¯ HF _ # k (S 1 S 2 ), t _ is zero unless c 1 (t) = 0, and for c 1 (t 0 ) = 0 we have ¯ HF _ # k (S 1 S 2 ), t 0 _ ∼ = Z 2 k 2 _ ∼ = H ∗ (T k ; Z 2 ) _ . 14.2. Heegaard Floer theory for 3- and 4-manifolds 243 α β Figure 14.5. Admissible Heegaard diagram for S 1 ×S 2 As we saw in Proposition 14.2.4, the groups are nontrivial only for ﬁnitely many spin c structures in Spin c (Y ). In fact, the particular geometry of Y provides a constraint for the nontriviality of the Ozsv´ath–Szab´o homology groups: Theorem 14.2.6 (Adjunction formula, [136]). Suppose that Σ ⊂ Y is an oriented surface of genus g in the 3-manifold Y with g > 0. If ¯ HF(Y, t) is nontrivial for a spin c structure t then [ ¸ c 1 (t), [Σ] _ [ ≤ −χ(Σ). If Σ ∼ = S 2 then ¯ HF(Y, t) ,= 0 implies that ¸ c 1 (t), [Σ] _ = 0. Similar ideas provide invariants for 4-dimensional manifolds. Suppose that (W, s) is a spin c cobordism between (Y 1 , t 1 ) and (Y 2 , t 2 ). Standard manifold topology implies that W can be decomposed as W = W 1 ∪W 2 ∪W 3 , where W i can be built using 4-dimensional i-handles only (i = 1, 2, 3). A homomorphism F W,s : ¯ HF(Y 1 , t 1 ) → ¯ HF(Y 2 , t 2 ) can be given as follows (for simplicity we drop the spin c stucture from the notation): deﬁne F W as the composition F W 3 ◦ F W 2 ◦ F W 1 , where the homomorphisms F W 1 and F W 3 are standard maps, since the cobordisms W 1 and W 3 depend only on Y 1 and Y 2 , and the number of 1-handles (3-handles) involved in the cobordism. For example, W 1 is a cobordism between Y 1 and Y ′ 1 = Y 1 # k (S 1 S 2 ) (where k is the number of 1-handles in W 1 ), and so F W 1 ,s is a map ¯ HF _ Y 1 , s[ Y 1 _ → ¯ HF _ Y ′ 1 , s[ Y ′ 1 _ = ¯ HF _ Y 1 , s[ Y 1 _ ⊗ ¯ HF _ # k (S 1 S 2 ), t 0 _ sending x ∈ ¯ HF _ Y 1 , s[ Y 1 _ to x ⊗θ k where θ k is the highest degree element in ¯ HF _ # k (S 1 S 2 ), t 0 _ ∼ = H ∗ (T k ; Z 2 ). Similar formula describes F W 3 . The cobordism W 2 , on the other hand, can be presented by a Heegaard triple, i.e., three g-tuples of curves α, β and γ as we discussed it in the preceding 244 14. Appendix: Heegaard Floer theory section. Counting speciﬁc holomorphic triangles in Sym g (Σ g ) with appro- priate boundary conditions (in a similar spirit as ∂ was deﬁned) we get F W 2 . As before, a long and tedious proof shows that F W,s is independent of the choices made (i.e., the decomposition of W, the chosen almost-complex structure, the base point, etc.). Theorem 14.2.7 (Ozsv´ath–Szab´o, [137]). The resulting map F W,s depends only on the oriented 4-dimensional spin c cobordism (W, s) and is indepen- dent of the choices made throughout the deﬁnition. Once again, F W,s ,= 0 holds only for ﬁnitely many spin c structures s ∈ Spin c (W), moreover Theorem 14.2.6 can be used to show Theorem 14.2.8 (Adjunction formula, [136]). If Σ ⊂ W is a closed, oriented, embedded surface with 0 ≤ 2g(Σ) − 2 < [Σ] 2 + [ ¸ c 1 (s), [Σ] _ [ or with g(Σ) = 0 and [Σ] 2 ≥ 0 then F W,s = 0. (For the detailed proof of a special case of this theorem see [101].) Simi- lar ideas result a variety of Ozsv´ath–Szab´o invariants of closed (oriented) 3-manifolds and oriented cobordisms between them. For the detailed dis- cussion of these variants of the theory we advise the reader to turn to [135, 136, 137]. 14.3. Surgery triangles homologies lies in the fact that there is a scheme for computing them once the 3-manifold is given by a surgery diagram. The key step in such computations is the application of an appropriate surgery exact sequence, which relates Ozsv´ath–Szab´o homologies of three 3-manifolds we get by doing surgeries on some knots. As we will see, the scheme does not produce the Ozsv´ath–Szab´o homology group of the 3-manifold given by surgery directly, but rather gives it as part of several exact sequences. In addition, maps in the sequences are usually induced by cobordisms, hence exactness provides information about the maps as well. Below we give the most important surgery exact sequence proved for the ¯ HF-theory. To state the theorems, let us assume that Y is a given 3-manifold with a knot K ⊂ Y in it. Fix a framing f on K and suppose that Y 1 is the result of an integral surgery on K with the given framing. Let X 1 denote 14.3. Surgery triangles 245 the resulting cobordism. Suppose that Y 2 is the result of a surgery along K with framing we get by adding a right twist to the framing f ﬁxed on K. Equivalently, Y 2 can be given by doing surgery on K with framing f and (−1)-surgery on a normal circle N to K. This alternative viewpoint also provides a cobordism X 2 from Y 1 to Y 2 given by the second surgery. Let t be a ﬁxed spin c structure on Y − νK, and let t(Y ), t(Y 1 ) and t(Y 2 ) denote the set of extensions of t to Y, Y 1 and Y 2 , resp. Denote the homomorphism ¯ HF _ Y, t(Y ) _ → ¯ HF _ Y 1 , t(Y 1 ) _ induced by the cobordism given by the ﬁrst surgery on K by F 1 . Here F 1 is the sum of F X 1 ,s for all spin c structure s ∈ Spin c (X 1 ) extending elements of t(Y ) and t(Y 1 ). We deﬁne F 2 for the cobordism X 2 in a similar fashion. Exercise 14.3.1. Perform a surgery along a (−1)-framed normal circle N ′ to N ⊂ Y 1 and denote the resulting cobordism from Y 2 by X 3 . Show that X 3 is a cobordism from Y 2 to Y . (Hint: Blow down N ′ and put a dot on the image of N. Finally cancel the resulting 1-handle/2-handle pair, see Figure 14.6.) K K K n n n N N 0 −1 −1 Figure 14.6. Identiﬁcation of a 3-manifold in the surgery exact triangle Let F 3 denote the homomorphism ¯ HF _ Y 2 , t(Y 2 ) _ → ¯ HF(Y, t) induced by the cobordism X 3 given by the 2-handle attachment along N ′ as it is given in Exercise 14.3.1. Consider the triangle of cobordisms as given by Figure 14.7. Theorem 14.3.2 (Surgery exact triangle, Ozsv´ath–Szab´o [136]). Under the above circumstances the surgery triangle induces an exact triangle ¯ HF _ Y, t(Y ) _ ¯ HF _ Y 1 , t(Y 1 ) _ ¯ HF _ Y 2 , t(Y 2 ) _ F 1 F 3 F 2 for the corresponding homology groups. 246 14. Appendix: Heegaard Floer theory Y 1 Y 2 X 1 X 3 X 2 < > n < > n K Y K n N N −1 <−1> K −1 N Figure 14.7. Cobordisms in the surgery exact triangle Remark 14.3.3. With Z 2 -coeﬃcients the map F W induced by a cobordism W is simply the sum F W,s for all spin c structures extending the ﬁxed ones on the boundaries of W. With Z-coeﬃcients, however, signs have to be attached to the various maps F W,s for exactness to hold. For a complete argument see [136]. By summing over all spin c structures on Y −K and denoting ⊕ t∈Spin c (Y ) ¯ HF(Y, t) by ¯ HF(Y ) as usual, we get Corollary 14.3.4. The triangle ¯ HF(Y ) ¯ HF(Y 1 ) ¯ HF(Y 2 ) F 1 F 3 F 2 induced by the surgery triangle of Figure 14.7 is exact. 14.3. Surgery triangles 247 As an example, we show Proposition 14.3.5. Suppose that the 4-ball genus of the knot K ⊂ S 3 is equal to g s . Then for n ≥ 2g s −1 > 0 ¯ HF _ S 3 n (K) _ ∼ = ¯ HF _ S 3 2gs−1 (K) _ ⊕Z n−2gs+1 2 . Proof. For n = 2g s − 1 the proposition obviously holds. The general case now follows by induction. To see this, consider the surgery triangle for Y = S 3 , and knot K with framing n: ¯ HF(S 3 ) ¯ HF _ S 3 n (K) _ ¯ HF _ S 3 n+1 (K) _ F 1 F 3 F 2 Since the ﬁrst cobordism contains a surface of genus g s with square n, the adjunction formula of Theorem 14.2.8 implies that F 1 = 0, hence ¯ HF _ S 3 n+1 (K) _ ∼ = ¯ HF _ S 3 n (K) _ ⊕Z 2 , concluding the proof. To see a more complicated example, suppose that Y ﬁbers over S 1 with ﬁber F of genus ≥ 2, and consider the canonical spin c structure t can ∈ Spin c (Y ) induced by the oriented 2-plane ﬁeld formed by the tangencies of the ﬁbers of Y → S 1 . Obviously ¸ c 1 (t can ), [F] _ = χ(F). The surgery exact triangle and the adjunction formula together imply Proposition 14.3.6. Under the above circumstances ¯ HF(Y, t can ) ∼ = Z 2 ⊕Z 2 . The proof of the proposition involves two lemmas, only one of which will be proved below. Lemma 14.3.7. If Y 1 , Y 2 both ﬁber over S 1 with equal ﬁber genus ≥ 2 then for the canonical spin c structures t i ∈ Spin c (Y i ) we have ¯ HF(Y 1 , t 1 ) ∼ = ¯ HF(Y 2 , t 2 ). Proof (sketch). Let m i be the monodromy of the ﬁbration Y i → S 1 (i = 1, 2), and factor m 1 m −1 2 into the product of k right-handed Dehn twists along homologically nontrivial simple closed curves. This factorization gives rise to a Lefschetz ﬁbration over the annulus, which is a cobordism between Y 1 and Y 2 . The proof will proceed by induction on k. By composing the cobordisms it is enough to deal with the case of k = 1. In that case we get 248 14. Appendix: Heegaard Floer theory Y 2 from Y 1 by doing surgery on the vanishing cycle of the singular ﬁber of the Lefschetz ﬁbration over the annulus. Writing down the surgery triangle for that surgery, we get ¯ HF _ Y 1 , t(Y 1 ) _ ¯ HF _ Y 2 , t(Y 2 ) _ ¯ HF _ Y 0 , t(Y 0 ) _ F 1 F 3 F 2 The third group vanishes by the adjunction formula of Theorem 14.2.6: Since Y 0 is the result of a surgery along the vanishing cycle with coeﬃcient 0 relative to the framing induced by the ﬁber, it contains a surface of genus (g − 1) in the homology class of the (old) ﬁber. Therefore F 1 is an isomorphism and it is not hard to see that the nonzero terms belong to the canonical spin c structures t i . Now the following lemma (which we give without proof) concludes the argument for Proposition 14.3.6. Lemma 14.3.8. For S 1 Σ g with the canonical spin c structure t can we have ¯ HF(S 1 Σ g , t can ) = Z 2 ⊕Z 2 . We close this section with an observation which will be useful in our ap- plications. A rational homology sphere Y is called an L-space if ¯ HF(Y ) = t∈Spin c (Y ) ¯ HF(Y, t) has dimension ¸ ¸ H 1 (Y, Z) ¸ ¸ . Since for a rational ho- mology sphere ¯ HF(Y, t) never vanishes, being an L-space is equivalent to ¯ HF(Y, t) = Z 2 for all spin c structures t ∈ Spin c (Y ). For example, lens spaces are all L-spaces. As an application of the above surgery exact trian- gles, we show a useful criterion for being an L-space. Proposition 14.3.9. Suppose that K ⊂ S 3 is a knot of 4-ball genus g s > 0. If there is n > 0 such that S 3 n (K) is an L-space then all S 3 m (K) with m ≥ min (2g s −1, n) is an L-space. Proof. Recall from Proposition 14.3.5 that if S 3 n (K) is an L-space and n ≥ 2g s − 1 then S 3 2gs−1 (K) is also an L-space. (Use the fact that [H 1 _ S 3 n (K); Z _ [ = [n[ for all n ,= 0.) In addition, by applying the surgery exact sequence for Y = S 3 , the knot K and framing m it is easy to see that if S 3 m (K) is an L-space then so is S 3 m+1 (K) (m ≥ 1). This observation concludes the proof. 14.4. Contact Ozsv´ath–Szab´o invariants 249 Example 14.3.10. If K denotes the right-handed trefoil knot then S 3 n (K) is an L-space for all n ≥ 1. This can be seen by the computation of Proposition 14.3.5 together with the fact that S 3 5 (K) is a lens space. Exercises 14.3.11. (a) Extend Proposition 14.3.9 to all rational m with m ≥ min(2g s −1, n). (Hint: Cf. [101].) (b) Using the fact that ¯ HF(Y ) ,= 0 holds for any 3-manifold (cf. Exam- ple 14.2.5(c)) verify that with K denoting the right-handed trefoil knot, ¯ HF _ S 3 0 (K) _ = Z 2 2 holds. (Hint: Use the surgery exact triangle and the fact that ¯ HF _ S 3 1 (K) _ = Z 2 .) (c) Let Y n denote the circle bundle over the torus T 2 with Euler number n > 0. Show that ¯ HF(Y n ) = Z 4n 2 . (Hint: Apply the surgery exact triangle induced by the cobordisms of Figure 14.8. Find a torus of self-intersection n in the coboridsm X and use induction on n. Find another triangle to handle the case of n = 1.) 14.4. Contact Ozsv´ ath–Szab´ o invariants One of the main applications of Heegaard Floer theory is in contact topol- ogy. Contact Ozsv´ath–Szab´o invariants can be fruitfully applied in deter- mining tightness of structures given by contact surgery diagrams, hence these invariants ﬁt perfectly in the main theme of the present notes. The deﬁnition of the invariant of a contact structure given by Ozsv´ath and Szab´ o is based on a compatible open book decomposition with connected binding. According to Giroux’s result discussed earlier, well-deﬁnedness of such an invariant requires the veriﬁcation that the quantity does not change under positive stabilization. The construction of Ozsv´ath and Szab´ o goes in the following way: Suppose that a compatible open book decomposition with connected binding is ﬁxed on (Y, ξ). Then 0-surgery on the binding of this open book decomposition produces a ﬁbered 3-manifold Y B and a cobordism W between Y and Y B . Notice that the contact structure ξ induces a spin c structure t ξ on Y , and Y B admits a natural spin c structure t can induced by the oriented 2-plane ﬁeld tangent to the ﬁbers. Exercise 14.4.1. Show that W admits a unique spin c structure s such that s[ Y = t ξ and s[ Y B = t can . 250 14. Appendix: Heegaard Floer theory n 0 0 0 0 n+1 X 0 0 Figure 14.8. 3-manifolds in a particular surgery triangle Turning W upside down to get W, we have a map F W,s : ¯ HF(−Y B , t can ) → ¯ HF(−Y, t ξ ), and ¯ HF(−Y B , t can ) has been computed to be isomorphic to Z 2 ⊕ Z 2 . By making use of the corresponding homology theory HF + , a nontrivial el- ement h ∈ ¯ HF(−Y B , t can ) can be distinguished: There is a long ex- act sequence connecting the related theories HF + (Y, t) and ¯ HF(Y, t) for any spin c 3-manifold (Y, t), and for a ﬁbered 3-manifold Y and t = t can we have (similarly to Proposition 14.3.6) that HF + (Y, t can ) = Z 2 . Now h ∈ ¯ HF(−Y B , t can ) is the element mapping to the nontrivial element in HF + (−Y B , t can ). Deﬁnition 14.4.2. The contact Ozsv´ ath–Szab´ o invariant c(Y, ξ) of the contact structure (Y, ξ) is deﬁned to be equal to F W,s (h) ∈ ¯ HF(−Y, t ξ ). 14.4. Contact Ozsv´ath–Szab´o invariants 251 The fundamental theorem concerning c(Y, ξ) is Theorem 14.4.3 (Ozsv´ath–Szab´o, [140]). The Ozsv´ath–Szab´o homology element c(Y, ξ) ∈ ¯ HF(−Y, t ξ ) does not depend on the chosen compatible open book decomposition, hence is an invariant of the isotopy class of the contact 3-manifold (Y, ξ). Remark 14.4.4. The deﬁnition given in [140] involves the Ozsv´ath–Szab´o knot invariant of the binding of a compatible open book decomposition — using this deﬁnition Ozsv´ath and Szab´ o veriﬁes independence of the open book decomposition and then proves that the two deﬁnitions (one relying on surgery along the binding and the one originating from the knot invariants) are the same. Since we will not make any use of the knot invariants, we do not discuss the details of the deﬁnition here. The main properties of the invariant c(Y, ξ) are summarized in the following statements Theorem 14.4.5 (Ozsv´ath–Szab´o, [140]; cf. also [100]). If (Y K , ξ K ) is given as contact (+1)-surgery along the Legendrian knot K ⊂ (Y, ξ) and W is the corresponding cobordism then by reversing the orientation on W and using the resulting cobordism −W we get F −W _ c(Y, ξ) _ = c _ Y (K), ξ(K) _ . Again, F −W stands for the sum F −W,s for all spin c structures on W. Proof. The proof below is an adaptation of [140, Theorem 4.2], cf. also [100]. Present (Y, ξ) by a contact (±1)-surgery diagram along the Legendrian link L ⊂ (S 3 , ξ st ) and add K to L. Applying the algorithm of Akbulut and the ﬁrst author [7] for L ∪ ¦K¦ we get an open book decomposition of Y compatible with ξ such that K lies on a page of it. Denote the results of the 0-surgeries along the bindings on Y and Y (K) with Y B and _ Y (K) _ B respectively. The cobordism W B of the handle attachment along the knot K gives rise to a map F −W B : ¯ HF(−Y B ) → ¯ HF( − _ Y (K) _ B ), which ﬁts into an exact triangle of the type encountered in Lemma 14.3.7. The same argument now provides that F −W B is an isomorphism, resulting in a commutative diagram ¯ HF(−Y B ) ¯ HF( − _ Y (K) _ B ) ¯ HF(−Y ) ¯ HF _ −Y (K) _ F −W B ∼ = F −W F W Y F Y (K) 252 14. Appendix: Heegaard Floer theory Since the distinguished generator h Y ∈ ¯ HF(−Y B ) maps to the distinguished generator h Y (K) ∈ ¯ HF(− _ Y (K) _ B ), the statement of the theorem follows from the commutativity of the above diagram and the deﬁnition of the contact invariant. Example 14.4.6. The contact Ozsv´ath–Szab´o invariant of the overtwisted structure (S 3 , ξ ′ ) depicted by Figure 11.3 vanishes. This can be veriﬁed by applying the above principle for (S 3 , ξ st ) and K as in Figure 11.3. The co- bordism −W inducing the map F −W with the property F −W _ c(S 3 , ξ st ) _ = c(S 3 , ξ ′ ) contains a sphere of self-intersection (+1), hence F −W = 0, there- fore c(S 3 , ξ ′ ) = 0 as claimed. This example can be generalized as Theorem 14.4.7 (Ozsv´ath–Szab´o, [140]). If (Y, ξ) is overtwisted then c(Y, ξ) = 0. Proof. Consider the oriented 2-plane ﬁeld ξ 1 on Y with the property that the oriented 2-plane ﬁeld (Y, ξ 1 )#(S 3 , ξ ′ ) is homotopic to the oriented 2- plane ﬁeld induced by (Y, ξ). (Here ξ ′ is the oriented 2-plane ﬁeld induced by the contact structure of Example 14.4.6.) By the classiﬁcation of overtwisted contact structures, there is a contact structure representing the oriented 2- plane ﬁeld ξ 1 . Consequently, the above argument shows that contact (+1)- surgery along the knot of Figure 11.3 located in a Darboux chart of some contact structure ξ 1 on Y provides an overtwisted structure homotopic, hence isotopic to (Y, ξ). Therefore c(Y, ξ) can be given as F −W _ c(Y, ξ 1 ) _ . Since −W contains a 2-sphere of self-intersection (+1), the adjunction formula provides F −W = 0 and therefore c(Y, ξ) = 0. Corollary 14.4.8. If c(Y, ξ) ,= 0 for (Y, ξ) and (Y K , ξ K ) is given as contact (−1)-surgery along the Legendrian knot K ⊂ (Y, ξ) then c(Y K , ξ K ) ,= 0, therefore it is tight. Proof. Let K ′ be a Legendrian push oﬀ of K in (Y, ξ), giving rise to a Legendrian knot (also denoted by K ′ ) in (Y K , ξ K ). By the Cancellation Lemma 11.2.6, contact (+1)-surgery on K ′ gives (Y, ξ) back, therefore The- orem 14.4.5 shows that for the cobordism W of the contact (+1)-surgery we have F −W _ c(Y K , ξ K ) _ = c(Y, ξ). If c(Y, ξ) ,= 0, then this shows that c(Y K , ξ K ) ,= 0. In the light of Theorem 14.4.7 this implies tightness of (Y K , ξ K ). 14.4. Contact Ozsv´ath–Szab´o invariants 253 Proposition 14.4.9 (Ozsv´ath–Szab´o, [140]). For (S 3 , ξ st ) the contact invariant c(S 3 , ξ st ) generates ¯ HF(S 3 ) = Z 2 . Proof (sketch). Recall that (S 3 , ξ st ) admits an open book decomposition with the unknot as binding. Therefore the map deﬁning the invariant ﬁts into the exact triangle ¯ HF(−S 1 S 2 ) ¯ HF(−S 3 ) ¯ HF(S 3 ) F G Since we know that ¯ HF(S 3 ) = Z 2 and ¯ HF(S 1 S 2 ) = Z 2 ⊕Z 2 , it follows that G = 0, and F is onto. Now by using a certain grading on Ozsv´ath–Szab´o homologies (cf. [138]) it is not hard to see that the element h ∈ ¯ HF(S 1 S 2 ) used in the deﬁnition of the contact invariant maps into the nonzero element of ¯ HF(S 3 ), concluding the proof. Lemma 14.4.10. Consider the contact structure η k on # k (S 1 S 2 ) given by contact (+1)-surgery on the k-component Legendrian unlink. The contact invariant c _ # k (S 1 S 2 ), η k _ does not vanish. Proof. The lemma will be proved by induction on k. For k = 0 we have the standard contact 3-sphere (S 3 , ξ st ) which has nonzero invariant by Proposition 14.4.9. By deﬁnition, η k is given as contact (+1)-surgery along a knot in η k−1 , therefore F −W (c _ # k−1 (S 1 S 2 ), η k−1 _ ) = c _ # k (S 1 S 2 ), η k _ for the cobordism we get by the handle attachment. Therefore the injectivity of F −W immediately provides the result. Now writing down the surgery exact triangle for the above handle attachment, for the Ozsv´ath–Szab´o homology groups we get ¯ HF _ # k−1 (S 1 S 2 ) _ ¯ HF _ # k (S 1 S 2 ) _ ¯ HF _ # k−1 (S 1 S 2 ) _ F −W Since dim Z 2 ¯ HF _ # k (S 1 S 2 ) _ = 2 k , injectivity of F −W follows from exact- ness and simple dimension count. Notice that the nonvanishing of the contact invariant shows that the con- tact 3-manifold (# k S 1 S 2 , η k ) is tight. It is known that # k (S 1 S 2 ) 254 14. Appendix: Heegaard Floer theory carries a unique isotopy class of tight contact structures, which is the Stein ﬁllable boundary of D 4 ∪ k 1-handles. In conclusion, (+1)-surgery on the k-component Legendrian unlink produces a contact 3-manifold contacto- morphic to the boundary of the Stein surface we get by attaching k 1-handles to D 4 , cf. Exercise 11.2.7. Exercise 14.4.11. Show that if (Y, ξ) is a Stein ﬁllable contact 3-manifold then c(Y, ξ) ,= 0. (Hint: Recall that any Stein ﬁllable contact structure can be given as Legendrian surgery along a link in _ # k (S 1 S 2 ), η k _ for some k. Use Lemma 14.4.10 and Corollary 14.4.8.) Making use of the Embedding Theorem 12.1.7 of weak symplectic ﬁllings and the nonvanishing of the mixed Ozsv´ath–Szab´o invariants for closed symplectic 4-manifolds [137, 141], the above exercise was generalized for a version of contact invariants in some “twisted coeﬃcient system” as follows: Proposition 14.4.12 (Ozsv´ath–Szab´o, [143]). If (Y, ξ) is a weakly sym- plectically ﬁllable contact 3-manifold then by using an appropriate twisted coeﬃcient system the contact invariant c(Y, ξ) does not vanish. 15. Appendix: Mapping class groups In this appendix we summarize some basic facts regarding algebraic prop- erties of mapping class groups. After discussing the presentation of these groups we recall the equivalence between certain words in some mapping class groups and geometric structures discussed in earlier chapters. We close this chapter with some theorems making use of those connections. 15.1. Short introduction Let Σ n g,r denote an oriented, connected genus-g surface with n marked points and r boundary components. Deﬁnition 15.1.1. The mapping class group Γ n g,r is deﬁned as the quotient of the group of orientation preserving self-diﬀeomorphisms of Σ n g,r (ﬁxing marked points and boundaries pointwise) by isotopies (ﬁxing marked points and boundaries pointwise). For n = 0 (r = 0, resp.) we use the notation Γ g,r (Γ n g , resp.), and in case n = r = 0 we write Γ g . For F = Σ g,r we will denote Γ g,r by Γ F . Simple closed curves in the surface give rise to special mapping classes: Deﬁnition 15.1.2. A right-handed Dehn twist t a : Σ n g,r → Σ n g,r on an em- bedded simple closed curve a in an oriented surface Σ n g,r is a diﬀeomorphism obtained by cutting Σ n g,r along a, twisting 360 ◦ to the right and regluing. More formally, we identify a regular neighborhood νa of a in Σ n g,r with S 1 I, set t a (θ, t) = (θ +2πt, t) on νa and smoothly glue into id Σ n g,r −νa . A left-handed Dehn twist is the inverse of a right-handed Dehn twist. 256 15. Appendix: Mapping class groups Remark 15.1.3. Notice that in the deﬁnition of the Dehn twist along a curve a we do not need to orient a even though the surface Σ n g,r has to be oriented. It is well-known that Dehn twists generate Γ n g,r — in fact we can choose a ﬁnite (fairly simple) set of generators, see [171] and Theorem 15.1.12. First we discuss relations which hold in Γ n g,r . In the following we will use the usual functional notation for products in Γ n g,r . Lemma 15.1.4. If f : Σ n g,r → Σ n g,r is an orientation preserving diﬀeomor- phism and a ⊂ Σ n g,r is a simple closed curve then ft a f −1 = t f(a) . Proof. Let a ′ = f(a). Since f maps a to a ′ we can assume that (up to isotopy) it also maps a neighborhood N of a to a neighborhood N ′ of a ′ . Let us examine the eﬀect of applying ft a f −1 . The homeomorphism f −1 takes N ′ to N, then t a maps N to N, twisting along a, and ﬁnally f takes N back to N ′ . Since t a is supported in N, the composite map is supported in N ′ and is a Dehn twist about a ′ . We say that a simple closed curve a ⊂ Σ n g,r is separating if Σ n g,r −a has two connected components — otherwise a is called nonseparating. Lemma 15.1.4 together with the classiﬁcation of 2-manifolds provides Lemma 15.1.5. Suppose that a and b are nonseparating simple closed curves in Σ n g,r . Then there is an orientation preserving diﬀeomorphism f : Σ n g,r → Σ n g,r which takes a to b. Consequently t a and t b are conjugate in Γ n g,r . In particular, if a and b are homologically essential simple closed curves in a surface with at most one boundary component then t a and t b are conjugate. Exercise 15.1.6. Verify that if a intersects b transversely in a unique point then t a t b (a) = b. (Hint: Use Figure 15.1.) a b t a t (a) b t (a) b b Figure 15.1. An identity for right-handed Dehn twists 15.1. Short introduction 257 Lemma 15.1.7. If a, b ⊂ Σ n g,r are disjoint then t a t b = t b t a . If a intersects b in a unique point then t a t b t a = t b t a t b . Proof. The commutativity relation t a t b = t b t a is obvious. To prove the braid relation t a t b t a = t b t a t b we observe that t a t b (a) = b (see Exercise 15.1.6). By Lemma 15.1.4 we get t a t b t a = t a t b t a t −1 b t −1 a t a t b = t tat b (a) t a t b = t b t a t b . Lemma 15.1.8. Let a 1 , a 2 , , a k be a chain of curves, i.e., the consecutive curves intersect once and nonconsecutive curves are disjoint. Let N denote a regular neighborhood of the union of these curves. Then the following relations hold: • The commutativity relation: t a i t a j = t a j t a i if [i −j[ > 1. • The braid relation: t a i t a j t a i = t a j t a i t a j if [i −j[ = 1. • The chain relation: If k is odd then N has two boundary components d 1 and d 2 , and (t a 1 t a 2 t a k ) k+1 = t d 1 t d 2 . If k is even then N has one boundary component d and (t a 1 t a 2 t a k ) 2k+2 = t d . d 2 1 d a a a 3 2 1 Figure 15.2. Chain relation for k = 3: (ta 1 ta 2 ta 3 ) 4 = t d 1 t d 2 The next lemma was ﬁrst observed by Dehn and then rediscovered by Johnson [78] who called it the lantern relation. Lemma 15.1.9. Let U be a disk with the outer boundary a and with 3 inner holes bounded by the curves a 1 , a 2 , a 3 . For 1 ≤ i ≤ 3, let b i be the simple closed curve in U depicted in Figure 15.3. Then the lantern relation t a t a 1 t a 2 t a 3 = t b 1 t b 2 t b 3 holds. 258 15. Appendix: Mapping class groups 3 3 2 2 1 1 a b a b a b a U Figure 15.3. The lantern relation Lemma 15.1.10. If i denotes the hyperelliptic involution (i.e., rotation of the standard embedded Σ g ⊂ R 3 by 180 ◦ around the x-axis, see Figure 15.4) and a is a curve in Σ g ∩ ¦xy −plane¦ then [i, t a ] = 1. Remark 15.1.11. The idea of the proofs of Lemmas 15.1.8, 15.1.9 and 15.1.10 is the following: We split the surface into a union of disks by cutting along a ﬁnite number of simple closed curves and properly embedded arcs. We prove that the given product of Dehn twists takes each one of these curves (arcs, resp.) onto an isotopic curve (arc, resp.). Then the product is isotopic to a homeomorphism pointwise ﬁxed on each curve and arc. But Alexander’s lemma says that a homeomorphism of a disk ﬁxing its boundary is isotopic to the identity, relative to boundary. Thus the given product is isotopic to the identity. Now a presentation of Γ g (and Γ g,1 ) can be given using the relations described above. It turns out that the mapping class groups Γ g and Γ g,1 are 15.1. Short introduction 259 Figure 15.4. The hyperelliptic involution i generated by t a 0 , . . . , t a 2g with curves a 0 , . . . , a 2g depicted in Figure 15.5. Let A ij = [t a i , t a j ] for all pairs (i, j) with a i ∩ a j = ∅. Let B i denote a 2g+1 a 1 a g 2 a 0 a a a 2 3 5 a 4 a 6 Figure 15.5. The simple closed curves inducing a generating system the braid relation t a i t a i+1 t a i t −1 a i+1 t −1 a i t −1 a i+1 for i = 1, . . . , 2g − 1 and B 0 = t a 0 t a 4 t a 0 t −1 a 4 t −1 a 0 t −1 a 4 . Finally C, D and E = [i, t a 2g+1 ] denote the appropriate chain, lantern and hyperelliptic relations, cf. Figures 15.6 and 15.7. Notice that there are a number of relations of type A and B, but the relations C, D and E are unique (as shown by the ﬁgures). Write all these relations in terms of the generators t a 0 , . . . , t a 2g and consider the normally generated subgroups R 1 = ¸A ij , B i , C, D) No and R = ¸A ij , B i , C, D, E) No in the free group F 2g+1 on 2g + 1 letters corresponding to the generators t a 0 , . . . , t a 2g . Now the presentation theorem of Wajnryb (see also [81]) reads as follows: Theorem 15.1.12 (Wajnryb, [171]). For g ≥ 3 the sequences 1 →R →F 2g+1 →Γ g →1 and 1 →R 1 →F 2g+1 →Γ g,1 →1 are exact; in other words, the above generators and relations provide a presentation of Γ g and Γ g,1 . 260 15. Appendix: Mapping class groups a 2 a 3 a 0 a a 1 Figure 15.6. The chain relation in the presentation a 1 a a 3 5 a Figure 15.7. The a-curves in the lantern relation of the presentation 15.1. Short introduction 261 Remark 15.1.13. For g = 2 omit the lantern relation to get the correct result. If we denote t a i by a i for simplicity, an alternative presentation of Γ 2 can be given by generators a 1 , a 2 , a 3 , a 4 , a 5 , the braid and commutativity relations for them (i.e., a i a i+1 a i = a i+1 a i a i+1 and a i a j = a j a i for [i−j[ ≥ 2), requiring that i = a 1 a 2 a 3 a 4 a 2 5 a 4 a 3 a 2 a 1 is central, i 2 = 1, and ﬁnally that (a 1 a 2 a 3 a 4 a 5 ) 2 = 1. Next we would like to discuss two exact sequences relating various map- ping class groups. By collapsing a boundary component to a point (or gluing a disk with marked center to a boundary component) we get an obviously surjective map Γ n g,r → Γ n+1 g,r−1 . It is easy to see that the Dehn twist ∆ = t δ along a curve δ parallel to the boundary we collapsed becomes trivial. In fact, 1 →Z →Γ n g,r →Γ n+1 g,r−1 →1 turns out to be an exact sequence, where Z is generated by t δ . Forgetting the marked point we get a map Γ n g,r →Γ n−1 g,r , and now the sequence 1 →π 1 (Σ n−1 g,r ) →Γ n g,r →Γ n−1 g,r →1 is exact (here π 1 (Σ n−1 g,r ) is the fundamental group of the (n − 1)-punctured surface with r boundary components). Using these exact sequences, pre- sentations for all Γ n g,r can be derived by starting with Wajnryb’s result and knowing presentations for the kernels in the above short exact sequences; for such results see [58]. It follows that Γ n g,r is generated by ﬁnitely many nonseparating Dehn twists plus Dehn twists along boundary-parallel curves. In fact, if g ≥ 2 then for each boundary component of Σ n g,r we can embed a lantern relation (as shown in Figure 15.9) in Σ n g,r in such a way that one of the boundary curves in the lantern relation is mapped onto that boundary component of Σ n g,r and all the other curves in the lantern relation are non- separating in Σ n g,r . It follows that Γ n g,r is generated by ﬁnitely many Dehn twists along nonseparating curves for g ≥ 2. Proposition 15.1.14 (Powell, [147]). For g ≥ 3 the commutator subgroup [Γ g , Γ g ] is equal to Γ g , i.e., Γ g is a perfect group. Proof. Let a be any nonseparating curve on Σ g . For g ≥ 3, there is an embedding of a sphere with 4-holes (one of which is bounded by a) into Σ g where all seven curves in the lantern relation t a t a 1 t a 2 t a 3 = t b 1 t b 2 t b 3 262 15. Appendix: Mapping class groups a a a b b b a 2 1 3 2 3 1 Figure 15.8. Appropriate lantern relation involving a with a nonseparating a a b b a a b 3 2 2 3 1 1 Figure 15.9. Appropriate lantern relation involving a with a separating of Lemma 15.1.9 are nonseparating, see Figure 15.8. Since the a i ’s are disjoint from the b j ’s we have t a = t b 1 t −1 a 1 t b 2 t −1 a 2 t b 3 t −1 a 3 . By Lemma 15.1.5, on the other hand, there are diﬀeomorphisms h i such that t b i = h i t a i h −1 i for i = 1, 2, 3. Substituting these expressions into the relation above we get t a = h 1 t a 1 h −1 1 t −1 a 1 h 2 t a 2 h −1 2 t −1 a 2 h 3 t a 3 h −1 3 t −1 a 3 = [h 1 , t a 1 ][h 2 , t a 2 ][h 3 , t a 3 ]. We showed that a nonseparating Dehn twist is a product of (three) commu- tators. This ﬁnishes the proof since Γ g is generated by Dehn twists along nonseparating curves (for g ≥ 3) and any two Dehn twists along nonsepa- rating curves are conjugate by Lemma 15.1.5. Note that the conjugate of a commutator is a commutator. 15.1. Short introduction 263 Remark 15.1.15. In fact, any mapping class group Γ n g,r is perfect, i.e., Γ n g,r /[Γ n g,r , Γ n g,r ] = 0 for g ≥ 3 (see [81], for example). For g = 1, 2 it is impossible to embed a lantern relation into Σ g with nonseparating boundary components, and hence the above proof breaks down from the beginning. Using the presentations of Γ n 1,r and Γ n 2,r , however, one can derive that • Γ n 1,0 /[Γ n 1,0 , Γ n 1,0 ] = Z 12 , • Γ n 1,r /[Γ n 1,r , Γ n 1,r ] = Z r for r ≥ 1, and • Γ n 2,r /[Γ n 2,r , Γ n 2,r ] = Z 10 . Lemma 15.1.16. Any element in Γ g can be expressed as a product of nonseparating right-handed Dehn twists. Proof. The following is a standard relation in the mapping class group Γ g : (t a 1 t a 2 t a 2g ) 4g+2 = 1, where the curves a i are depicted in Figure 15.5. We deduce that t −1 a 1 is a product of nonseparating right-handed Dehn twists. Therefore any left- handed nonseparating Dehn twist — being conjugate to t −1 a 1 — is a product of nonseparating right-handed Dehn twists. This ﬁnishes the proof of the lemma combined with the fact that Γ g is generated by (right and left- handed) nonseparating Dehn twists. Exercises 15.1.17. (a) Show that any element in Γ g,1 can be expressed as a product of nonseparating right-handed Dehn twists plus left-handed Dehn twists along a boundary-parallel curve. (Hint: Use the same argument as above with the relation (t a 1 t a 2 t a 2g ) 4g+2 = t δ in Γ g,1 where δ denotes a curve parallel to the boundary.) (b) Show that a separating right-handed Dehn twist in Γ g,1 can be expressed as a product of nonseparating right-handed Dehn twists. 264 15. Appendix: Mapping class groups 15.2. Mapping class groups and geometric structures As our earlier discussion indicated, the geometric objects we discussed in the preceding chapters have counterparts in various mapping class groups. To clarify the situation, below we summarize these relations. • A product Π k i=1 [a i , b i ] of k commutators in Γ g gives a Σ g -bundle over the surface Σ k,1 with one boundary component. The mapping classes a i and b i specify the monodromy along the obvious free generating system ¸α 1 , β 1 , . . . , α k , β k ) of π 1 (Σ k,1 ). If Π k i=1 [a i , b i ] = 1 in Γ g , we get a Σ g -bundle X →Σ k . (The bundle is uniquely determined by the word once g ≥ 2.) In case Π k i=1 [a i , b i ] = 1 holds in Γ 1 g , the bundle X → Σ k admits a section. In this case Π k i=1 [a i , b i ] = (t δ ) n in Γ g,1 for some n ∈ Z, and it is not hard to see that the self-intersection of the section given by this word is exactly −n. • An expression Π k i=1 t i ∈ Γ g with t i right-handed Dehn twists provides a genus-g Lefschetz ﬁbration X →D 2 over the disk with ﬁber Σ g closed. If Π k i=1 t i = 1 in Γ g then the ﬁbration closes up to a ﬁbration over the sphere S 2 and the closed up manifold is uniquely determined by the word Π k i=1 t i once g ≥ 2. Once again, a lift of the relation Π k i=1 t i = 1 to Γ 1 g shows the existence of a section, and its self-intersection is −n if Π k i=1 t i = (t δ ) n in Γ g,1 for the Dehn twist t δ along the boundary- parallel simple closed curve δ ⊂ Σ g,1 . • By combining the above two constructions, a word w = Π k ′ i=1 t i Π k j=1 [a i , b i ] gives a Lefschetz ﬁbration over Σ k,1 and if w = 1 in Γ g we get a Lefschetz ﬁbration X → Σ k . Sections can be captured in the same way as above. • An expression Π n i=1 t i = t δ 1 t δ k in Γ g,k (where all t i stand for right- handed Dehn twists and t δ i are right-handed Dehn twists along circles parallel to the boundary components of the Riemann surface at hand) naturally describes a Lefschetz pencil: The relation determines a Lef- schetz ﬁbration with k section, each of self-intersection (−1), and after blowing these sections down we get a Lefschetz pencil. Conversely, by blowing up the base locus of a Lefschetz pencil we arrive to a Lef- schetz ﬁbration which can be captured (together with the exceptional divisors of the blow-ups, which are all sections now) by a relator of the above type. 15.3. Some proofs 265 • If we allow the Dehn twists t i to have negative exponents in the previ- ous constructions, we can also encounter achiral Lefschetz ﬁbrations in this way. • An element h ∈ Γ g,r (r > 0) speciﬁes a 3-manifold equipped with an open book decomposition by considering the mapping cylinder of h and collapsing the boundaries to the core circles. Notice that the binding has r components. Through the equivalence discussed in Section 9 the mapping class h ∈ Γ g,r determines a contact 3- manifold. All closed contact 3-manifolds can be given in this way; h fails to be unique though, since by positively stabilizing the open book decomposition (and so leaving the contact structure unchanged) we can change g and r. • Since Π n i=1 t i ∈ Γ g,r gives a Lefschetz ﬁbration with nonclosed ﬁbers over the disk D 2 , and these manifolds can be equipped with Stein structures, a factorization h = Π n i=1 t i in Γ g,r into right-handed Dehn twists gives a Stein ﬁlling of the contact 3-manifold determined by h ∈ Γ g,r . All Stein ﬁllings arise in this manner, although we might need to pass to a stabilization of h to recover certain ﬁllings of the contact 3-manifold speciﬁed by h. 15.3. Some proofs We close this chapter with a few results which show an interesting bridge between Lefschetz ﬁbrations, contact structures and mapping class groups. For g ≥ 3, Proposition 15.1.14 shows that Γ g is a perfect group, i.e., every element of Γ g is a product of commutators. The minimal number of commutators one has to use to express an element as a product in a group is called the commutator length of that element. Theorem 15.3.1 ([83]). The commutator length of a Dehn twist in Γ g (g ≥ 3) is equal to two. Proof. Consider a sphere X with four holes with boundary components a, a 1 , a 2 , a 3 . Since the genus of Σ g is at least three, X can be embedded in Σ g in such a way that a 1 , a 2 , a 3 , b 1 , b 2 , b 3 are all nonseparating. The simple closed curve a can be chosen either nonseparating or separating bounding a subsurface of arbitrary genus (cf. Figures 15.8 and 15.9). Furthermore, 266 15. Appendix: Mapping class groups the complement of a 1 ∪ b 1 and of a 2 ∪ b 2 are connected. Hence, there is an orientation preserving diﬀeomorphism f of Σ g such that f(a 1 ) = b 2 and f(b 1 ) = a 2 . Let h be another orientation preserving diﬀeomorphism of Σ g such that h(a 3 ) = b 3 . Then the lantern relation combined with the above choices implies t a = t b 1 t −1 a 1 t b 2 t −1 a 2 t b 3 t −1 a 3 = t b 1 t −1 a 1 t f(a 1 ) t −1 f(b 1 ) ht a 3 h −1 t −1 a 3 = t b 1 t −1 a 1 ft a 1 f −1 ft −1 b 1 f −1 ht a 3 h −1 t −1 a 3 = [t b 1 t −1 a 1 , f][h, t a 3 ]. Next we show that the commutator length of a Dehn twist is not equal to one. Suppose that a right-handed Dehn twist is equal to a single commu- tator. Then there is a 4-manifold X which admits a (relatively minimal) genus-g Lefschetz ﬁbration over the torus T 2 with only one singular ﬁber. It is easy to see that χ(X) = 1. Since the ﬁbration is relatively minimal, and so by Proposition 10.3.8 the 4-manifold X is a minimal symplectic 4-manifold, we have the inequality 0 ≤ c 2 1 (X) = 3σ(X) + 2χ(X) which implies that σ(X) ≥ − 2 3 . This gives σ(X) ≥ 0 since σ(X) is an integer. Recall that the holomorphic Euler characteristic is deﬁned by χ h (X) = 1 4 _ σ(X) +χ(X) _ and it is an integer for any closed almost-complex, hence for any closed symplectic 4-manifold. Rewriting the above equality we get χ h (X) = 1 4 _ σ(X) + 1 _ . Therefore σ(X) = 4χ h (X) −1 and so c 2 1 (X) = 3σ(X) + 2χ(X) = 12χ h (X) −1. On the other hand, by [155] it follows that c 2 1 (X) ≤ 10χ h (X) since X admits a Lefschetz ﬁbration over T 2 . Since the holomorphic Eu- ler characteristic χ h (X) is an integer, it follows that χ h (X) ≤ 0 imply- ing σ(X) + 1 = 4χ h (X) ≤ 0. This last inequality, however, contradicts σ(X) ≥ 0, which has been shown earlier. 15.3. Some proofs 267 Recall from Proposition 15.1.14 that Γ g is a perfect group for g ≥ 3. The mapping class group Γ g is, however, not uniformly perfect, that is, there is no constant K such that any element of Γ g can be written as a product of at most K commutators. This statement can be proved by using the correspondence between certain words in mapping class groups and Lefschetz ﬁbrations. (For a diﬀerent proof see [13].) Theorem 15.3.2 (Endo–Kotschick [36], Korkmaz [82]). Let c ⊂ Σ g be a separating simple closed curve. If t n c = Π kn i=1 _ α i (n), β i (n) ¸ then the sequence ¦k n ¦ cannot be bounded. In conclusion, the mapping class group Γ g is not uniformly perfect. Proof. Notice that a commutator expression of the type of the theorem gives a relator which gives rise to a Lefschetz ﬁbration X n →Σ kn . Suppose that ¦k n ¦ is bounded, say k n ≤ K. By adding trivial monodromies if necessary, this assumption provides a sequence f n : X n → Σ K (n ∈ N) of Lefschetz ﬁbrations over the ﬁxed base Σ K . It is easy to see that χ(X n ) = χ(Σ g )χ(Σ K ) +n = 4(K −1)(g −1) +n, while by Novikov additivity and the signature calculation for a separating vanishing cycle (cf. [130]) we get σ(X n ) = −n +σ _ X −∪ n i=1 νf −1 n (q i ) _ . On the other hand one can show that σ _ X −∪ n i=1 νf −1 n (q i ) _ ≤ C for some constant C depending on K and g only. (The points q i denote the critical values of the Lefschetz ﬁbration f n .) This implies that c 2 1 (X n ) = 3σ(X n ) + 2χ(X n ) ≤ −3n + 2n +C ′ = −n +C ′ , where C ′ = 3C+8(K−1)(g−1) and hence for n large enough the expression c 2 1 (X n ) will be negative. This observation contradicts the result of [155] where it is proved that a relatively minimal Lefschetz ﬁbration over a base of positive genus is minimal, hence its c 2 1 invariant is nonnegative, cf. Corollary 10.3.10. The contradiction shows that the sequence k n is unbounded, verifying the statement of the theorem. 268 15. Appendix: Mapping class groups Remark 15.3.3. In fact, using the exact sequences in Section 15.1 one can show that the mapping class group Γ n g,r is not uniformly perfect. As discussed in [36, 82], the fact that Γ n g,r is not uniformly perfect has inter- esting corollaries regarding the second bounded cohomology of Γ n g,r . Also, the proof given above can be reﬁned to get explicit lower bounds for the commutator lengths for certain elements in Γ n g,r ; for details see [36, 82]. Similar question can be raised for the length of expressions writing a given element as product of right-handed Dehn twists. Since by Lemma 15.1.16 1 ∈ Γ g can be written as a nontrivial product of right-handed Dehn twists there is no bound for the length of such expression for h ∈ Γ g . The situation, however, is diﬀerent in Γ g,r once r > 0. Theorem 15.3.4 ([7, 157]). If r ≥ 1 then 1 ∈ Γ g,r admits no nontrivial factorization 1 = t 1 t n into a product of right-handed Dehn twists. Proof. Suppose that 1 ∈ Γ g,r admits a nontrivial factorization 1 = t 1 t n into a product of right-handed Dehn twists. Now cap oﬀ all but one of the boundary components with disks to get a relation in Γ g,1 where identity is expressed as a nontrivial product of right-handed Dehn twists. Thus we re- duce the problem to show that 1 ∈ Γ g,1 admits no nontrivial factorization 1 = t 1 t n into a product of right-handed Dehn twists. Clearly we can as- sume that g ≥ 1. Moreover we can assume that all the t i ’s are nonseparating Dehn twists since any separating right-handed Dehn twist in Γ g,1 is a prod- uct of nonseparating right-handed Dehn twists. Then we can express t −1 1 , and hence any nonseparating left-handed Dehn twist, as a product of right- handed Dehn twists. We know that any element in Γ g,1 can be expressed as a product of nonseparating Dehn twists. Now replace every left-handed Dehn twist in this expression by a product of right-handed Dehn twists to conclude that any element in Γ g,1 can be expressed as a product of (non- separating) right-handed Dehn twists. 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Index achiral Lefschetz ﬁbration, 155 adjunction – equality, 51 – formula, 243 – inequality, 13, 226 almost-complex structure, 50, 106 almost-K¨ahler structure, 50 basic class, 226 Bennequin inequality, 21, 77 binding, 131 blackboard framing, 29 botany, 53 boundary connected sum, 26 branch set, 150 bypass, 183 characteristic foliation, 78, 87 classiﬁcation, 179 compatible, 191 complex point, 127 connected sum, 26 – boundary, 26 contact – 1-form, 63 – Dehn surgery, 185 – framing, 68 – invariant, 230 – structure, 63 – coorientable, 67 – ﬁllable, 254 – isotopic, 66 – overtwisted, 21, 76 – positive, 67 – standard, 66 – tight, 21, 76 – universally tight, 76 – virtually overtwisted, 77 – type, 111 – type boundary, 229 – vector ﬁeld, 85 contactomorphic, 66 continued fraction, 35 convex, 86 Darboux theorem, 70 Dehn – surgery, 31 – contact, 185 – twist, 158, 193, 255 destabilization – negative, 137 – positive, 137 Dirac operator, 224 dividing set, 86 dotted circle, 39, 169 elimination lemma, 82 elliptic, 128 – singularity, 80 ﬁbered link, 132 ﬁllable – holomorphically, 201 – Stein, 201, 254 – strongly symplectically, 201 – weakly symplectically, 201 ﬁlling – Stein, 230, 265 Floer homology, 241 four-ball genus, 18 framing, 27 282 Index – blackboard, 29, 74 – contact, 68 – Seifert, 29 – Thurston–Bennequin, 68 Fredholm map, 225 Frobenius theorem, 64 front projection, 72 gauge group, 224 geography, 53 gordian number, 18 handle, 28 handlebody, 28 – relative, 28 h-cobordism theorem, 11 Heegaard – decomposition, 27, 96 – diagram, 30, 236 Hirzebruch signature theorem, 104 Hodge star operator, 224 holomorphic convex hull, 121 holomorphically convex, 121 Hopf link, 133 hyperbolic, 128 – singularity, 80 hyperelliptic involution, 258 Kirby calculus, 38 Kodaira dimension, 59 Lagrangian – neighborhood theorem, 57 – submanifold, 51 Lefschetz – ﬁbration, 156, 264 – achiral, 155 – allowable, 163 – relatively minimal, 155 – pencil, 156, 264 – achiral, 155 Legendrian – isotopy, 72 – knot, 68 – realization principle, 90 – unknot, 76 lens space, 34, 216 Levi–Civita connection, 223 Liouville vector ﬁeld, 113 longitude, 32 mapping class group, 131, 255 – presentation, 259 meridian, 32 minimal model, 58 monodromy, 131 Moser’s method, 55 Murasugi sum, 134 neighborhood theorem – contact, 70 – Lagrangian, 57 – symplectic, 56 nonisolating, 90 normal connected sum, 114 ω-concave, 111 ω-convex, 111 open book decomposition, 96, 131 – binding, 131 – compatible, 138, 191 – monodromy, 131 – page, 131 – standard, 133 overtwisted – contact structure, 21, 76 – disk, 76 Ozsv´ath–Szab´o invariant, 22 – contact, 205, 249 page, 131 PALF, 163 perfect, 267 – uniformly, 267 plumbing, 134 plurisubharmonic, 122 pseudo-holomorphic submanifold, 51 pseudoconvex, 123 rational surgery, 31 real point, 127 Reeb vector ﬁeld, 67 regular ﬁber, 155 relation Index 283 – braid, 257 – chain, 257 – commutativity, 257 – hyperelliptic, 258 – lantern, 257 relatively minimal, 155 Rolfsen twist, 35 rotation number, 74 Sard–Smale theorem, 225 Seiberg–Witten – invariant, 223 – moduli space, 224 – parametrized, 224 – simple type, 226, 228 Seifert – ﬁbered manifold, 45 – framing, 29 self-linking number, 82 simple – (ADE) singularity, 218 – cover, 150 – elliptic singularity, 218 – type, 226 singular ﬁber, 155 singularity – elliptic, 80 – hyperbolic, 80 slam-dunk, 35 slice genus, 18 slope, 91 sobering arc, 148 spin – group, 99 – structure, 99 – induced, 101 spin c – group, 100 – structure, 100, 240 – induced, 101 stabilization – negative, 137 – positive, 137 state traversal, 205 Stein – cobordism, 124 – domain, 124, 162 – manifold, 121 – surface, 122 surface bundle, 264 surgery, 27 – Dehn, 31 – exact triangle, 244 – rational, 31 symplectic – cut-and-paste, 111 – dilation, 111, 123 – form, 49 – manifold, 49 – minimal, 58, 177 – neighborhood theorem, 56 – structure, 49 – deformation equivalent, 54 – equivalent, 54 – singular, 54 – standrad, 49 – submanifold, 51 Symplectic Thom conjecture, 227 symplectization, 71 3-dimensional invariant, 105 Thom conjecture, 232 Thurston–Bennequin – framing, 68 – invariant, 166 tight contact structure, 21 totally real submanifold, 51 transverse knot, 68 2-plane ﬁeld, 102 twisted coeﬃcient system, 254 twisting, 138 unknotting number, 18 vanishing cycle, 156 Vitushkin’s conjecture, 17 Weinstein handle, 115 Whitney – disk – holomorphic, 240 – trick, 11 writhe, 73