B. Ozbagci and A. I. Stipsicz- Surgery on contact 3-manifolds and Stein surfaces

Surgeryon

contact3-manifolds

and

Stein surfaces

B. Ozbagci and A. I. Stipsicz

Surgeryon

contact3-manifolds

and

Stein surfaces

B. OzbagciA. I. Stipsicz

JANOS BOLYAI MATHEMATICAL SOCIETY

Budapest, Fo u. 68., H–1027, Hungary

c© BOLYAI JANOS MATEMATIKAI TARSULAT

Budapest, Hungary, 2004

ISBN: ??? ???? ???

Published by

JANOS BOLYAI MATHEMATICAL SOCIETY

Budapest, Fo 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 symplectic4-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. Spinc structures on 3- and 4-manifolds . . . . . . . . . . . . . . . . . . . . . . 99

6.1. Generalities on spin and spinc structures . . . . . . . . . . . . . . . . . . . . 99

4 Contents

6.2. Spinc structures and oriented 2-plane fields . . . . . . . . . . . . . . . . . . 102

6.3. Spinc 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 S1 ×D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Ozsvath–Szabo 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 topologicalaspect is strengthened by the existing results of Weinstein and Eliashberg(and Gompf in dimension 4) on handle attachment in the symplectic andStein category, and by Giroux’s theory of convex surfaces, enabling us toperform surgeries on contact 3-manifolds. The main objective of these notesis to provide a self-contained introduction to the theory of surgeries one canperform on contact 3-manifolds and Stein surfaces. We will adopt a verytopological point of view based on handlebody theory, in particular, onKirby calculus for 3- and 4-dimensional manifolds.

Surgery is a constructive method by its very nature. Applying it in anintricate way one can see what can be done. These results are nicely com-plemented by the results relying on gauge theory — a theory designed toprove that certain things cannot be done. We will freely apply recent resultsof gauge theory without a detailed introduction to these topics; we will becontent with a short introduction to some forms of Seiberg–Witten theoryand 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 andcontact topology can be very fruitfully capitalized when coupled with someform of Seiberg–Witten theory. On the other hand, Lefschetz pencils onsymplectic, and open book decompositions on contact manifolds are well-suited for the newly invented contact Ozsvath–Szabo invariants. Undersome fortunate circumstances these dual viewpoints provide interesting re-sults in the subject. As a preview, Chapter 1 is devoted to the descriptionof 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 turnto the symplectic cut-and-paste operation, which enables us to glue sym-plectic 4-manifolds along contact type boundaries. To put this operationin the right perspective, in Chapters 3 and 4 we first briefly review someparts 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 makea little digression and discuss spinc 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 toω-convex boundaries along Legendrian knots. After having these prepara-tions, we can turn to the discussion of the famous result of Eliashberg thatshows how to attach a Stein 2-handle to the pseudoconvex boundary of aStein domain along a Legendrian knot. For the convenience of the moretopologically minded reader, in Chapter 8 a short recollection of rudimentsof the theory of Stein manifolds is included. Once the gluing constructiongiven, we can turn to its applications, including the search for Lefschetz fi-bration structures on Stein domains, embeddability of Stein domains intoclosed surfaces with extra (symplectic or complex) structures, or the studyof Stein fillings of contact 3-manifolds (Chapters 10 and 12). In the contactsetting, the most important technique for being able to do surgery is theconvex surface theory developed by Giroux. After recalling relevant partsof this beautiful theory, and proving the neighborhood theorems we need inthis subject, in Chapter 11 we will be able to do contact surgeries. Withthis construction at our disposal, now we can seek for applications: we willbe able to draw explicit diagrams of many contact 3-manifolds, show waysto distinguish them and to determine the homotopy type of contact struc-tures given by various constructions. These results — together with variousversions of gauge theories, including Seiberg–Witten theory and HeegaardFloer theory — provide ways to examine tightness and fillability propertiesof numerous contact structures, which are given in Chapter 12. To makethe presentation more complete, we include Chapter 9 on open book de-compositions and their relation to contact structures. The appearance ofmapping class groups in these theories, together with some nice applicationsallows us to conclude the discussion with a short recollection of definitionsand results in that field.

To guide the interested reader, we close this preface by listing somemonographs 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 andtopology is provided by [111]. For additional reading on contact topology,the reader is advised to turn to [1, 2, 57]. Seiberg–Witten theory is coveredby many volumes, including for example [119, 126, 149].

These notes are based on two lecture series given by the second author atthe Banach Center (Warsaw, Poland) and at the University of Lille (France).He wants to thank these institutions for their hospitality. The final form ofthe notes were shaped while the authors visited KIAS (Seoul, Korea); theywish to thank KIAS for its hospitality. The authors would like to thankSelman Akbulut, John Etnyre, Sergey Finashin, David Gay, Paolo Lisca,Gordana Matic and Robert Szoke for many enlightening conversations.Special thanks go to Hansjorg Geiges for suggesting numerous correctionsand improvements of an earlier version of the text. The second authoralso wants to express his thanks to his family — without their support thisvolume would not have come into existence. The first author acknowledgessupport from the Turkish Academy of Sciences and from Koc University.The second author acknowledges partial support by OTKA T034885 andT037735.

Istanbul and Budapest, 2004. Burak Ozbagci and Andras Stipsicz

1. Introduction

1.1. Why symplectic and contact?

The intense interest of 4-manifold topologists in symplectic geometry andtopology 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) thealgebraic and geometric intersection numbers of two submanifolds can bemade equal by isotoping one of them. In other words, by isotopy we canget rid of “excess intersections”, which are present in the geometric picturebut are invisible for algebra. After eliminating these intersections “algebrawill govern geometry”, and the smooth classification problem of manifoldscan 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 thismap is an embedding (with the appropriate normal bundle), we get a localmodel showing us the required isotopy. Once the dimension is high enough(at least 5), any map from the disk admits a perturbation such that theresult is an embedding. In dimension four, however, the disk might haveself-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 fourprovided we allow topological isotopies. In fact, as Donaldson’s theorems onthe failure of the smooth h-cobordism theorem in dimension 4 show, in someexamples the excess intersections persist if we allow smooth maps only.

It is a standard fact that in a complex manifold complex submanifoldsintersect (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 submanifoldsstill intersect positively — with the usual restrictions of not sharing com-mon components, see [109]. The problem with almost-complex manifolds isthat although the existence of the structure is guaranteed by some simpleproperties of the cohomology ring H∗(X4; Z), it is very hard to show thatalmost-complex submanifolds exist (in general they do not), i.e., that theabove 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 alreadyguarantee 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 classificationproblem. The spectacular results built on Taubes’ work nevertheless showthe above described principle in action. For this reason we chose to studysymplectic 4-manifolds (and their topological counterparts, Lefschetz fibra-tions) in more detail. According to Donaldson’s result, symplectic manifoldsalways decompose along a circle bundle into a union of a disk bundle andanother piece which can be endowed with a Stein structure. Conversely, anyStein surface embeds into some closed symplectic 4-manifold. The analogybecomes even deeper if we study the topological counterparts of symplec-tic and Stein manifolds: these are Lefschetz fibrations with closed or withbounded fibers. Therefore it appears natural to study topological propertiesof 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 bookdecompositions on the 3-manifolds) fits perfectly into this picture sinceopen book decompositions can be interpreted as boundaries of Lefschetzfibrations. (In the general case we allow achiral Lefschetz fibrations aswell.) The fascinating, and still not completely well-understood interplay ofthe above notions provides the leading theme of these notes.

Topological questions regarding symplectic 4-manifolds and Lefschetzfibrations are fairly well-treated in the literature ([66, 111]); in these in-stances we merely restrict ourselves to quoting the necessary results. Forcontact surgeries and open book decompositions the available sources areless 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 understandingtopological properties of Stein surfaces and contact 3-manifolds. In orderto attack such a problem we need two major tools, which provide existenceand nonexistence results. Complex geometry (e.g., complex surfaces, Mil-nor fibers, links of singularities) provides a rich source of examples, givingthe needed existence results. A more systematic way of studying the ex-istence problem is provided by the theory of handlebodies — initiated bySmale, Milnor and Kirby, and extended to the symplectic and Stein cate-gory by Weinstein, Eliashberg and Gompf. On the boundary, the handleattachment translates into contact surgery, showing existence of a varietyof contact structures. By suitably generalizing the attachment scheme de-scribed by Weinstein (and incorporating achiral Lefschetz fibrations intothe theory), in fact all contact 3-manifolds can be treated in this way. Onthe other hand, gauge theory (more specifically, Seiberg–Witten theory andOzsvath–Szabo invariants) can be used to prove that manifolds or diffeo-morphisms with certain properties do not exist. Therefore Seiberg–Wittenand Ozsvath–Szabo 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 specificcontact 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 itturns out, the existence of a Stein structure on a 4-manifold X considerablyconstrains its differential topology. The most apparent constraint can besummarized by the adjunction inequality given in Theorem 1.2.1. Closelyrelated formulae appear in many other branches in 3- and 4-dimensionaltopology, and these type of results always play a central role in the theory athand. (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 ⊂ Xis a closed, connected, oriented, embedded surface of genus g in it, then

[Σ]2 + |⟨c1(X), [Σ]⟩ | ≤ 2g − 2

unless Σ is a sphere with [Σ] = 0 in H2(X; Z).

14 1. Introduction

Remark 1.2.2. Note that if C is a (smooth) connected complex curve ina complex surface X then the Whitney product formula for Chern classesimplies 2g(C) − 2 = [C]2 −

⟨c1(X), [C]

⟩; this equation is frequently called

the adjunction equality. Its generalization for closed complex surfaces andsmooth submanifolds Σ ⊂ X was proved first by Kronheimer and Mrowka(in the case [Σ]2 ≥ 0) and in general by Ozsvath and Szabo [133]. Formore about the adjunction inequality see Section 13.3, where we will indi-cate how such formulae for a closed symplectic 4-manifold X follow fromSeiberg–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 X1, X2 defined by theknots K1, K2 of Figure 1.1 are homeomorphic but nondiffeomorphic.

−1 −1

−4

K K1 2

Figure 1.1. Homeomorphic but nondiffeomorphic 4-manifolds


The meaning of such Kirby diagrams will be explained in Chapter 2; for amore thorough treatment see [66]. Here we just note that the knots (togetherwith the numbers) indicate how to glue a 4-dimensional 2-handle D2 ×D2

along ∂D2 ×D2 to D4 in order to get X1 and X2, respectively.

Proof (sketch). Using some simple operations on the diagrams one canshow that the 3-manifolds ∂X1 and ∂X2 are diffeomorphic (see [66, Fig-ure 11.4]). Now the signatures σ(X1) and σ(X2) are both equal to −1, theEuler characteristics are both equal to 2, and since ∂X1 = ∂X2 is an in-tegral homology sphere, the extension of Freedman’s famous theorem (see[51]) implies that X1 and X2 are homeomorphic. Next we show that X1

carries a Stein structure. This follows from the theory of gluing symplectichandles (developed by Weinstein and Eliashberg), once we realize that K1

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 X1 from X2 itis enough to prove that X2 does not admit any Stein structure. This state-ment follows from the observation that the generator of H2(X2; Z) (whichhas self-intersection −1) can be represented by a sphere. Such a represen-tative can be easily found once we get a disk D2 ⊂ D4 with ∂D2 = K2 —glue the core of the 2-handle to this disk. The existence of such a disk isshown 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 D4 as r grows from 0 to1. These intersections start with two circles (which are boundaries of twodisks) which get tangled as “time” passes (Figures 1.3 (1)–(2)), and thena ribbon is added to connect the disks, resulting an embedded disk in D4

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 ∂X1

∼= ∂X2.) Now the application of the adjunction inequality withg = 0 and [Σ]2 = −1 would give

−1 + |⟨c1(X2, J), [Σ]⟩ | ≤ −2,

a contradiction for any Stein structure J on X2. Therefore X2 cannot carryany Stein structure, implying that X1 and X2 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 proofis due to Akbulut and Matveyev [6].


Existence of Stein neighborhoods

Theorem 1.2.5. Let S ⊂ CP2 be a smoothly embedded sphere in CP2

which is nontrivial in homology. Then there is no open set U containing aneighborhood 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] 6= 0 in H2(CP2; Z) we have that [S]2 > 0, providing the result.

Remark 1.2.6. The same argument works for any smooth 2-dimensionalsubmanifold Σ in a complex surface X with 2g(Σ) − 2 < [Σ]2. Surprisinglyenough, if the inequality [Σ]2 + |⟨c1(X), [Σ]

⟩ | ≤ 2g(Σ)−2 does hold (noticethat because of the absolute value this is, in fact, the union of two inequal-ities) then there is a Stein neighborhood U ⊂ CP2 of Σ, see [49]. For theoutline of this latter argument see also Section 8.3.

This application of the adjunction inequality leads to the solution of aseemingly unrelated problem in complex analysis.

Corollary 1.2.7 (Nemirovski, [125]). Suppose that S ⊂ CP2 is a smoothlyembedded sphere in CP2 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 Vcontaining U such that every holomorphic function on U extends holomor-phically to V . Denote this envelope of holomorphy by U . According to aresult of Fujita, in our case U is either CP2 or it is Stein. Since S ⊂ U ⊂ U ,Theorem 1.2.5 shows that U cannot be Stein. Therefore U = CP2, henceall holomorphic functions on U (and so on U) are constant.

Remark 1.2.8. Notice that if S is a complex submanifold generatingH2(CP2; Z) then the statement is obvious: if f is a holomorphic function onU then by restricting it to U ∩ (CP2 − CP1) ⊂ C2 and applying a theoremof Hartogs we get an extension of f to CP2, implying that it is constant.The question answered by the above theorem was raised by Vitushkin, andsimilar results (for higher genus and immersed surfaces) are still in the focusof current research, see [49, 125].

18 1. Introduction

The four-ball genus of knots in S3

Let K ⊂ S3 be a given knot. The genus g(K) is defined as

ming(Σ) | Σ ⊂ S3 is a Seifert surface for K

.

For example, it is fairly easy to see that g(K) = 0 holds if and only if K isthe unknot. The four-ball genus (or slice genus) g∗(K) can be defined as

ming(F ) | F ⊂ D4, ∂F = K

,

where F denotes a smoothly embedded connected surface in D4 transverseto ∂D4. Obviously g∗(K) ≤ g(K), and as the proof of Corollary 1.2.3showed, g∗(K) can be equal to 0 for a nontrivial knot K, e.g. for K2.(Knots with vanishing four-ball genus g∗ are called smoothly slice.) Theadjunction inequality provides a nontrivial lower bound for g∗(K) in thefollowing way. Approximate K with a Legendrian knot L and glue a 2-handle to D4 along K with surgery coefficient one less than the contactframing of L. The resulting 4-manifold X will contain a surface F withg(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 [F ]2

and⟨c1(X), [F ]

⟩admit expressions purely in terms of data of the Legendrian

knot L as [F ]2

= tb(L) − 1 and⟨c1(X), [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 withtb(L) = 1 and rot(L) = 0 (see Figure 1.4), hence the adjunction inequalitytranslates as 0 ≤ 2g∗(K) − 2, implying 1 ≤ g∗(K). It is not hard to finda genus-1 Seifert surface for K, therefore we see that the four-ball genus ofthe trefoil knot is 1.

The unknotting number (or gordian number) u(K) of a knot K is definedas the minimal number of crossing changes in any projection which untiethe knot.

Exercise 1.2.10. Show that u(K) ≥ g∗(K) for any knot K ⊂ S3.


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 K2 of Figure 1.1, which has g∗(K2) = 0 butu(K2) > 0 since it is not the unknot. Heegaard Floer theory provides knotinvariants which can be fruitfully used to get new constraints on the 4-ballgenus of a knot, see [129, 142].

Topological characterization of Stein domains

According to a recent result of Loi and Piergallini [104], Stein domains admita nice topological description in terms of Lefschetz fibrations.

Theorem 1.2.11 (Loi–Piergallini, [104]). If S is a complex 2-dimensionalStein domain then it admits a Lefschetz fibration structure over D2.

This result — similarly to Donaldson’s result on existence of Lefschetzpencils on closed symplectic 4-manifolds — brings a topological flavour intothe study of Stein domain. The original proof of Theorem 1.2.11 relies onan approach of presenting the 4-manifolds at hand as branched covers ofD4 along fairly complicated branch sets. A conceptually simpler proof ofthe same statement was given by Akbulut and the first author [7], makinguse of the handle decomposition of a Stein domain and relating it to handledecompositions of 4-manifolds admitting Lefschetz fibrations. The detaileddescription 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 domaincannot be arbitrary.

Theorem 1.2.12 (Lisca, [94]). Let E denote the boundary of the (+E8)-plumbing W (as shown by the plumbing diagram of Figure 1.5). There is

2 2 2 2 2 22

2

Figure 1.5. The (+E8)-plumbing

no Stein domain S with ∂S = E.

Proof. Using standard pull-apart arguments in Seiberg–Witten theory (seeChapter 13) it can be shown that if X = X1 ∪E X2 and X is symplecticthen b+2 (X1) = 0 or b+2 (X2) = 0. (This argument uses the fact thatE admits a positive scalar curvature metric, since it is diffeomorphic tothe Poincare 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 canbe embedded into a closed symplectic 4-manifold X with b+2 (X−S) > 0. Inconclusion, from the above principle we get b+2 (S) = 0. Therefore the closed4-manifold Z = S ∪E (−W ) is negative definite. Since the intersection formof −W (which is the famous negative definite E8-form) does not embedinto any diagonal intersection form, the intersection form QZ cannot bediagonalized over the integers. This last consequence, however, contradictsDonaldson’s famous result about diagonalizability of definite intersectionforms of smooth 4-manifolds, showing that S cannot exist.

Remark 1.2.13. Analogous statements have been proved for the bound-aries of the (+E7)- and (+E6)-plumbings [96]. Results of this type willbe 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 moreof 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 differentclasses. Overtwisted structures were classified by Eliashberg, and the clas-sification scheme depends only on homotopic properties of the underlying3-manifold. On the other hand, tight structures are expected to containmore geometric information about the manifold. The contact counterpartof the adjunction formula (frequently called the Bennequin inequality) char-acterizes tight structures. This inequality reads as follows: Suppose that Σis 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 tbΣ(L) ∈ Z denote the framing induced by the contactstructure on L with respect to the framing Σ defines on L and rotΣ(L) therelative Euler number of ξ|Σ with ξ trivialized along ∂Σ by the tangentsof L. Now

Theorem 1.3.1 (Eliashberg, [26]). The inequality

tbΣ(L) +∣∣ rotΣ(L)

∣∣ ≤ −χ(Σ)

is satisfied for all L and Σ if and only if the contact 3-manifold (Y, ξ) istight.

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 givesa characterization of contact 3-manifolds in terms of open books. Girouxproved that there is a one-to-one correspondence between open books andcontact 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 decompositionof Y there is a compatible contact structure ξ on Y . Contact structurescompatible 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 fixedcontact structure admit common positive stabilization.

22 1. Introduction

The reinterpretation of contact structures provided by this theorem enablesus to treat them as topological objects. The nicest manifestation of thisprinciple is probably the definition and application of contact Ozsvath–Szabo invariants discussed in Chapter 14. It is still an open (and veryintriguing) question how the monodromy of the open book decompositionencodes 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 fillablecontact structure if and only if it admits a positive stabilization for whichthe 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 6= 0. In thiscase 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 Bennequininequality 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 beingthe appropriate boundary of any symplectic 4-manifold. The first suchstructures were found by Etnyre and Honda [44]; we will give a variety ofsuch contact 3-manifolds, cf. also [100, 101]. Next we give a sample of theseresults. Contact surgery provides a simple way for constructing contact 3-manifolds. Because of its topological character, the surgery diagram can beused 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 surgerydiagram of Figure 1.6 is tight but not fillable.

Proof (sketch). Nonfillability of this contact structure follows from thefact that the underlying 3-manifold Y is diffeomorphic to the boundary ofthe (+E7)-plumbing (see Exercise 2.3.5(e) together with Remark 1.2.13).Tightness follows from the fact that the contact Ozsvath–Szabo invariantc(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 withat least n distinct tight, nonfillable contact structures.

These examples will be given by Figure 12.6. Once again, tightness willbe proved by (partially) determining the contact Ozsvath–Szabo invariants,while we will show that the structures are nonfillable through determininghomotopic properties of the contact structures via analyzing the diagramand then apply a version of Seiberg–Witten theory. This last step is astraightforward 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-manifoldis 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 contact3-sphere (S3, ξst) then W is diffeomorphic to the 4-dimensional disk D4.

Proof (sketch). Gauge theory as applied in the proof of Theorem 1.2.12implies that b+2 (W ) = b−2 (W ) = 0. The surgered manifold Z = W ∪(CP2−D4) is a symplectic 4-manifold containing a symplectic sphere of self-intersection (+1), hence Z is symplectomorphic to CP2. Since a symplecticsphere representing the generator of H2(CP2; Z) is isotopic to the complex

24 1. Introduction

line CP1 ⊂ CP2, we get that W is diffeomorphic to D4. For more details ofthis argument see Section 12.3.

Similar strong classification results of Stein fillings have been obtainedfor a variety of 3-manifolds (see [96, 110] or Section 12.3), but the generaldescription of all Stein fillings of a contact 3-manifold is still missing. Herewe restrict ourselves to two statements along these lines:

Theorem 1.3.7 ([132]). There are contact 3-manifolds with infinitely manynondiffeomorphic Stein fillings.

The proof of this theorem (see in Section 12.3) makes use of the connectionbetween Stein structures and Lefschetz fibrations. Using symplectic cut-and-paste technique and applying Seiberg–Witten theory we will get somerestrictions 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 existsa 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 ) fora Stein filling W of (Y, ξ) — which resembles the c21-invariant of a closedcomplex surface — is bounded from below.

A little elaboration of the above result together with some specific casesgives evidence for the following

Conjecture 1.3.9. For any contact 3-manifold (Y, ξ) there is a constantK such that if W is a Stein filling of (Y, ξ) then for its Euler characteristicχ(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 Steinsurgeries. 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 furtherexplanation. The manifolds appearing in these notes are all assumed to besmooth (i.e., C∞-) manifolds, possibly with nonempty boundary. The gen-eral discussion of handlebodies will be followed by a short overview of Dehnsurgeries in dimension three, and an outline of Kirby calculus concludes thechapter. 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 X1,X2 are given n-dimensional man-ifolds with boundaries and Zi ⊂ ∂Xi are (n− 1)-dimensional submanifolds(with possibly nonempty boundary). For a diffeomorphism f : Z1 → Z2 wecan glue the two manifolds X1 and X2 together along Zi via f , and get anew n-manifold X = X1#f(Z1)=Z2

X2 (with possibly nonempty boundary).In the following we will always assume that Xi and Zi are compact (andthen so is X), and that the Xi are oriented. Note that an orientation of Xi

induces one for ∂Xi and so orients Zi as well. In order to have a canonicalorientation for X, we assume that f reverses orientation.

Remark 2.1.1. In order to give a manifold structure to X we have toround off the corners created by gluing along Zi (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 bymultiplying this picture with the extra dimensions, the same can be carriedout in arbitrary dimensions, see [66].

The reason why the above construction works is that the boundaries ∂Xi

and so Zi admit “canonical” neighborhoods (by the collar neighborhoodtheorem), hence once the map f is fixed, neighborhoods of Zi can beidentified and so the smooth structures can be patched together. The samescheme will work for other structures (like symplectic, contact, and so on)once the right assumptions ensuring canonical neighborhoods have beenmade. The drawback of this general construction is that usually it is quitehard to describe and identify f , although — as we will see — in many casesthe particular choice of the identification is crucial. Here are a few simpleexamples of this operation:

Examples 2.1.2. (a) Suppose that X1, X2 are compact manifolds withboundaries ∂X1, ∂X2 orientation reversing diffeomorphic via a smooth mapf : ∂X1 → ∂X2. Then X = X1 ∪f X2 is a closed manifold.

(b) Suppose that Xi are closed n-manifolds. Consider Xi − intDn andglue them with an orientation reversing map f : Sn−1 = ∂(X1 − intDn) →Sn−1 = ∂(X2 − intDn) which extends to the disk Dn. (This latter require-ment specifies f up to isotopy.) The resulting manifold X = X1#X2 iscalled the connected sum of the two manifolds X1 and X2. For Xi con-nected, the result can be proved to be independent of the choice of thedisks.

(c) Another special case of this general construction is when Zi ⊂ ∂Xi

(i = 1, 2) are both diffeomorphic to the (n − 1)-dimensional disk Dn−1.The resulting manifold X is usually denoted by X1X2 and is called theboundary connected sum of X1 and X2 along Z1 and Z2. As in the previouscase, the result can be proved to be independent of the choices provided theboundaries ∂X1 and ∂X2 are connected.

(d) Suppose that X = X1 ∪ΣXX2 and Y = Y1 ∪ΣY

Y2 are closed manifolds(Xi ⊂ X and Yi ⊂ Y , i = 1, 2, are compact codimension-0 submanifoldswith boundaries and disjoint interiors). If there is an orientation preservingdiffeomorphism f : ∂X1 → ∂Y1 then we can use it to glueX1 and Y2 togetheralong their boundaries to get Z = X1 ∪ Y2 (and similarly V = Y1 ∪X2), seeFigure 2.1. The choice of f is usually crucial in this construction.

Exercises 2.1.3. (a) Let X1 = X2 = [0, 1] × [0, 1] and Z1 = Z2 =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

XΣ YΣ

Figure 2.1. Flipping X2 with Y2

manifold Xf is diffeomorphic to S1 × [0, 1] and g such that the result Xg isa Mobius band.

(b) Show that both S3 and S1 × S2 can be built by gluing two solid toriS1 ×D2 together (using different identifications of the boundary tori).

(c) More generally, show that every closed, oriented 3-manifold Y can begiven as Y = H1 ∪H2 where H1

∼= H2 are solid genus-g three-dimensionalhandlebodies with ∂H1

∼= ∂H2 = Σg, where Σg stands for the genus-gsurface. (Hint: Use a Morse function.) Such a decomposition is usuallycalled a Heegaard decomposition of Y .

(d) Verify that ∂(X1X2) = ∂X1#∂X2.

Notice that ∂(Sk ×Dn−k) = ∂(Dk+1 × Sn−k−1) = Sk × Sn−k−1, henceif Sk ⊂ Xn is a submanifold with trivial normal bundle νSk ∼= Sk ×Dn−k

then cutting out Sk × Dn−k and gluing back in Dk+1 × Sn−k−1 we geta new manifold. Once again, the chosen identifications do matter. Atrivialization of νSk in X is called a framing of the submanifold. By fixinga framing ϕ we get an embedding ϕ : Sk × Dn−k → X. (If k ≤ 3 thenthe parametrization of Sk ⊂ X is unique, otherwise we think of Sk ⊂ Xas given by the image of a map.) Now we can use ϕ|∂(Sk×Dn−k) to glue

Dk+1 × Sn−k−1 back in; the new manifold is the result of the surgery of Xalong Sk (with the given framing). Notice that the connected sum operationof Example 2.1.2(b) is just a special case of this surgery scheme: Embedthe disconnected manifold S0 × Dn = ±1 × Dn into X1 ∪ X2 in such away that −1 ×Dn ⊂ X1 and 1 ×Dn ⊂ X2 and then do surgery on it,i.e., replace S0 ×Dn with the cylinder D1 × Sn−1. The effect of the abovesurgery 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


handle of index (k + 1)) Dk+1 × Dn−k along the part ∂Dk+1 × Dn−k ofits boundary to ϕ(Sk) ⊂ X × 1 with the specified framing — the imageϕ(Sk) and the framing completely determine the gluing map f . During thisconstruction the tubular neighborhood ϕ(Sk)×Dn−k sinks in the interior ofthe cobordism and Dk+1 × Sn−k−1 appears on the surface. More generally,if X is an (n + 1)-manifold with boundary ∂X and ϕ : Sk × Dn−k → ∂Xis a given embedding, i.e., a framed sphere ϕ(Sk) is given, then we canglue the (n + 1)-dimensional (k + 1)-handle Dk+1 × Dn−k to X using ϕand get a new manifold. The repeated application of the above process(starting with a given closed n-manifold M and considering M × [0, 1]) iscalled a (relative) handlebody built on M . Notice that M might be theempty manifold, in which case we get a handlebody. (In that case, to startthe process, we first glue a 0-handle D0 ×Dn+1 along ∂D0 ×Dn+1 = ∅ tothe empty manifold.) It can be shown that any compact smooth manifoldadmits 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 spherecan 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. Byassuming linearity on the fibers (which can be achieved by an isotopy),any other framing defines a linear map at every point of the sphere (whichlinear map matches up the chosen bases in the fiber of the normal bundle),so at the end we get a map Sk → GLn−k(R). Since homotopy does notchange the framing, and GLn−k(R) retracts to O(n − k), we conclude thatthe different framings of the k-dimensional sphere Sk in an n-manifoldare parametrized by πk

(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 aretwo possible framings, corresponding to the two components of O(n). Onegives rise to an orientable, while the other to a nonorientable manifold.Since we restrict our attention to the study of orientable manifolds, we getuniqueness of framings even for k = 0. Notice that if we are dealing withoriented 3- and 4-manifolds (so n = 2 or 3), then there is only one morecase 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 frameembedded circles in 3-manifolds, and the set of framings is parameterized byπ1

(O(2)

) ∼= π1(S1) ∼= Z. In this special case the normal D2-bundle can be

regarded as a complex line bundle, hence it can be trivialized by a nowherevanishing 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 orderto set up an actual isomorphism between the set of framings and Z, we needto choose a preferred framing first (which we will call the 0-framing). Thischoice, however, is canonical only in some special cases: for example, in S3

or if the knot is null-homologous in the 3-manifold Y . Another instanceof the existence of a canonical framing is provided by the situation whenthe knot is in Legendrian position in a contact 3-manifold, or if the knotis naturally contained in a surface (which induces a natural framing bypushing the knot off of itself inside the surface) — such a surface can beprovided by a fiber of a fibration, or a page of an open book decomposition,for example.

Exercises 2.1.4. (a) For K ⊂ S3 fix a Seifert surface and consider the 0-framing to be the push-off of the knot along the Seifert surface. Show thatthis framing — called the Seifert framing — does not depend on the chosenSeifert surface, and that the isomorphism between the space of framingsand Z is given by the linking number of K and the push-off of it along theframing. (For the linking number to make sense, fix an orientation on Kand orient any push-off accordingly.)

(b) Generalize the uniqueness of the Seifert framing for any null-homologousknot in an arbitrary 3-manifold Y . (Hint: Argue that if two differentframings come from Seifert surfaces then their difference vanishes in thefirst homology of the knot complement, contradicting the fact that the knotis null-homologous.)

(c) Verify that the push-off K ′ of a null-homologous knot K ⊂ Y definesthe Seifert framing if and only if the homology class [K ′] of K ′ vanishes inH1(Y −K; Z).

Remark 2.1.5. When drawing the projection of a knot K ⊂ R3 in R2

(with the usual genericity assumptions and the conventions of over- andunder-crossings), there is one more natural framing we can consider: it isthe blackboard framing bb(K) induced by the particular projection. We getthe blackboard framing bb(K) by pushing K off along a vector field parallelto the plane of the projection, see Figure 2.2. Notice that this framingheavily depends on the chosen projection. The conversion between theSeifert framing and the blackboard framing is given by the linking numberof K and its parallel push-off.

Exercise 2.1.6. For a given knot K ⊂ S3 compute the blackboard framingbb(K) of one of its projections with respect to the Seifert framing of the knot.


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. Forthis to make sense, we need to fix an orientation on K, but w(K) can beproved to be independent of this choice of orientation. For the sign of acrossing see Figure 2.3.)

_+Figure 2.3. Positive and negative crossings

As a consequence of our framing computation above, we get that inorder to build 3-dimensional orientable handlebodies one only needs to keeptrack of the attaching spheres. For 1-handles these are essentially uniquesince S0 = −1 ∪ 1 embeds into a connected manifold uniquely up toisotopy. For 2-handles we get embedded circles in a genus-g surface — orequivalently, in the plane with 2g holes, which are glued together in pairs.For convenience, we identify S2 with R2 ∪ ∞ and draw the diagrams inits “finite” part R2. 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 eachother. In case of more pairs of shaded disks we connect the pairs to beidentified with dotted lines. The curve of Figure 2.4 might seem to havemany components, but after identifying the shaded disks it becomes aconnected 1-manifold.) Finally, a 3-handle can be attached uniquely to

Figure 2.4. A Heegaard diagram

a 3-manifold with boundary diffeomorphic to S2. In dimension four the2-handle attachment is somewhat more complicated; we will return to thedetailed discussion of this question in Section 2.3.

2.2. Dehn surgery

There is one more — purely 3-dimensional — construction we would like todiscuss, frequently called Dehn or rational surgery. The basic idea is againpretty simple: consider a 3-manifold Y (for simplicity we assume that it isclosed), and fix a knot K ⊂ Y in it. By deleting a tubular neighborhoodνK (∼= S1 × D2) of K and regluing it via a diffeomorphism f : ∂νK →∂(Y − νK) we get a new 3-manifold. Obviously, the resulting manifold willdepend on the chosen gluing map f . Notice that we reglue S1 ×D2 alonga (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 mapf∗ : H1(T

2; Z) → H1(T2; Z) ∼= Z2, i.e., (after fixing a basis of H1(T

2; Z))by a 2 × 2 integer matrix. Since f (and so f∗) is invertible, the matrixis of determinant ±1; the fact that f reverses orientation implies thatdet f∗ = −1. Consequently, after fixing a basis of H1(T

2; Z), four integersspecify f . Different matrices might yield diffeomorphic 3-manifolds, forexample, if for the maps f1, f2 : ∂(S1×D2) → ∂(Y −int νK) the composition


f−12 f1 : ∂(S1 ×D2) → ∂(S1 ×D2) extends to a diffeomorphism of S1 ×D2

then the surgered manifolds using f1 or f2 will be diffeomorphic. Actually,two of the four numbers already determine the surgery; we show this factfrom a slightly different point of view. Notice that S1 ×D2 can be thoughtof as the union of a 3-dimensional 2-handle and a 3-handle, and rememberthat the gluing of 3-handles is unique, while for 2-handles one only needsto 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 homologyclass a ∈ H1

(∂(Y − int νK); Z

) ∼= Z2. This is the curve which bounds thedisk pt. ×D2 in the surgered manifold.

Definition 2.2.1. For a fixed closed 3-manifold Y , knot K ⊂ Y andprimitive element a ∈ H1


)the manifold (Y − int νK) ∪f

(S1 ×D2) will be denoted by Ya(K), where f : ∂(S1 ×D2) → ∂(Y − int νK)is specified by f∗

[pt.× ∂D2

]= a ∈ H1


). The resulting

manifold Ya(K) is called the Dehn surgery of Y along K with slope a. Noticethat since

[pt.×∂D2

]is nondivisible and f∗ is invertible, the chosen class

a should be a primitive class. It can be proved that the simple closed curverepresenting such a homology class a is unique up to isotopy.

A choice of a basis of H1


) ∼= Z2 converts a into a pairof relatively prime integers. A canonical choice for one basis element is pro-vided by the meridian µ ∈ H1


)of the knot K — i.e., a

nontrivial primitive element which vanishes under the embedding of ∂νKinto νK. By fixing an orientation on K, the element µ is uniquely deter-mined by the requirement that it links K with multiplicity +1 (otherwiseµ is determined only up to sign). Informally, µ is the homology class of thecircle which is the boundary of a small normal disk to K (i.e., a small diskintersecting K transversely in a unique point), see Figure 2.5(a). The choiceof a longitude λ (another basis element in H1


)) is, how-

ever, not canonical. For an example see Figure 2.5(b). The longitude can befixed without further choices only if K admits a canonical framing. In fact,fixing a longitude or a framing is equivalent, since the normal R2-bundle(regarded as a C-bundle) is trivialized by a nonvanishing section, i.e., by alongitude. Therefore, if K ⊂ Y is null-homologous (for example, Y = S3)or if K is a Legendrian knot in a contact manifold, then using the canonicallongitude λ the homology class a can be converted into a pair of relativelyprime integers (p, q) by setting a = pµ+ qλ, but in general the integers will


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 bothµ 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 ∞ = 10 is also allowed. Therefore after fixing a lon-

gitude if needed, the rational number pq ∈ Q ∪ ∞ encodes all the gluing

information we need. For this reason Dehn surgery is also called rationalsurgery. Notice finally that by definition the difference of two framings λ1

and λ2 is some integral multiple of the class µ.

Remark 2.2.2. The two integers p and q can be easily recovered from thematrix f∗, since it maps the meridian of ∂(S1 × D2) into pµ + qλ. Thisshows 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 pq ∈ Z (i.e., q = ±1)

then Ya(K) = Y pq(K) can be given by an ordinary surgery, i.e., by a (4-

dimensional) 2-handle attachment. If pq = ∞ (i.e., q = 0) then Ya(K) = Y

for any knot K.

Proof. The coefficient pq 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.


This shows that integral Dehn surgery has the same effect as 4-dimensionalhandle attachment. If q = 0 then a = µ, so we simply glue back the 2-handle of S1 ×D2 in the way it was before the surgery.

Remark 2.2.4. Notice that the fact that pq 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 inthis case a can be represented by a meridian. Alternatively, observe thatfor a = p1µ + q1λ1 = p2µ + q2λ2 we have that q1 = q2 and (providedλ1 − λ2 = kµ) that p2 = p1 + q1k. This argument shows again that thevalue 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 S3. In fact, all rationalnumbers used in the surgery presentation can be assumed to be integers.

Proof. Since the third cobordism group Ω3 vanishes, for a 3-manifold Ythere is an oriented 4-manifold W such that ∂W = Y . Surgering out the1- and 3-handles we get a presentation of Y as the boundary of D4 ∪ some2-handles, hence as an integral surgery on a link.

For a particular example of 3-manifolds we consider lens spaces. For thismatter, for coprime integers p > q ≥ 1 take the group

Gp,q =

(z 00 zq

) ∣∣∣ z ∈ C, zp = 1

⊂ U(2)

and denote the factor S3/Gp,q by L(p, q). (By viewing S3 ⊂ C2 as vectors ofunit length, the action of U(2) on S3 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 areexactly those 3-manifolds which admit Heegaard decompositions along the2-dimensional torus T 2.

Exercise 2.2.6. Show that the lens space L(p, q) is diffeomorphic to theresult of a −p

q -surgery on the unknot in S3. (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 exercisewe list those moves which will be useful in our subsequent discussion.


Exercises 2.2.7. (a) Verify the slam-dunk operation, i.e., that the twosurgeries given by Figure 2.6 give diffeomorphic 3-manifolds. Here it isassumed that n ∈ Z and r ∈ Q ∪ ∞. (Hint: Perform surgery on K2 firstand isotope K1 into the glued-up solid torus T . Since first we performed anintegral surgery, K1 will be isotopic to the core of T , hence when performing

K

r

2

n1

K

K

n

r1

2

Figure 2.6. The slam-dunk operation

the second surgery we cut T out again and reglue it. Therefore it can bedone by one surgery; the coefficient can be computed by first assuming n = 0and then adding n extra twists. For more details see [66, pp. 163–164].)

(b) Turn a rational surgery in S3 with coefficient r into a sequence ofintegral surgeries. (Hint: Use the continued fraction expansion of r andapply (a) above. For the convention regarding continued fraction expansionssee Section 11.1.)

(c) Using the above result transform the Dehn surgery diagram of a lensspace into an integral surgery on a linear chain of unknots. Using thisdiagram 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 performedonly for n ∈ Z. Take, for example the 3-manifold given by the diagram ofFigure 2.7. Applying a slam-dunk on the −1

2 -framed circle we get L(2, 1) =RP3. But if we perform the illegal slam-dunk on the (−4)-framed circle, weget S3 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 = pq on the left and (1

r +n)−1

= pq+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


RP3

S3

1

2

1

4

−4−2

Figure 2.7. Warning with slam-dunks

intersecting the spanning disk of K changes from ri to ri + n(ℓk(K,Ki)

)2.

The term ℓk(K,Ki) denotes the linking number of the two knots K and Ki.(Hint: See [66, page 162].)

q+np

p

n

Kq

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.)


(c) Show that surgery on the disjoint union of two framed links yields theconnected sum of the two corresponding 3-manifolds. (We say that twolinks 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 theunknot provides S3.)

(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 linkpresenting Y and multiply the framings by (−1).)

In fact, there is a complete set of moves which determines when theresulting 3-manifolds are diffeomorphic:

Theorem 2.2.10 (Kirby, [80]). Two links L, L′ with rational coefficients inS3 determine diffeomorphic 3-manifolds through Dehn surgery if and only ifL can be transformed into L′ by a finite sequence of Rolfsen twists, isotopiesand inserting and deleting components with coefficient ∞.

We have to note here that in particular cases it might be quite difficult tofind the actual finite sequence of moves transforming one surgery picture ofa 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 handfrom its rational surgery diagram. For this matter, suppose that Y is givenby rational surgery on the n-component link L = (K1, . . . ,Kn) ⊂ S3 withsurgery coefficients pi

qiwith respect to the meridians µi and longitudes λi,

where these latter provide the Seifert framings for Ki in S3. It is not hardto see that H1(S

3−∪ni=1 int νKi; Z) = Zn, freely generated by the homologyclasses of the meridians: simply use the long exact homology sequence ofthe pair (S3, S3−∪ni=1 int νKi). Next, as the surgery procedure dictates, weadd a 3-dimensional 2- and a 3-handle to every T 2-boundary component ofS3−∪ni=1 int νKi. Notice that if Σi is a Seifert surface for Ki (containing thelongitude λi) then it provides the relation λi =

∑j 6=i ℓk(Ki,Kj)µj, where

ℓk(Ki,Kj) stands for the linking number of the two knots in S3. Noweach 2-handle provides a relation among the µi’s: by definition piµi + qiλibecomes zero after the surgery (since this is the curve which bounds thecore of the new 2-handle). Therefore we conclude

Theorem 2.2.11. If Y is given by Dehn surgery along (K1, . . . ,Kn) ⊂ S3

with surgery coefficients pi

qi(i = 1, . . . , n) then H1(Y ; Z) can be presented


by the meridians µi (i = 1, . . . , n) as generators and the expressions

piµi + qi∑

j 6=i

ℓk(Ki,Kj)µj = 0

as relators.

Corollary 2.2.12. If Y is given by pq -surgery along a knot K ⊂ S3 then

H1(Y ; Z) ∼= Zp. (Here Z0 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 H1(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∣∣H1(Y ; Z)

∣∣ <∞.Alternatively, Y is a rational homology sphere if and only if its first Bettinumber b1(Y ) is zero.

Exercises 2.2.13. (a) Show that the 3-manifold S3r (K) we get by r-surgery

on K ⊂ S3 is an integral homology sphere if and only if r = 1k for some

k ∈ Z.

(b) Suppose now that Y is given by (n1, . . . , nk)-surgery on the link L =(K1, . . . ,Kk) ⊂ S3 (ni ∈ Z). Verify that Y is an integral homology sphere ifand only if the determinant of the linking matrix of L is ±1. Show that Yis a rational homology sphere if this determinant is nonzero. The diagonalentries 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 the4-dimensional 2-handle attachment along L and the 3-manifold Y = ∂X.)

2.3. Kirby calculus

Suppose that Xn is a given smooth n-dimensional manifold. By choosingan appropriate Morse function on X we see that it admits a handlebodydecomposition and we can always assume that our handlebody is builtby attaching handles in the order with increasing index to the 0-handleDn. In this section we will focus on the n = 4 case. If X4 is closed then(according to a result of Laudenbach and Poenaru [91]) the gluing of theunion of 3- and 4-handles (which union is diffeomorphic to kS

1 × D3 forsome k) is unique. Therefore, in order to present closed 4-manifolds, we mayrestrict 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 surfacealways admits a handle decomposition involving 0-, 1- and 2-handles only,hence the study of 2-handlebodies is sufficient for the purposes of thesenotes. 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 waysof picturing the attachment of a 1-handle to the boundary S3 of the unique0-handle D4. (For convenience we identify S3 with R3 ∪ ∞ and use onlyits “finite part” R3). We can draw a pair of D3’s in R3, indicating where thefeet of the 1-handle are attached, or alternatively we can draw an unknotwith a dot on it, symbolizing that we consider the 4-manifold D4 − aneighborhood of a spanning disk for the above unknot in D4, i.e., a dottedcircle refers to the compact 4-manifold

(D2 − νp

)× D2, where νp

denotes a small tubular neighborhood of a point p ∈ intD2. Obviously, inboth ways we get a 4-manifold diffeomorphic to S1 × D3. We will followthe latter convention, therefore the subhandlebody X1 =union of 1-handles∼= k(S

1 × D3) will be symbolized by a k-component unlink in S3 with adot on every component: the unknots in S3 simply denote the boundariesof the disks D2 × pi (i = 1, . . . , k) deleted from D4. The 2-handles areattached along a framed link in ∂(kS

1×D3). By the above convention thislink can be regarded as lying in S3, therefore (using the Seifert framings)the surgery coefficients can be naturally converted into integers. A 2-handlepasses through a 1-handle exactly when its attaching circle links with thedotted circle of the 1-handle. Such a link presentation of the 4-dimensional2-handlebody is called a Kirby diagram.

Remark 2.3.1. One can easily convert a handle picture using the firstconvention into the dotted circle notation. To do this, first isotope allattaching circles away from the region between the two feet D3 ⊂ R3 ofthe 1-handle. Then delete the embedded 3-balls, connect the attachingcircles of the 2-handles and link them with a dotted circle. An example forthis procedure is given by Figure 2.9. The first convention (which uses theattaching balls of the 1-handle) is probably conceptually clearer, but whenmanipulating the diagram of an explicitly given 4-manifold, the dotted circlenotation — introduced by Akbulut in [3] — is much more convenient.

Exercises 2.3.2. (a) Verify that Figure 2.10 gives a diagram for D2 × 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. Sinceany two Morse functions can be joined by a path of functions, by analyzing


(1) (2)

(3)

Figure 2.9. Converting 1-handle into dotted circle

0

Figure 2.10. D2× T 2


0

Figure 2.11. An alternative diagram for D2× 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 repeatedapplications of the following moves:

• isotopies of the link in S3,

• handle slides and

• adding/deleting cancelling 1/2- and 2/3-handle pairs. (A pair ofhandles is cancelling if their union amounts to a connected sum withD4.)

In the diagram we visualize a 2-handle slide corresponding to circles K1,K2

by connect summing K1 to a push-off of K2 corresponding to its framingalong an arbitrary band. The new surgery coefficient K ′

1 becomes the sum ofthe old coefficients of the two knots ± twice their linking number — the signdepends on whether the connecting band respects or disrespects a chosenorientation on K1 and K2. 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 latteras a 0-framed 2-handle. When sliding a 1-handle over an other 1-handle wemust be careful with the choice of the band, since the resulting dotted circlesshould still form an unlink. A 1-handle/2-handle pair cancels if the 2-handleintersects the spanning disk of the 1-handle in a single point; in this casefirst we slide off all the 2-handles geometrically linking the dotted circle inquestion (using, for example, the cancelling 2-handle) and then erase the1/2-handle pair from the picture. The process can of course be reversed


by introducing a pair of knots geometrically linking once (and one is theunknot); then by putting a dot on the unknot and an arbitrary surgerycoefficient to the other knot the 4-manifold remains the same. Finally, a2-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 thepicture. Notice that on an unknot (disjoint from the other dotted circles)we can have surgery coefficient 0 or a dot — such a change corresponds tosurgery along the sphere given by the 0-framed unknot.

The operations listed above obviously do not change the boundary ∂Xof a 4-manifold X given by a diagram. Changing a dot to 0-framing (or viceversa) changes the 4-manifold but leaves the boundary intact. Besides thesemoves, we can also insert or delete a (±1)-framed unknot disjoint form therest of the picture — which corresponds to adding and removing a copy ofCP2 or CP2 — without changing the boundary of the 4-manifold. For moredetails about these operations see [66]. The art of manipulating diagramsusing the above rules and understanding the structure of smooth 3- and 4-manifolds in this way is frequently called Kirby calculus. Here we restrictedourselves to outline the very basics of this theory, and highlighted only theaspects which are important in our contact geometric studies. For a morecomplete 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] providesa proof for the fact that the 4-manifolds X1, X2 of Corollary 1.2.3 havediffeomorphic boundaries.

Suppose that X admits a handlebody decomposition with a single 0-handle and some 1- and 2-handles. The homology groups Hi(X; Z) andHi(∂X; Z) can be easily read off from a diagram corresponding to sucha handle decomposition; this method will be discussed in the following.Consider the Abelian groups C1 and C2 freely generated by [K ′

1], . . . , [K′t]

and [K1], . . . , [Kn], corresponding to the t dotted circles and the n attachingcircles of the 2-handles respectively, and define the map ϕ : C2 → C1 by

[K] 7→t∑

i=1

ℓk(K,K ′i)[K

′i]

on the generators and extend linearly. As for CW -complexes, we getC1/ imϕ ∼= H1(X; Z) and kerϕ ∼= H2(X; Z). This latter identity followsfrom C3 = 0, which is the consequence of the absence of 3-handles. Nowthe universal coefficient theorem and Poincare duality allows us to com-pute all homologies and cohomologies of X. In fact, the ring structure of


H∗(X,∂X; Z) can also be read off from the picture. We restrict ourselvesto the case when there are no 1-handles in the decomposition: if the homo-logy classes α1, . . . , αn ∈ H2(X; Z) are represented by Seifert surfaces of Ki

together with the cores of the 2-handles (i = 1, . . . , n) then we can easilysee that PD(αi)∪PD(αj) = ℓk(Ki,Kj); as before, PD(αi)∪PD(αi) = ni,the framing of Ki. In other words, in the basis PD(α1), . . . , PD(αn) ofH2(X,∂X; Z) ∼= H2(X; Z) the intersection form of X is represented by thelinking matrix of the link Kini=1. Here PD denotes the Poincare dualityisomorphism between H2(X; Z) and H2(X,∂X; Z). By considering surfacesFi in D4 with ∂Fi = Ki and gluing the core disks to them we might findlower genus representatives of the homology class αi ∈ H2(X; Z). The geo-metric intersections of these Fi’s are, however, harder to visualize.

Next we discuss the computation of Hi(∂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 ∂Xunchanged. Notice that Z is the union of a 0-handle and m(= t + n) 2-handles which are attached (after renaming) along the knots K1, . . . ,Km

with framings n1, . . . , nm. According to the above said, H2(Z; Z) is freelygenerated by the closed surfaces Σi we get by gluing an orientable Seifertsurface of the knot Ki and the core of the 2-handle together. After fixingan orientation on Ki these surfaces are canonically oriented: fix the ori-entation making Ki the oriented boundary of the Seifert surface. Let Di

denote a small meridional disk to Ki. It is fairly straightforward to see thatH2(Z, ∂Z; Z) is generated by the relative homologies represented by [Di](i = 1, . . . ,m). Here we choose an orientation on these disks in such a waythat Ki intersects Di positively when we use the orientation on Ki fixedabove. The long exact homology sequence of the pair (Z, ∂Z) reduces to

0 → H2(∂Z; Z) → H2(Z; Z)ϕ1−→ H2(Z, ∂Z; Z)

ϕ2−→ H1(∂Z; Z) → 0

(since H3(Z, ∂Z; Z) ∼= H1(Z; Z) = 0 and H1(Z; Z) = 0 by the simpleconnectivity of Z). As Theorem 2.2.11 shows, the map ϕ1 is given by

ϕ1

([Σi]

)= ni[Di] +

∑

j 6=i

ℓk(Ki,Kj)[Dj ],

while ϕ2 is simply ϕ2

([Di]

)= [∂Di] = µi, where µi denotes the homology

class of the linking normal circle of the knot Ki oriented in such a way thattheir linking number is (+1). The exact sequence (with the maps describedabove) provides an explicit presentation for both H1(∂Z; Z) = H1(∂X; Z) ∼=H2(∂X; Z) and H2(∂Z; Z) = H2(∂X; Z) ∼= H1(∂X; Z).


With introducing more notation, in fact we can picture cobordismsinvolving only 1- and 2-handles. To this end, consider the cobordism Wfrom Y1 to Y2. First present Y1 as ∂X4 for some 4-manifold X and drawa 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 linkproducing 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 RP3 to S3

is one rule we have to obey with handleslides and cancellations in such acobordism: handles in X cannot be slid over handles in the cobordism Wand handles in X cannot be cancelled against handles in W . On the otherhand, 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 thatW is a given cobordism from Y1 to Y2. Fix a 4-manifoldX with ∂X = Y1, and suppose that it is given by attaching 2-handles to D4

along a framed link L. For the sake of simplicity, suppose furthermore thatW is given by a single 2-handle attachment to Y1. Denote the 4-manifoldX ∪W by X ′.

Exercises 2.3.4. (a) Determine the homology class inH2(X′; Z) generating

H2(W,∂W ; Z). (Hint: Consider a primitive homology class α ∈ H2(X′; Z)

such that QX′(α, β) = 0 for all β ∈ H2(X; Z) ⊂ H2(X′; Z).)

(b) Determine the self-intersection QW (α,α) of this generator.

(c) Find a surface in W representing the above α ∈ H2(W ; Z). (Hint:Use the above computation to represent α ∈ H2(X

′; Z) with a surface. Byadding extra handles make sure that the surface is disjoint from the coresof all the 2-handles defining X. Now show that the surface is in W .) Noticethat different presentations of Y1 as ∂X4 might provide different estimateson the genus of a surface representing α.

(d) Go through the above computations for the cobordism provided byFigure 2.12. Find a torus of square (−2) in this cobordism.


(e) Let K ⊂ S3 be a given knot with 4-ball genus gs(K). Perform n-surgeryalong K and denote the resulting 3-manifold S3

n(K) by Y . Let K ′ be ameridian to K and define the cobordism W by attaching a 2-handle alongK ′ with surgery coefficient k. Let α ∈ H2(W,∂W ; Z) denote a generator.Compute the self-intersection of α and give an estimate for the genus of asurface 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

.........

.........

.........

.........

ng

rk

r1

1 1

r1

2

Figure 2.13. A Seifert fibered 3-manifold

Seifert invariants (g, n; r1, . . . , rk) (g, n ∈ N, ri ∈ Q). Notice that the dottedcircles form an unlink in the diagram. If ri ≥ 1 then we say that this setof invariants is in standard form. Note that by applying Rolfsen twists anysuch diagram can be transformed into standard form. When g = n = 0,the 3-manifold with Seifert invariants (g, n; r1, . . . , rk) is usually denotedby M(r1, . . . , rk). Notice that according to this convention the surgerycoefficients are negative reciprocals of the given data.


Exercises 2.3.5. (a) Determine the intersection matrix of the 4-manifoldX given by Figure 2.14. (This manifold is frequently called the Gompfnucleus.) What is H1(∂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 : D3,n → Σ3 over the genus-3 surface Σ3 with Euler number n.Draw the diagram of Dg,n for an arbitrary positive integer g and n ∈ Z.Compute the intersection form, signature and Euler characteristic for Dg,n.

(c) By inverse slam-dunks find a 4-manifold X such that ∂X = M(g, n; r1,. . . , rk). (Hint: Use the continued fraction expansions of ri ∈ Q, cf. Exer-cise 2.2.7(b).)


(d) Verify that the boundary of the (+E7)-plumbing (a truncation of thelong leg of the diagram of Figure 1.5) is diffeomorphic to the 3-manifoldwe 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 (+E7)-plumbing.)

(f) Show that r-surgery on the right-handed trefoil knot gives the Seifertfibered manifold M( − 1

2 ,13 ,− 1

r−6). Use this fact to reprove (c) above.Determine the Seifert invariants of the result of (+6)- and (+7)-surgerieson the trefoil knot.

(g) Generalize the above result to a (2, 2n+ 1) torus knot T(2,2n+1). (Hint:

S3r(T(2,2n+1)) is diffeomorphic to M(− 1

2 ,n

2n+1 ,− 1r−4n−2), cf. [102].)

Another family of 3–manifolds is provided by the Brieskorn spheresΣ(p, q, r) (p, q, r ∈ N). Such a 3–manifold can be most conveniently definedas the oriented boundary of the compactified Milnor fiber V (p, q, r), where

V (p, q, r) =

(x, y, z) ∈ C3 | xp + yq + zr = 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 ofthe isolated singularity xp + yq + zr = 0. By perturbing the equation werather consider the smoothing of this singularity — the introduction of theperturbing term ǫ leaves the topology of the link Σ(p, q, r) unchanged.


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) ∈ N3 canbe rather involved.

In order to fix our convention, we remark here that we orient Σ(2, 3, 5)(i.e., the Poincare homology sphere) as the boundary of the negative defi-nite E8-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 definiteE8-plumbing. Thisorientation convention is consistent with complex geometry — the Poincaresphere 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 theorientation 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 ,13 ,

15) and in a similar vein −Σ(2, 3, 4) = M( − 1

2 ,13 ,

14) and

−Σ(2, 3, 7) = M(− 12 ,

13 ,

17).

(b) In general, however, the transition from Σ(p, q, r) to M(r1, r2, r3) is lesssimple, for example −Σ(2, 3, 6n − 1) = M(− 1

2 ,13 ,− 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 theproof of numerous fundamental statements discussed in the text. The chap-ter concludes with a short review on what is known about the classificationof symplectic 4-manifolds. For a more detailed treatment of symplectic geo-metry and topology the reader is advised to turn to [111]; here we restrictour 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 symplecticform if ω is closed (i.e., dω = 0) and nondegenerate (i.e., for any nonzerotangent vector v there is w with ω(v,w) 6= 0). The pair (X,ω) is called asymplectic manifold.

Since any antisymmetric form is degenerate on an odd dimensional vectorspace, a symplectic manifold is necessarily even dimensional.

Examples 3.1.2. (a) For R2n with coordinates (x1, y1, . . . , xn, yn) the 2-form ωst =

∑ni=1 dxi ∧ dyi is symplectic, called the standard symplectic

structure on R2n.

(b) The above form is invariant under translations, hence defines a sym-plectic form on the 2n-torus T 2n = R2n/Z2n.

(c) Let g denote the Fubini–Study metric on the complex projective spaceCPn. Then ωFS(u, v) = g(iu, v) is a symplectic form on CP2.

50 3. Symplectic 4-manifolds

(d) If (Xi, ωi) are symplectic (i = 1, 2) then their product X1 × X2 withany of the pulled back forms π∗1ω1 ± π∗2ω2 is a symplectic manifold. (Themap πi : X1 ×X2 → Xi denotes the projection to the ith 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 tothe nonvanishing of ωn = ω ∧ . . . ∧ ω (n times). Notice that in this wayω provides an orientation for X; for oriented manifolds we require the twoorientations to agree.

(b) Show that the sphere Sn admits a symplectic structure only if n = 2.

(c) Prove that S1 × S3 does not carry any symplectic structure. Showthe same for CP2. Here CP2 denotes the complex projective plane with itsnatural (complex) orientation reversed. (Hint: Note that dω = 0 impliesthat [ω] represents a cohomology class in H2(X; R), and [ω]n > 0 followsfrom nondegeneracy and the compatibility with the given orientation. Usecompactness of the above manifolds.)

(d) Prove that a smooth projective variety (i.e., a complex manifold witha holomorphic embedding into some complex projective space) admits asymplectic structure.

(e) Verify that for any smooth manifold V the cotangent bundle T ∗V withthe 2-form dλ is a symplectic manifold, where the Liouville 1-form λ isdefined as λp(v) = p(π∗v) for p ∈ T ∗V , v ∈ Tp(T ∗V ) and π : T ∗V → V .

It turns out that symplectic manifolds are “close” to complex manifoldsin the sense that their tangent bundles can be equipped with complexstructures. 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 J2 = − idTX . An almost-complex structure is said to be compatiblewith a given symplectic structure ω if ω(Ju, Jv) = ω(u, v) and for u 6= 0 wehave ω(u, Ju) > 0, that is g(u, v) = ω(u, Jv) is a Riemannian metric. If ωand J are compatible then (X,ω, J, g) is called an almost-Kahler manifold.

For any symplectic structure ω there exists a compatible almost-complexstructure J , moreover the space of such J ’s is contractible. (This state-ment can be proved fiberwise.) In conclusion, the tangent bundle TX of asymplectic manifold (X,ω) carries a complex structure. In fact, all compat-ible almost-complex structures are homotopic to one another, therefore theChern classes ci(X,ω) ∈ H2i(X; Z) are well-defined.

3.1. Generalities about symplectic manifolds 51

Definition 3.1.5. A submanifold Σ of an almost-Kahler manifold(X,ω, J, g) is symplectic if the restriction ω|TΣ is a symplectic form onΣ. The submanifold is J-holomorphic (or pseudo-holomorphic) if TΣ is J-invariant, that is, v ∈ TΣ ≤ TX implies Jv ∈ TΣ. The submanifold L ⊂ Xis Lagrangian if ω|L = 0. Finally, L ⊂ X is totally real if TlL ∩ JTlL = 0for all l ∈ L.

Example 3.1.6. Recall that for a symplectic manifold (X,ω) the productX ×X with ω × (−ω) is a symplectic manifold. It is not hard to see thatthe submanifolds X × pt. and pt. ×X are symplectic submanifolds of(X×X,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 ifω|TΣ is nondegenerate.

(b) Suppose that ω and J are compatible. Show that a J-holomorphicsubmanifold is symplectic. Find a counterexample for the converse.

(c) Show that a submanifold Σ is symplectic if and only if there is acompatible J for which it is J-holomorphic.

(d) Show that if L is Lagrangian and J is an ω-compatible almost-complexstructure then L is totally real. (Hint: Show that if V is a complex subspaceof TxX then ω|V 6= 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 Σ1 and Σ2 are J-holomorphic submanifolds of the almost-complex 4-manifold X. If Σ1 andΣ2 do not share a component then [Σ1] · [Σ2] ≥ 0, with equality if and onlyif the submanifolds are disjoint.

One of the most important formulae in the study of symplectic manifoldsis the following adjunction equality, which is just a simple manifestation ofthe Whitney product formula for characteristic classes:

Theorem 3.1.9. If Σ2 ⊂ X4 is a symplectic submanifold then −χ(Σ) =[Σ]2 −

⟨c1(X), [Σ]

⟩.

Proof. Notice that in order for c1(X) to make sense we need to fix an ω-compatible almost-complex structure. If Σ is symplectic, one can choosea compatible J such that Σ becomes a J-holomorphic curve. Then thesplitting TX|Σ = TΣ ⊕ νΣ (as complex bundles) gives

⟨c1(X), [Σ]

⟩=⟨c1(TΣ), [Σ]

⟩+⟨c1(νΣ), [Σ]

⟩.


The identity ⟨c1(TΣ), [Σ]

⟩=⟨e(TΣ), [Σ]

⟩= χ(Σ)

for the Euler characteristic is fairly straightforward, while

⟨c1(νΣ), [Σ]

⟩= [Σ]2

needs only a little argument realizing that a push-off of Σ gives rise to asection 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 inequivalentstructures do they carry. Using Gromov’s h-principle it can be shown thatevery open 2n-manifold admits a symplectic structure (see [33, 56]); thequestion is more subtle for closed manifolds. In order to understand top-ological properties of symplectic 4-manifolds, first we have to understandobstructions to the existence of symplectic structure and describe construc-tions of symplectic manifolds. According to the following proposition, theexistence of an almost-complex structure depends only on the homotopytype of a 4-manifold. Recall that the existence of a symplectic structureimplies the existence of an almost-complex structure.

Proposition 3.1.10 (Wu, [175]). A closed, oriented 4-manifold X carriesan almost-complex structure if and only if there is a class h ∈ H2(X; Z)such that h ≡ w2(X) (mod 2) and h2 = 3σ(X) + 2χ(X). In particular,a simply connected, closed 4-manifold X is almost-complex if and only ifb+2 (X) is odd.

Remark 3.1.11. If X is not closed then the 4-dimensional cohomologyclass h2 ∈ H4(X; Z) might have noncompact support, hence it might notbe integrable on X. Therefore

⟨h2, [X]

⟩might not be defined. If X is

compact with nonempty boundary ∂X and the restriction of h to ∂X istorsion then h2 can be defined as a rational number as follows: the multiplenh will vanish on the boundary ∂X, hence the square (nh)2 has compactsupport, so the expression h2 = 1

n2 (nh)2 ∈ Q is a well-defined quantity. Thedifference h2−3σ(X)−2χ(X), however, is not necessarily zero anymore foran almost-complex structure. It provides an invariant of the oriented 2-plane field induced by the complex tangencies on ∂X; for more about thistopic see Section 6.2. We note here that since the congruence h ≡ w2(X)(mod 2) always admits a solution (which implies, in particular, the existenceof a spinc 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 theoremof Taubes. (For the brief definition of the Seiberg–Witten function SWX

and more on Taubes’ work see Chapter 13.)

Theorem 3.1.12. If X carries a symplectic structure ω then for theSeiberg–Witten invariant SWX we have that

SWX

(± c1(X,ω)

)= ±1.

Moreover, if K ∈ H2(X; Z) and SWX(K) 6= 0 then∣∣K · [ω]

∣∣ ≤∣∣c1(X,ω) · [ω]

∣∣ ,with equality if and only if K = ±c1(X,ω).

It can be shown, for example, that SW3CP2 ≡ 0, hence although 3CP2 ad-mits an almost-complex structure, it cannot be equipped with a symplecticstructure.

Above we saw obstructions to the existence of symplectic structures,in the following we will describe some constructions to produce symplecticmanifolds. As we already mentioned, all Kahler surfaces are symplectic.One of the most effective ways of constructing symplectic manifolds isthe symplectic normal connected sum operation, which we will describein Section 7.1. Another source of examples comes from surface bundles oversurfaces, since we have

Theorem 3.1.13 (Thurston, [166]). If the 4-manifold X admits a fibrationX → Σg such that the fiber has genus different from 1 then X admits asymplectic structure.

Remark 3.1.14. If the fiber is a torus, similar result cannot be expected,since S1 × S3 admits a torus fibration over the sphere: multiply the Hopffibration S3 → S2 by a circle. On the other hand, as a theorem of Geiges[55] shows, torus bundles over tori are all symplectic. A generalization ofTheorem 3.1.13 to more general Lefschetz fibrations will be discussed inSection 10.1.

These constructions give partial results regarding the existence of a sym-plectic structure on a given smooth 4-manifold. Such studies are usuallycalled “geographic” questions of symplectic 4-manifolds. For overviews ofvarious aspects of such geographic questions see [14, 144, 152, 156]. Thenext 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.


Definition 3.1.15. Let X be a given 4-manifold and ω0, ω1 two symplecticforms on it. The forms ω0 and ω1 are said to be deformation equivalent ifthere is a smooth path of symplectic forms interpolating between them. Theform ω0 is the pullback of ω1 if there is a diffeomorphism f : X → X suchthat f∗ω1 = ω0. Finally, ω0 and ω1 are equivalent if they lie in the sameequivalence class under the equivalence relation generated by the above tworelations.

Theorem 3.1.16 ([112, 150, 170]). For any n ∈ N there is a simplyconnected 4-manifold Xn which carries at least n inequivalent symplecticstructures.

The construction of the manifolds and the symplectic structures usesGompf’s symplectic normal connected sum operation (given in Theorem7.1.10). By computing c1 of the various symplectic structures it is easy toshow that they are deformation inequivalent (since the c1’s are distinct).By proving that the different c1’s lie in different orbits of Diff+(X) it fol-lows that the symplectic structures are inequivalent. In this last step eitherthe divisibilities of the integral cohomology classes show the nonexistenceof certain diffeomorphisms [150], or the Seiberg–Witten equations pose re-strictions on the action of Diff+(X) on H2(X; Z), see [112, 170].

The spectacular success of the results of Taubes on Seiberg–Witteninvariants of symplectic 4-manifolds indicates that appropriate extensionsof these techniques might lead to new results for a much broader class of4-manifolds. Such a potential extension was initiated by Taubes [165] byconsidering singular symplectic forms, that is, closed 2-forms nondegenerateonly away from a subset of the given closed 4-manifold X.

Exercise 3.1.17. Suppose that for a given 4-manifold X the conditionb+2 (X) ≥ 1 holds. Show that there exists a closed 2-form ω which isnondegenerate away from the closed 1-manifold Z = x ∈ X | ωx = 0 ofits zeros. (Hint: Fix a metric on X and consider harmonic representativesof a second cohomology class of positive square. Choose generic metric.)

The analysis for setting up a correspondence between J-holomorphic curvesin X−Z (with appropriate boundary conditions) and Seiberg–Witten solu-tions on X is much more complicated than in the symplectic case, and it isin the focus of current research. A fairly explicit way of finding a singularsymplectic form on a closed 4-manifold X with b+2 (X) ≥ 1 is given by Gayand Kirby [54]. This procedure makes use of symplectic surgery in the spiritit 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 asMoser’s method. Using this method we can prove that symplectic manifolds“have no local invariants”. This last statement can be interpreted in twoways: (i) a symplectic manifold is locally standard, or (ii) a small deforma-tion of the symplectic structure produces symplectomorphic manifold. Thefirst statement actually generalizes to neighborhoods of special submani-folds, while the second holds for deformations keeping the cohomology classdefined by the symplectic form fixed (see Moser’s Stability Theorem 3.2.1).

The main idea can be easily summarized: Suppose that X is compactand ωt ∈ Ω2(X) is a family of symplectic forms with exact derivative:ddtωt = dσt. We claim that in this case there is a family Ψt ∈ Diff(X)of diffeomorphisms such that Ψ∗

tωt = ω0. The diffeomorphisms Ψt can beconstructed via the flow of the family of vector fields Xt they induce by

d

dtΨt = Xt Ψt, Ψ0 = id .

The key point is that if Ψ∗tωt = ω0 then for Xt we have

0 =d

dtΨ∗tωt = Ψ∗

t

(d

dtωt + ιXtdωt + dιXtωt

)= Ψ∗

td(σt + ιXtωt),

since dωt = 0 and ddtωt = dσt. Therefore a vector field Xt satisfying

σt = −ιXtωt

will be appropriate for our purposes. This equation is, indeed, easy tosolve for Xt since ωt is nondegenerate. Then solving d

dtΨt = Xt Ψt forΨt we get the family Ψt with the desired property. (This last step canbe achieved without any problem provided the manifold X is compact; thegeneral case needs some more care.) Applying this principle, one can deducethe following (see [111]):

Theorem 3.2.1 (Moser’s Stability Theorem). Suppose that ωt(t ∈ [0, 1]

)

is a family of symplectic forms on the closed manifold X and [ωt] = [ω0].Then there is an isotopy ϕt such that ϕ0 = idX and ϕ∗

tωt = ω0.

Proof. In order to apply the above principle we need to show the existenceof a family σt with d

dtωt = dσt. Since [ωt] is constant, we obviously get

that ddtωt is exact; applying Hodge theory, for example, a smooth family of

appropriate σt can be chosen.


More interestingly, Moser’s method shows that symplectic manifolds arelocally the same. This principle rests on the following result:

Theorem 3.2.2. Suppose that X is a smooth manifold with Y ⊂ X acompact submanifold and ω1, ω2 ∈ Ω2(X) two closed 2-forms which areequal and nondegenerate on TyX for all y ∈ Y . Then there are openneighborhoods N1, N2 of Y in X and a diffeomorphism ψ : N1 → N2 suchthat ψ|Y = id and ψ∗ω2 = ω1.

Proof (sketch). By applying Moser’s argument described above, the the-orem reduces to finding a 1-form σ ∈ Ω1(N1) with dσ = ω2 − ω1 andσ|TY X

= 0. In fact, with such σ the family ωt = (1 − t)ω1 + tω2 = ω1 + tdσwill be symplectic on a neighborhood of Y . This follows from the fact thatnondegeneracy is an open condition, while dωt = (1 − t)dω1 + tdω2 = 0.Therefore the argument of Moser provides an appropriate vector field Xt

which vanishes along Y . By possibly shrinking N1, this implies the exis-tence of ψ and N2 with the properties given in the theorem. For the explicitconstruction 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 ⊂ R2n equipped with the standard symplecticform ωst|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(R2n, ωst) and the transition functions between such charts respect the sym-plectic structures on the charts. This approach turns out to be equivalentto Definition 3.1.1.

In fact, symplectic structures are standard not only around points, but alsonear symplectic and Lagrangian submanifolds. In the following we formulatethese theorems only for 4-dimensional symplectic manifolds.

Theorem 3.2.5 (Symplectic neighborhood theorem, Weinstein [172]). Sup-pose that (Xi, ωi) is a symplectic 4-manifold with 2-dimensional closed sym-plectic submanifolds Σi ⊂ Xi for i = 1, 2. Suppose furthermore that there isan isomorphism F : ν(Σ1) → ν(Σ2) of the normal bundles ν(Σi) → Σi cov-ering a symplectomorphism f :

(Σ1, ω1|Σ1

)→(Σ2, ω2|Σ2

). Then f extends

to a symplectomorphism on some tubular neighborhoods of the surfaces Σi.

3.2. Moser’s method and neighborhood theorems 57

Proof (sketch). The symplectomorphism f guarantees that ω1 and f∗ω2

coincide on TΣ1 ⊂ TX1. By choosing appropriate neighborhoods we canassume that the two structures are equal in the normal direction as wellusing the isomorphism of the normal bundles. Then an application ofTheorem 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 adiffeomorphism φ : U → V such that φ∗ω = −dλ and φ|L = id, where λ isthe Liouville form on T ∗L.

Recall that the Liouville form on T ∗L is defined by λp(v) = p(π∗v) forπ : T ∗L → L and v ∈ Tp(T

∗L). For more details of the proofs see [111]or McDuff’s lectures in [34]. Notice that for symplectic submanifolds weneed the existence of a diffeomorphism F : ν(Σ1) → ν(Σ2) and a symplecto-morphism f :

(Σ1, ω1|Σ1

)→(Σ2, ω2|Σ2

); the topology around a symplectic

submanifold Σ is not unique and symplectic structures might be differentfor 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 ⊂ X4 is determinedby the self-intersection number [Σ]2 ∈ Z. After possibly rescaling ω2 on X2

there exists a symplectomorphism f :(Σ1, ω1|Σ1

)→(Σ2, ω2|Σ2

)once Σ1

and Σ2 are diffeomorphic, that is, the genera g(Σ1) and g(Σ2) are equal.In conclusion, the assumptions of Theorem 3.2.5 can be checked from thetopology of the situation.

In the Lagrangian case, on the other hand, ω|L = 0 holds, so the topologyof L already determines its neighborhood, as the following exercise shows.

Exercise 3.2.7. Show that for L2 ⊂ X4 Lagrangian we have [L]2 = −χ(L).(Hint: Fix a compatible almost-complex structure J and show that JTpLis the orthogonal complement of TpL with respect to the metric gJ inducedby ω and J .) In conclusion, two orientable Lagrangian 2-manifolds in asymplectic 4-manifold admit symplectomorphic neighborhoods if and onlyif the genera of the surfaces are equal.


3.3. Appendix: The complex classification scheme for

symplectic 4-manifolds

In the following we give a short overview about the present status of thesmooth classification of closed symplectic 4-manifolds. The classificationscheme tries to imitate the classification results obtained for compact com-plex surfaces (for a detailed description of the latter see [10]), hence firstwe introduce the notion of minimality and Kodaira dimension of symplec-tic 4-manifolds. Since this discussion falls outside the main theme of thisvolume, we mainly give the statements without proofs.

Definition 3.3.1. A symplectic 4-manifold X is minimal if it does notcontain a symplectic submanifold S ⊂ X diffeomorphic to the 2-sphere S2

with [S] · [S] = −1.

Remark 3.3.2. A detailed analysis of the Seiberg–Witten invariants showsthat minimality is equivalent to requiring that X does not contain anysmoothly embedded 2-sphere S with [S] · [S] = −1.

A symplectic 4-manifold can always be blown up in a point by imitatingthe 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#CP2. In this latter symplectic 4-manifold the generator ofthe H2(CP2; 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 ⊂ Xis a symplectic sphere with square (−1) then X is the blow-up of anothersymplectic manifold. This implies

Lemma 3.3.3. A symplectic 4-manifold X can be written as Y#nCP2

where Y is a minimal symplectic manifold. Y is called a minimal modelof X.

Proof. If S ⊂ X is a symplectic sphere of self-intersection (−1) thenX = X1#CP2 since νS is diffeomorphic to CP2 − intD4. Taking thesymplectic sum of X and CP2 along S and CP1 ⊂ CP2 (as it is discussed inTheorem 7.1.10) we find that X1 is symplectic. Repeating the above processcompletes the proof. Notice that each step reduces b2(X) by 1, hence thisprocedure 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 exampleCP2#2CP2 ∼= S2 × S2#CP2 admits different minimal models (CP2 andS2 ×S2) according to the order of blow-downs. For a related discussion seeRemark 3.3.9.

Let us assume that (X,ω) is a minimal symplectic 4-manifold. Followingthe complex analogy, its Kodaira dimension is defined as follows: Fix anω-compatible almost-complex structure J and consider c1(X,ω) = c1(X,J).

Definition 3.3.5.

• If c1(X,ω)[ω] > 0 or c21(X,ω) < 0 then the Kodaira dimension κ(X)of X is −∞.

• In case c1(X,ω)[ω] = 0 we say that X is of Kodaira dimension 0.

• For c1(X,ω)[ω] < 0 and c21(X,ω) = 0 we define κ(X) = 1.

• Finally, if c1(X,ω)[ω] < 0 and c21(X,ω) > 0 then κ(X) = 2.

• If X is nonminimal then κ(X) is defined as the Kodaira dimension ofits minimal model.

Theorem 3.3.6 (Liu, [103]). If (X,ω) is a minimal symplectic 4-manifoldwith c21(X,ω) < 0 then X is a ruled surface, that is, an S2-bundle over aRiemann 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 mutuallydisjoint. In principle κ(X) might depend on the minimal model chosen,since Xmin might not be unique.

Proposition 3.3.7. If the symplectic 4–manifold has a minimal modelXmin

with κ(Xmin) ≥ 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 c21(X,ω) is equal to 3σ(X) + 2χ(X), hence dependsonly 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 ofsymplectic 4-manifolds with κ = −∞:

Theorem 3.3.8 (Liu, [103]). If X is minimal and κ(X) = −∞ then X isdiffeomorphic either to CP2 or to a ruled surface.


Remark 3.3.9. All these manifolds carry complex, in fact, Kahler struc-tures. According to the classification of complex surfaces, these are all theKahler surfaces with (complex) Kodaira dimension −∞ [10]. Suppose thatwe blow up a fiber of a ruled surface X → Σg. When constructing theminimal model of this symplectic 4-manifold, we can choose which (−1)-sphere to blow down: the exceptional sphere of the blow-up or the propertransform of the fiber. It is not very hard to see that the result of oneblow-down is spin, while the other is not. Therefore the minimal model ofthe 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 nonuniqueminimal 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 c1(X,ω) = 0.In the case b+2 (X) = 1, the assumption κ(X) = 0 implies 2c1(X,ω) = 0.

Examples of such manifolds are provided by the K3-surface, T 2-bundlesover T 2 (which are all symplectic by the quoted result of Geiges) and theEnriques surface. This latter manifold is the quotient of a K3-surface byan appropriate free Z2-action, therefore its fundamental group is Z2 andthe 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 assumptionc1(X,ω) = 0 implies that for a manifold X with b+2 (X) > 1 there is a uniquebasic class (i.e., K ∈ H2(X; Z) with SWX(K) 6= 0), and this unique class isequal to 0. According to a result of Morgan and Szabo, a simply connected4-manifold X with SWX(0) odd is homeomorphic to the K3-surface, henceit can be proved that

Theorem 3.3.11 (Morgan–Szabo, [120]). If (X,ω) is a simply connectedsymplectic 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-surfacesand Enrique surfaces. For more detail see [10, pp. 188–189]. It seemsreasonable to expect that all symplectic 4-manifolds with κ = 0 admit agenus-1 Lefschetz fibration, hence these are essentially torus bundles, theK3-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 thenthere is a symplectic 4-manifold (X,ω) with κ(X) = 1 and π1(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 beconstructed from E(1) = CP2#9CP2 and torus bundles using fiber sum andan additional operation called logarithmic transformation. For additionaldiscussion on the topology of elliptic surfaces see [66].

If the symplectic 4-manifold (X,ω) has κ(X) = 2 then we say that it isof general type. Such examples are provided by complex surfaces of generaltype (since these complex manifolds are all algebraic, hence Kahler). Recallthat for a complex surface the condition κ(X) = 2 implies c21(X) ≤ 3c2(X),the famous Bogomolov–Miyaoka–Yau inequality. (For a related discussionsee [153].) This inequality, or some similar relation between c21 and c2 isconjectured to hold for symplectic 4-manifolds of general type. We do notknow too much about the topology of symplectic 4-manifolds of generaltype. The following conjecture (usually attributed to Gompf) would providea strong topological restriction:

Conjecture 3.3.15. If a symplectic 4-manifold (X,ω) satisfies κ(X) ≥ 0then 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 theneither κ(X) = −∞ or b1(X) ∈ 0, 2.Proof. By the existence of an almost-complex structure we get that 1 −b1 + b+2 = 2 − b1 is even, therefore b1(X) is even. Now by κ(X) ≥ 0 andTheorem 3.3.8 we have that 0 ≤ c21(Xmin, ω) = 3σ(Xmin) + 2χ(Xmin)

= 3(b+2 (Xmin) − b−2 (Xmin)

)+ 2(2 − 2b1(Xmin) + b+2 (Xmin) + b−2 (Xmin)

)

= 9 − b−2 (Xmin) − 4b1(Xmin),

showing that 4b1(Xmin) = 4b1(X) ≤ 9, which yields the result.


Conjecture 3.3.17. There is no symplectic 4-manifold X with b+2 (X) = 1,b1(X) = 2 and b−2 (X) = 0. (Notice that such a 4-manifold would provide acounterexample to Conjecture 3.3.15.)

4. Contact 3-manifolds

This chapter is devoted to the recollection of basic facts about contactmanifolds. As before, we start with the general case, but very quicklyspecialize to 3-manifolds. To understand the topology of contact 3-mani-folds we consider submanifolds and the contact structures near them. Thecontact version of Darboux’s theorem says that every point in a contact3-manifold has a neighborhood which is standard regardless of the contactstructure. Then we consider knots which are always tangent or alwaystransverse to the contact planes and examine their classical invariants. Itturns out that the contact structures near these types of knots are essentiallyunique. For an arbitrary surface embedded in a contact 3-manifold we lookat the characteristic foliation induced by the contact structure to extractinformation. It is typical to move a surface by a small isotopy to modify itscharacteristic foliation to get a generic picture and/or to eliminate certaintype of singularities. As it turns out, the characteristic foliation determinesthe contact structure near the surface. A more complete treatment of theideas 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 ∈ Ω1(M) is a contact form if α ∧ (dα)n is nowhere zero. The2n-dimensional distribution ξ ⊂ TM is a contact structure if locally it canbe defined by a contact 1-form α as ξ = kerα.

Example 4.1.2. The standard contact structure ξst on R2n+1 can be givenin the coordinates (x1, y1, . . . , xn, yn, z) as ker (dz+

∑ni=1 xi dyi). The com-

plex tangents to S2n−1 ⊂ Cn also form a contact structure.

64 4. Contact 3-manifolds

According to a classical result of Frobenius, the plane field ξ = kerαis integrable if and only if α ∧ dα = 0. Integrability is equivalent to be-ing closed under Lie bracket, hence ξ = kerα is integrable if and only ifα([v1, v2]

)= 0 whenever α(vi) = 0 (i = 1, 2). Recall that dα

([v1, v2]

)=

Lv2α(v1) − Lv1α(v2) + α([v1, v2]

), hence if ξ is integrable then dα vanishes

on ξ = kerα. So ξ is “maximally nonintegrable” if dα is nondegenerate onξ, i.e., α∧ (dα)n 6= 0. Therefore the contact condition can be interpreted as“maximally nonintegrable”. In other words, α is a contact form if dα is asymplectic form on the hyperplane distribution kerα. As we will see, Dar-boux’s theorem generalizes to the contact setting (Theorem 4.1.13), hence inconjunction with Remark 3.2.4 it can be shown that contact structures canbe given by patching open subsets of R2n+1 together with transition func-tions respecting ξst. The existence of contact structures on open manifolds(similarly to the symplectic case) follows from an appropriate h-principlesee [33, 56], for example. The question becomes more subtle on closed (odddimensional) manifolds.

From now on we will assume that Y is a 3-manifold, that is, in the abovedefinition n = 1. In the following we describe a few examples of contactstructures on R3, T 3 and S3 just to illustrate how contact structures maylook like in 3-manifolds.

Exercise 4.1.3. Verify that α1 = dz + x dy and α2 = dz − y dx definecontact structures on R3. Visualize the contact planes ξi = kerαi, fori = 1, 2. (Hint: For the latter see Figure 4.1.)

Examples 4.1.4. (a) The form α3 = dz + r2 dθ (with polar coordinates(r, θ) on the (x, y)-plane) gives a contact structure ξ3 = kerα3 on R3.To check this, realize that in (x, y, z)-coordinates the form α3 is equal todz+x dy−y dx, and so α3 ∧dα3 = 2 dx∧dy∧dz = 2r dr∧dθ∧dz. One caneasily see that the contact planes are spanned by ∂

∂r , r2 ∂∂z − ∂

∂θ. Theseplanes are horizontal (i.e., parallel to the xy-plane) along the z-axis andas we move out along any ray perpendicular to the z-axis the planes willtwist in a clockwise manner. The contact structure ξ3 is obviously invariantunder 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 isan 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 R3.To check this we calculate

dβ = − sin r 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 R3

β ∧ dβ = −r sin2 r dθ ∧ dr ∧ dz + (sin r cos r + r cos2 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 rr > 0 for r > 0. Again, as in the

previous example, ker β admits the same symmetries as ξ3 and the contactplanes (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 rayperpendicular to the z-axis. This time, however, the contact planes willmake infinitely many full twists as r goes to ∞. More generally, the 1-formβn = cos fn(r) dz+ sin fn(r) dθ, where fn(r) is a strictly monotone functionequal to r2 near r = 0 and asymptotic to nπ + π

2 (as r → ∞) provides acontact form on R3 for every nonnegative integer n. For n = 0 this formgives the standard contact structure on R3.


(c) Let us identify the 3-torus T 3 with R3/Z3. For any positive integern the 1-form sin(2πnx) dy + cos(2πnx) dz defined on R3 induces a contactstructure θn on T 3.

(d) Consider the smooth map f : R4 → R defined by f(x1, y1, x2, y2) =x2

1 + y21 + x2

2 + y22 . Let p denote the point (x1, y1, x2, y2). It is clear that

S3 = f−1(1) and TpS3 = ker dfp = ker(2x1 dx1+2y1 dy1+2x2 dx2+2y2 dy2).

By identifying R4 with C2 we can define a complex structure J on eachtangent space of R4 by J ∂

∂xi= ∂

∂yiand J ∂

∂yi= − ∂

∂xifor i = 1, 2. Let ξ be

the plane field of complex tangencies of J along S3, i.e.,

ξp = TpS3 ∩ J(TpS

3).

We claim that ξ is a contact structure on S3. To show this we will find acontact 1-form α on S3 such that ξ = kerα. Consider the 1-form df Jon R4. By evaluating on the basis vectors ∂

∂x1, ∂∂y1

, ∂∂x2

, ∂∂y2

it is easy tocheck that −dfp J = 2x1 dy1 − 2y1 dx1 + 2x2 dy2 − 2y2 dx2. Moreover wehave J(TpS

3) = ker(−df J) since J2 = − id. Let α = −12(df J)

∣∣S3. It is

clear that ξ = kerα. It is straightforward to check that α ∧ dα is nonzeroon S3. We will check this only at a point p = (x1, y1, x2, y2) on S3 wherex1 6= 0, y1 6= 0, y2 6= 0. A basis for the tangent space TpS

3 can be chosen as

∂

∂x1− x1

y1

∂

∂y1,∂

∂x2− x2

y2

∂

∂y2,∂

∂x1− x1

y2

∂

∂y2

.

Now it is easy to see that α ∧ dα > 0 on this basis. Hence we concludethat α = (x1 dy1 − y1 dx1 + x2 dy2 − y2 dx2)

∣∣S3 is a contact form. We define

ξ = kerα as the standard contact structure on S3 and denote it by ξst.

(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 gϕ(u, v) =−dJ∗dϕ(u, Jv) is a Riemannian metric. (Here J denotes multiplication byi on TX and J∗ is the induced map on T ∗X.) Then the 1-form αϕ given byαϕ(v) = −dJ∗dϕ(∇gϕϕ, v) defines a contact form on ϕ−1(a) for a regularvalue a of ϕ. We will return to these examples in Chapter 8.

Exercise 4.1.5. Show that the 1-form βn defined in Example 4.1.4(b) is acontact 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 = kerα and ξ′ = kerα′, this is equivalent to the existence of a nowherezero function g : Y → R such that f∗α′ = gα. Two contact structures ζ andζ ′ on a manifold Y are said to be isotopic if there is a contactomorphismh : (Y, ζ) → (Y, ζ ′) which is isotopic to the identity.


In fact, two contact structures on a closed manifold are isotopic if and onlyif they are homotopic through contact structures by Theorem 4.1.16 below.Notice also that there exist contact structures which are contactomorphicbut not isotopic, see Exercise 11.3.12(c).

Exercises 4.1.7. (a) Prove that α1 = dz + x dy, α2 = dz − y dx and α3 =dz + x dy − y dx = dz + r2 dθ define contactomorphic contact structures onR3. (Hint: For identifying the contact structures given by α1 and α3, use thediffeomorphism ϕ(x, y, z) = (x, y2 , z+

xy2 ) or φ(x, y, z) = (x+y2 , y−x2 , z+ xy

2 ).)

(b) Let p be any point in S3. Show that (S3 − p, ξst|S3−p) is contacto-

morphic to (R3, ξst). (Hint: See [57] for a complete solution.)

In case ξ can be defined by a global 1-form, we say that the contactstructure is coorientable. This implies that ξ is orientable as a 2-planefield on Y . In fact, the 2-plane field underlying the contact structure ξ isorientable if and only if ξ can be defined by a global 1-form. Given anoriented 3-manifold Y , we say that ξ = kerα is a positive contact structureon 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 ofthe contact form α defining the contact structure ξ. In the following wewill always assume that the contact structures at hand are positive andcooriented. We orient the normal direction to the contact planes by α, orequivalently the contact planes are oriented by dα. If we choose −α asour contact form then the normal orientation and hence the orientation ofthe contact planes will be reversed but −α ∧ d(−α) will induce the sameorientation on the 3-manifold.

Definition 4.1.8. Suppose that the contact structure ξ is given as kerαfor the contact 1-form α ∈ Ω1(Y ). The vector field Rα on Y satisfyingdα(Rα, .) = 0 and α(Rα) = 1 is called the Reeb vector field of α. In otherwords, at each point p ∈ Y , the Reeb vector field points in the directionwhere the skew-symmetric 2-form dαp (of rank 2) degenerates in the tangentspace TpY and it is uniquely determined by the normalization conditionα(Rα) = 1. Notice that Rα is transverse to the contact planes and dependson the contact 1-form, not just on the contact structure. In general theReeb vector field Rfα will be very different from Rα for a nowhere vanishingfunction f : Y → R.

Exercises 4.1.9. (a) Suppose that β a 1-form on Y such that β(Rα) = 0.Show that there is a unique vector field X with X(p) ∈ ξp on Y such that


ιXdα = β. (Hint: Choose X = β(e2)e1−β(e1)e2dα(e1,e2)

with respect to some (local)

frame e1, e2 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 Rα3 = ∂

∂z , and

Rβ = (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. Aknot 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 transverseif TK is transverse to ξ along the knot K, i.e., if α(TK) 6= 0. The contactframing 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 anonzero 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 natural0-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 beconverted into an integer tb(K) ∈ Z: measure the Thurston–Bennequinframing 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, thereforethe resulting number tb(K) will be independent of Σ. Also notice that bythe coorientation of ξ a transverse knot T ⊂ (Y, ξ) comes with a naturalorientation: choose the nonzero tangent vector v to be positive if α(v) > 0for 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 ξ0, ξ1 on Y which coincide ascooriented contact structures on N , i.e., ξ0|N = ξ1|N as oriented 2-planefields. Then there exists a neighborhood U of N and a contactomorphismϕ :(U, ξ0|U

)→(U, ξ1|U

)which is isotopic to idU rel N .


Proof (sketch). The proof makes use of Moser’s method again (see Sec-tion 3.2). Assume that α0, α1 are contact forms giving rise to the contactstructures ξ0, ξ1. Since ξ0|N = ξ1|N , there exists a function f : N → R+

such that α1 = fα0 on the compact set N . Then consider the familyαt = (1 − t)α0 + tα1 of 1-forms on Y . To see that αt is a contact form onN for every t ∈ [0, 1], we calculate that

αt ∧ dαt =[(1 − t)2 + 2f(1 − t)t

]α0 ∧ dα0 + t2α1 ∧ dα1 > 0.

In addition, one can show that there is a neighborhood U of N such that αtis a contact form on U for every t. (Notice that αt is not necessarily contacton the entire Y .) We would like to represent the map ϕ in the theoremas the time–1 map of a family of diffeomorphisms ϕt with (ϕt)∗ξ0 = ξtand ϕ0 = id. Equivalently, we start with the equation ϕ∗

tαt = λtα0 anddifferentiate it with respect to t, yielding

d

dt(ϕ∗

tαt) = ϕ∗t

(d

dtαt + LXtαt

)=

d

dtλtα0 =

ddtλtλt

ϕ∗tαt.

(The first equality is an exercise in differential forms and its proof can befound, for example, in [12].) As before, Xt denotes the family of vectorfields induced by ϕt, i.e., d

dtϕt = Xt ϕt. Taking

µt =d

dt(log λt) ϕ−1

t ,

the above equation gets the form

ϕ∗t

(d

dtαt + d

(αt(Xt)

)+ ιXt dαt

)= ϕ∗

t (µtαt),

which is solved by Xt ∈ kerαt provided

(4.1.1)d

dtαt + ιXt dαt = µtαt.

Now consider the Reeb vector field Rαt of the contact 1-form αt and plugit into the above equation to get

d

dtαt(Rαt) = µt.

This defines the function µt, hence Equation (4.1.1) above can be uniquelysolved for Xt, since dαt is nondegenerate on kerαt. (Recall that this non-degenracy is equivalent for αt being a contact 1-form.) Now Xt integratesto the desired flow ϕt.


Applying this principle for N = pt. we get

Theorem 4.1.13 (Darboux’s theorem for contact structures). For everyy ∈ Y in the contact 3-manifold (Y, ξ) there is a neighborhood U ⊂ Y suchthat

(U, ξ|U

)is contactomorphic to

(V, ξst|V

)for some open set V ⊂ R3.

As in the symplectic case, similar argument extends to special sub-manifolds. For stating the relevant results, consider the contact structureζ1 = ker

(cos(2πφ) dx − sin(2πφ)dy

)and ζ2 = ker(cos rdφ + r sin r dθ) on

S1 × R2. (Here φ is the coordinate in the S1-direction, while (x, y) areCartesian and (r, θ) are polar coordinates on R2.)

Exercise 4.1.14. Show that S1 × 0 is Legendrian for ζ1 and transversefor ζ2.

By taking N = S1, the neighborhood theorems 3.2.5 and 3.2.6 now translateto

Theorem 4.1.15 (Contact neighborhood theorems). If K ⊂ (Y, ξ) is aLegendrian knot then there are neighborhoods U1 ⊂ Y of K and U2 ⊂S1×R2 of S1×0 such that

(U1, ξ|U1

)and

(U2, ζ1|U2

)are contactomorphic

via a contactomorphism mapping K to S1 × 0. If K is transverse,then some neighborhood of it is contactomorphic to some neighborhoodof S1 × 0 in (S1 × R2, ζ2) — again K is mapped to S1 × 0.

For a detailed proof of the Legendrian case — which will be more importantfrom our present point of view — see [57] or [1, Section 2.2]. The proofrelies on Theorem 4.1.12 after finding a map f taking K to S1 × 0 insuch a way that f∗ maps ξ|K to ζi|S1×0. Notice that since both the1-manifold K and its normal bundle N(K) is topologically unique, notopological assumption is needed for the neighborhood theorems to hold.Another important corollary of the principle of Theorem 4.1.12 is Gray’sstability theorem, the contact version of Moser’s stability Theorem 3.2.1 —the other manifestation of the principle that contact structures admit nolocal invariants.

Theorem 4.1.16 (Gray, [71]). If αt (t ∈ [0, 1]) is a smooth family ofcontact forms on a closed 3-manifold Y then there is an isotopy ϕt of Ysuch that ϕ0 = id and ϕ∗

tαt = λtα0 for some smooth family of smoothfunctions λt : Y → R+. In particular, (ϕt)∗ξ0 = ξt for ξt = kerαt.

Remark 4.1.17. Notice that the theorem deals with contact structures asopposed to contact forms; in general one cannot find ϕt satisfying ϕ∗

tαt =α0.


There is an intimate relationship between symplectic and contact man-ifolds, which will not be discussed in full detail here. (See Definition 12.1.1as an example of this relationship.) In our cut-and-paste construction wewill frequently refer to a symplectic manifold Symp(Y, ξ) associated to acontact manifold (Y, ξ) — called the symplectization of (Y, ξ). In order todefine Symp(Y, ξ), choose a contact form α for ξ and consider

Symp(Y, ξ) = v ∈ T ∗mY | v = tαm for some t > 0.

It is easy to see that Symp(Y, ξ) is diffeomorphic to Y × (0,∞) and thatfor any other contact 1-form β with kerβ = ξ the 1-form β or −β issimply a section of Symp(Y, ξ). (This trivially follows from the fact thatif kerα = kerβ 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 λp(v) = p(π∗(v)

)for

p ∈ T ∗Y, v ∈ Tp(T∗Y ) and projection π : T ∗Y → Y . We claim that ω is a

symplectic form, that is, ω ∧ ω 6= 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)

)6= 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 ωon Symp(Y, ξ) is to take the 1-form µ = tα on Y × (0,∞) and define ω asdµ = t dα+ dt ∧ α; the result is clearly the same. Notice that the resultingsymplectic form is exact.

Exercise 4.1.20. Show that L ⊂ (Y, ξ) is Legendrian if and only if L×R ⊂Symp(Y, ξ) is Lagrangian.


4.2. Legendrian knots

In order to have a better understanding of the topological constructions wewill introduce in Chapter 8, we discuss a way to visualize Legendrian knots inthe standard contact S3 (or, equivalently, in R3) equipped with the standardcontact structure ξst = ker(dz+x dy). See [40] for more on Legendrian knots.Consider a Legendrian knot L ⊂ (R3, ξst) and take its front projection, i.e.,its projection to the yz-plane. Notice that the projection has no verticaltangencies (since − dz

dy = x 6= ∞), and for the same reason at a crossingthe strand with smaller slope is in front. A straightforward computation(see [57]) shows that L can be C2-approximated by a Legendrian knotfor which the projection has only transverse double points and (2, 3)-cuspsingularities (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 crossingsdepicted by Figure 4.3(a)) gives rise to a unique Legendrian knot in (R3, ξst)— define x from the projection as x = − dz

dy . Since any projection can beisotoped to satisfy the above properties, we can easily show that every knotcan be isotoped to Legendrian position. (This knot is, however, far frombeing unique up to Legendrian isotopy.)

Lemma 4.2.1. Any knot K ⊂ S3 can be isotoped to a Legendrian knot.

Proof. Consider a generic projection of K ⊂ R3 ⊂ S3 onto the yz-plane.Isotope the knot near the finitely many points where dz

dy = ∞ by addingcusps. At each crossing make sure that the strand with more negative slopecrosses 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 C0-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 Seifertframing in S3.) Define w(L) (the writhe of L) as the sum of signs of thedouble points (see Figure 4.4) — for this to make sense we need to fix anorientation 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–Bennequinframing tb(L) given by the contact structure is equal to w(L)− 1

2c(L) withrespect to the framing given by a Seifert surface.


_+Figure 4.4. Positive and negative crossings

Proof. The equality tb(L) = w(L) − 12c(L) can be seen by noting that ∂

∂zis transverse to ξ = ker(dz + x dy) hence tb(L) is just the linking numberℓk(L,L′) where L′ is a small vertical push-off of L. Now Figure 4.5 showsthat the canonical framing differs from the blackboard framing by a lefthalf-twist for each cusp, and this verifies our formula for tb(L). Recall fromExercise 2.1.4 that the blackboard framing of a knot differs from the framinggiven 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 followedby one pointing left and vice versa. Consequently c(L) = 2cr(L) = 2cl(L)where cr(L) and cl(L) denote the number of right and left cusps, resp.

Another invariant, the rotation number rot(L) can be defined by trivializingξst along L and then taking the winding number of TL. For this invariantto make sense we need to orient L, and the result will change sign whenreversing orientation. Since H2(S3; Z) = 0, this number will be independentof the chosen trivialization.

Lemma 4.2.4. For the rotation number we have rot(L) = 12

(cd(L)−cu(L)

)

where cd(L) (and cu(L)) denotes the number of down (and up) cusps in theprojection.

Proof. Notice that the vector field ∂∂x gives rise to a trivialization of ξst,

hence the rotation number can be computed as the winding number withrespect to this vector field. In conclusion, we have to count how manytimes 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 orientedupward/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 takingthe 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 ⊂ (S3, ξst) the sumtb(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 anycontact 3-manifold (Y, ξ): Suppose that for an embedded orientable (com-pact) surface Σ ⊂ Y we have ∂Σ = L. Then the contact framing can bemeasured with respect to the framing on L induced by Σ — the resultingnumber tbΣ(L) is the Thurston–Bennequin invariant of L with respect toΣ. (As we already pointed out, this quantity is independent of Σ.) Byconsidering the SO(2)-bundle ξ|Σ with the trivialization along ∂Σ given bythe tangents of L (after fixing an orientation on it), we get a relative Eulernumber 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 whichindeed induces a trivialization of ξ over ∂Σ = L. Also fix a vector field


v of tangents to L inducing the given orientation of L. Then the windingnumber of v along L with respect to the fixed trivialization of ξ on L is therotation 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 alsomight 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 Σ1,Σ2 such that rotΣ1(L) 6= rotΣ2(L). (Hint: Start witha contact structure ξ and closed surface Σ such that

⟨e(ξ), [Σ]

⟩6= 0 and

find L on Σ separating it.)

Recall that any knot K ⊂ (Y, ξ) can be C0-approximated by a Leg-endrian knot. It has been extensively studied recently to what extent aLegendrian knot is determined by the knot type in R3 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 tothe unknot then the above classical invariants determine L ⊂ (S3, ξst) up toLegendrian isotopy. In particular, up to Legendrian isotopy there is a uniqueknot L which is smoothly the unknot and has tb(L) = −1, rot(L) = 0. ThisL 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]. Theanswer to the above question in general is negative, though: according toresults of Chekanov [15, 37] there are Legendrian knots which have the sameclassical invariants but are not Legendrian isotopic. For further reading onthis 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 overtwisteddisk in the contact 3-manifold (Y, ξ) if ∂D = L is a Legendrian knot withtbD(L) = 0, i.e., if the contact framing of L coincides with the framinggiven by the disk D.

(b) The contact manifold (Y, ξ) is overtwisted if it contains an overtwisteddisk; (Y, ξ) is called tight otherwise. The contact structure ξ on Y isuniversally 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 iscalled virtually overtwisted.

Remark 4.3.2. A contact structure covered by a tight contact structure istight.

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(π2 − r2),

r ≤ π

is an overtwisted disk for sufficiently small |ε|.)(b) More generally, show that βn in Example 4.1.4(b) is overtwisted forn ≥ 1. (Hint: For n ≥ 1 consider the overtwisted disk z = 0, r ≤ r0 forfn(r0) = π.)

(c) Prove the assertion in Remark 4.3.2.

According to a fundamental result of Eliashberg [24], overtwisted contactstructures on closed 3-manifolds can be classified using homotopy theory,since

Theorem 4.3.4 (Eliashberg, [24]). Two overtwisted contact structures areisotopic if and only if they are homotopic as oriented 2-plane fields. More-over, every homotopy class of oriented 2-plane fields contains an overtwistedcontact structure.

In summary, the classification of overtwisted contact structures reduces to ahomotopy theoretic problem which is not very hard to solve. We will returnto the discussion of the homotopy classification of oriented 2-plane fields inSection 6.2. In fact, using contact surgery we will verify the second assertionof the theorem, usually attributed to Lutz and Martinet. Notice that so farwe do not have any example of tight contact structures. In general it is veryhard to show that there is no overtwisted disk present in a given contact3-manifold. This fact gives particular interest to the following result.

Theorem 4.3.5 (The Bennequin inequality, [11]). If L is a Legendrianknot in (R3, ξst) or in (S3, ξst) and Σ ⊂ Y is a Seifert surface for L thentb(L) +

∣∣ rot(L)∣∣ ≤ −χ(Σ).

Since an overtwisted disk D has χ(D) = 1 and tbD(L) = 0, this theoremimplies

Corollary 4.3.6. The standard contact structures (S3, ξst) and (R3, ξst)are tight.


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 andonly if for all Σ ⊂ Y with ∂Σ = L Legendrian we have tbΣ(L)+

∣∣ rotΣ(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 theappropriate adjunction inequality. Later we will see that the analogy be-tween the adjunction inequality of Theorem 1.2.1 and the above inequalityis 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 tightcontact structure ξ then

∣∣⟨e(ξ), [Σ]⟩ ∣∣ ≤ −χ(Σ) for any closed embedded

surface Σ 6= S2 and⟨e(ξ), [S2]

⟩= 0.

Notice that since [Σ]2 = 0 in H2(Y ; Z), this formula can again be regardedas an analogue of the adjunction inequality for 4-manifolds. Once again,this inequality fails to hold for overtwisted structures, in general. In orderto sketch the proofs of these inequalities, we need a tool for studying contactstructures near surfaces. Notice that by the nonintegrability of the planefield ξ, a surface Σ generically intersects the plane field (through the tangentplanes TΣ) in lines.

Definition 4.3.9. Fix a contact structure ξ on Y . For a surface Σ ⊂ Y wecan consider ξ∩TΣ, and for generic Σ this intersection is a line field exceptat 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 FΣ of Σ in (Y, ξ).

Examples 4.3.10. (a) Consider the unit sphere S in the contact manifold(R3, ξ3) where ξ3 = ker(dz+r2dθ) as in Example 4.1.4(a). Since the contactplanes 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 atthese points. By visualizing the contact planes as they slowly twist whilemoving 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 characteristicfoliation will “spiral” around the sphere connecting the two singular pointsas shown in Figure 4.6.

(b) Consider the disk D of radius π in the (r, θ)-plane in (R3, β) as in Exam-ple 4.1.4(b). Recall that the contact planes are spanned by ∂

∂r , r tan r ∂∂z −


Figure 4.6. Characteristic foliation on S2⊂ (R3, 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 characteristicfoliation is a line segment connecting the center of the disk to a boundarypoint. This gives an example of a nongeneric characteristic foliation on asurface. Notice that D is an overtwisted disk since tbD(∂D) = 0. Nowimagine 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 planestangent to D′ along its boundary are no longer horizontal. It is clear thatthe boundary of D′ becomes a closed leaf of the characteristic foliation withonly one singularity in the center of D′, see Figure 4.7.

Figure 4.7. The overtwisted disk, before and after


Definition 4.3.11. Consider the eigenvalues λ1, λ2 of the linearization ofthe flow at a generic isolated singular point p. We define the index of p tobe equal to +1 if λ1λ2 > 0 and −1 if λ1λ2 < 0. A generic isolated singularpoint of index +1 (resp. −1) is called an elliptic (resp. hyperbolic) singularpoint. 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 minimaof a Morse function on a surface, while hyperbolic points correspond tosaddle points. This analogy gets even deeper when recognizing that for ageneric flow hyperbolic points cannot be connected by a leaf — similar tothe saddle points of a Morse–Smale function. In addition, we can assign asign to each (isolated) singular point p of the characteristic foliation: Thesingularity is positive (resp. negative) if the orientation of ξp agrees (resp.disagrees) with the orientation of TpΣ. Notice that this makes sense onceΣ and ξ are both oriented. In Example 4.3.10(a) both singular points areelliptic with opposite signs. See Section 8.3 for similar notions in dimensionfour. The characteristic foliation FΣ can be oriented as follows: If p is anonsingular point of a leaf L of FΣ, then we choose v ∈ TpL so that (v, n)is an oriented basis for TpΣ, where n ∈ TpΣ is an oriented normal vector toξp. With this choice of orientation a positive elliptic point becomes a sourceand a negative elliptic point becomes a sink.


In order to understand the topology of contact 3-manifolds we need tohave a good grasp on how to cut and paste contact structures along surfaces.It turns out that the characteristic foliation determines the contact structurenear the surface. The following result can be obtained as an application ofTheorem 4.1.12; for a proof see for example [57].

Theorem 4.3.12. If Σi ⊂ (Yi, ξi) (i = 1, 2) embedded surfaces are diffeo-morphic through a diffeomorphism f : Σ1 → Σ2 which preserves the char-acteristic foliations then f extends to a contactomorphism on some neigh-borhood of Σ1.

Using the concept and count of positive and negative elliptic/hyperbolicpoints 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 thecharacteristic foliation FΣ is generic, i.e., the singular points are isolated andno 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 FΣ. Let e± and h± denote the number of ± elliptic/hyperbolicpoints of FΣ. Fix a vector field w which directs FΣ. Now it easily followsfrom the Poincare–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 byconsidering them as embedded oriented surfaces in the total space of thebundle ξ|Σ. We choose the section of ξ|Σ given by w which also gives asection 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 ). Thecount of oriented intersection number of sections to calculate

⟨e(ξ), [Σ]

⟩

will differ from the calculation of χ(Σ) exactly at those intersection pointswhere the orientations of the contact planes disagree with the orientationsof the tangent planes. So we need a sign reversal in the count exactly atthe negative singular points of FΣ 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+).


It is a theorem of Giroux (called the Elimination lemma) that if an ellipticand 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 suchthat both singular points disappear. (For the corresponding phenomenon indimension four see Section 8.3.) Conversely we can always create a pair ofelliptic and hyperbolic points of the same sign on a given leaf. Therefore wecan assume that there is no closed leaf in FΣ. We will call the new surfacewe 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 usedthe tightness of the contact structure. Suppose that p is a positive ellipticpoint on the surface Σ. Now let Op be the union of all leaves limiting to pand let Dp be the closure of it. Suppose that Dp is an embedded disk sothat ∂Dp = Dp −Op. Then all the singular points of FΣ on Dp other thanp will be on ∂Dp. Since FΣ is oriented, there is no positive elliptic point on∂Dp and no two elliptic points can be adjacent. This is because a positiveelliptic point is a source and a negative elliptic point is a sink, so a leafconnecting two elliptic points is directed form the positive to the negative.Therefore the arcs on ∂Dp between elliptic points are divided by hyperbolicpoints and, by the assumption we made at the beginning of the proof aboutFΣ, no two hyperbolic points are adjacent. Suppose that there is no posi-tive hyperbolic point on ∂Dp. Then we can eliminate all the singular pointson ∂Dp using the Elimination lemma and thus Dp becomes an overtwisteddisk which cannot exist in a tight contact manifold. Hence there has to bea positive hyperbolic point q on ∂Dp. But then we can eliminate the posi-tive elliptic point p using this positive hyperbolic point q. The difficult partof the proof is to show that we can eliminate a positive elliptic point evenif Dp is not embedded. For details of this part of the proof the reader isadvised to turn to [26, 39]. By completing this last step we conclude thate+ = 0 can be assumed, trivially implying

⟨e(ξ), [Σ]

⟩≤ −χ(Σ).

Moreover by subtracting the above equations and eliminating the negativeelliptic 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 numberslΣ(γ) of γ as the linking number of γ and γ′, where γ′ is a push-off obtainedby a nonzero vector field in the contact planes. That is, slΣ(γ) is the oriented


intersection number of γ′ with Σ. If γ ⊂ R3 or S3 then slΣ(γ) can be shownto 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 pushingL in opposite directions in a sufficiently small annulus neighborhood ofL to obtain positive and negative transverse push-offs L± of L. If L ⊂(R3, ξst) then it is easy to obtain the front projections of the transversepush-offs L± from the front projection of a Legendrian knot L: For L+

just smooth out the upward cusps and replace downward cusps by negativekinks. 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-linkingnumber of a transverse knot in (R3, ξst) is equal to its writhe in its frontprojection, we get that for a Legendrian knot L ⊂ (R3, ξst)

sl(L+) = w(L) − cd(L) = w(L) − 1

2

(cd(L) + cu(L)

)− 1

2

(cd(L) − cu(L)

)

= tb(L) − rot(L).

This equation holds for a null-homologous transverse knot in an arbitrarycontact 3-manifold, leading us to the equation

slΣ(L±) = tb(L) ∓ rotΣ(L).

Proof of Theorem 4.3.7. Notice that one direction of this equivalenceis 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


consider the positive and negative transverse push-offs L± of the Legendrianknot L. We can interpret the self-linking number of a transverse knot γas a relative Euler number and by the use of the method of the proof ofTheorem 4.3.8 we derive the equation

slΣ(γ) = −(e+ − h+) + (e− − h−).

In conclusion, we get a relation between slΣ(γ) and the number of differenttypes of singular points of the characteristic foliation on the surface Σbounded by γ. Combining this result with

χ(Σ) = (e+ + e−) − (h+ + h−)

we get slΣ(L±) + χ(Σ) = 2(e− − h−). By using the Elimination lemmaand the tightness of the contact structure as in the proof of Theorem 4.3.8we can assume that e− = 0 and thus slΣ(L±) ≤ −χ(Σ), clearly implyingEliashberg’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–Bennequininvariants. 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 tbΣ(L′) = n. It turns out that by taking enough copiesof 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 followsby an additional connect sum. For Legendrian knots with tight complementthe situation is more complicated.

5. Convex surfaces in contact 3-manifolds

When trying to do surgery on contact 3-manifolds we need to understandcontact structures in neighborhoods of embedded surfaces. As we alreadypointed out in Chapter 4, for a given surface Σ ⊂ (Y, ξ) the characteristicfoliation FΣ determines the contact structure near Σ. But it is not easyto describe or relate characteristic foliations. It turns out that the sameinformation can be captured by certain configurations of curves on thesurface at hand once the surface is in a special position with respect to thecontact structure. This theory has been developed and fruitfully appliedby Giroux and Honda in various circumstances in 3-dimensional contactgeometry. For the sake of completeness, in this Chapter we recall thefundamental definitions and results regarding convex surfaces and dividingsets. These statements will be used in our study of contact Dehn surgeryin 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 calledcontact if its flow ϕt preserves the contact planes, i.e., (ϕt)∗ξ = ξ.

The following lemma gives a convenient characterization of contact vectorfields on contact 3-manifolds.

Lemma 5.1.2. Let (Y, ξ) be a contact 3-manifold, where ξ = kerα for somecontact 1-form α. A vector field v on Y is contact if and only if Lvα = fαfor some smooth function f : Y → R.

Exercises 5.1.3. (a) Show that for each smooth function H : Y → R thereis a unique vector field VH ∈ ξ such that vH = HRα + VH is a contact

86 5. Convex surfaces in contact 3-manifolds

vector field, where Rα denotes the Reeb vector field for α. (Hint: VH isthe unique vector field in ξ satisfying ιVH

dα = dH(Rα)α − dH, cf. alsoExercise 4.1.9(a).)

(b) Verify that for each contact vector field v there is a unique smoothfunction H : Y → R such that v = vH .

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 contactvector field v transverse to Σ. If ∂Σ 6= ∅ then we also require that ∂Σ isLegendrian.

Remark 5.1.5. Notice that the direction of the contact vector field v inthe definition is irrelevant, therefore there is no distinguished side of Σ. Inthat respect the term “convex” is unfortunate, since there is no concavitypresent.

It can be shown that Σ ⊂ (Y, ξ) is convex if and only if it has a neighborhoodN = νΣ = Σ×I such that ξ|N is invariant in the I-direction. Consequently,in the neighborhood νΣ = Σ× I of the convex surface Σ the contact 1-formα can be written as f dt+ β, where f is a function, β is a 1-form on Σ andt denotes the I-coordinate.

Proposition 5.1.6 (Giroux, [61]). Any closed surface admits a C∞-smallperturbation which puts it into convex position.

Remark 5.1.7. In [76] it was shown that this is also true for a surface withboundary as long as the surface has Legendrian boundary and the twistingof the contact planes with respect to the surface is not positive.

We should point out that even though every surface in a contact 3-manifoldcan be perturbed into a convex surface it is the existence of non-convexsurfaces which makes the theory interesting. In Example 5.1.11 we willdescribe a non-convex surface.

Definition 5.1.8. Suppose that Σ ⊂ (Y, ξ) is a convex surface with thecontact vector field v. Define Γ =

x ∈ Σ | v(x) ∈ ξx

⊂ 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 disconnectedand possibly with boundary. We will often refer to this set Γ of finite unionof disjoint simple closed curves and properly embedded arcs on Σ as thedividing curves.

5.1. Convex surfaces and dividing sets 87

Proposition 5.1.9 (Giroux, [61]). The dividing set Γ is a 1-dimensionalsubmanifold of the surface Σ transverse to the characteristic foliation FΣ

and Σ − Γ = Σ+ ∪ Σ− where the flow of a vector field w which directs FΣ

expands (contracts) a volume form Ω on Σ+ (on Σ−, resp.) and w pointsoutward 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 thesecoordinates the 1-form α defining ξ can be expressed as α = f dt+ β. Thevertical vector field v = ∂

∂t is a contact vector field for Σ since ∂∂t is clearly

transverse to Σ and L ∂∂tα = 0. Then for a point x ∈ Σ we have αx( ∂

∂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 Σ×I.) In particular,f(x) = 0 implies that β ∧ df 6= 0 and hence df 6= 0. Consequently, Γ is asubmanifold of Σ, transversely cut out by f . Let u be a vector tangent to Γ.Then df(u) = 0 and thus β ∧ df 6= 0 implies that β(u) 6= 0. That is, u is notin TFΣ = ker β and it follows that Γ is transverse to FΣ. Let w be a vectorfield which directs FΣ. The vector field w can be defined by the equationιwΩ = α|Σ = β for a volume form Ω on Σ. Notice that w vanishes exactlyat the zeros of β. Moreover if we take a different volume form on Σ we geta positive multiple of w directing FΣ. 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 thisfirst 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 andwe can see that the normal orientation of the planes agree with v = ∂

∂t ifand only if f > 0. Furthermore the vector field w points outward from theboundary of Σ+. To see that the flow of w expands Ω on Σ+ we observethat Lw

fΩ = dιw

fΩ + ιw

fdΩ = dιw

fΩ = d(βf ) = 1

f2 (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 curvesis independent of the choice of the contact vector field.


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 = R2×S1 induced bythe contact structure ξ3 = ker(dz+r2 dθ) on R3 (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 = Tk = (r, θ, z) ∈ Y | r = k is not convex.Recall that the contact planes of the given contact structure ξ3 are spannedby ∂

∂r , r2 ∂∂z − ∂

∂θ (see Example 4.1.4(a)). Therefore at any point p on Tthe intersection of the tangent plane to T and the contact plane ξ3 is givenby the line generated by the vector k2 ∂

∂z − ∂∂θ . Here we can view this line

in the (θ, z)-plane when we consider T as obtained by the identificationsz ∼ z+1 and θ ∼ θ+2π. Thus we conclude that the characteristic foliationon T is linear as shown in Figure 5.1(a). Suppose that T is convex. Thenthe contact 1-form on Y can be written as f dt+ β in a vertically invariantneighborhood of T as explained above, where f is a smooth function andβ is a 1-form on T . The form β is given by (dz + r2 dθ)|T = dz + k2 dθand hence dβ = 0. On the other hand the contact condition implies thatβ ∧ df + fdβ > 0 as shown in the proof of Proposition 5.1.9. It follows thatdf(w) < 0 for some vector field w directing the characteristic foliation on Twhich is a contradiction since the function f on T has to be periodic in θand z and thus f can not be decreasing along a linear foliation on T .

Exercises 5.1.12. (a) Perturb the torus T = Tk in the example above intoa convex torus in (Y, ξ3). (Hint: First consider the two disjoint annuli in thecomplement of two orbits of the characteristic foliation on T . Then pushslightly one of the annuli (fixing its boundary) towards the z-axis whilepushing the other one slightly in the opposite direction to get a smoothembedded torus. Show that the dividing curves look like the dashed linesin Figure 5.1(b).)

(b) Show that the unit sphere S2 in (R3, ξ3) is convex. Determine thedividing set on S2. (Hint: Try the vector field v = z ∂

∂z + r2∂∂r .)

Definition 5.1.13. Let L be a Legendrian curve on a convex surface Σ ina contact 3-manifold (Y, ξ). Then tw(L,Σ) denotes the twisting number ofthe contact planes ξ along L measured with respect to the surface framingon 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 surfaceΣ 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


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 eachpoint in Γ ∩ L contributes −1

2 to tw(L,Σ).)

Proposition 5.1.15. Suppose that Σ is a closed convex surface in a contactmanifold (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 pointswill be in Σ+ (resp. Σ−). Then using the Poincare–Hopf theorem fora vector field on a manifold which is transverse to the boundary we getχ(Σ+) = e+ − h+ and χ(Σ−) = e− − h−.

If F is any singular foliation on the surface Σ then a multicurve Γ onΣ is said to divide F if the pair (F ,Γ) satisfies the properties proved inProposition 5.1.9, where FΣ is replaced by F . The power of studying thedividing set comes from the fact that Γ (rather than the full characteristicfoliation) already determines the contact structure near Σ:

Theorem 5.1.16 (Giroux’s flexibility, [61]). If F is another singular fo-liation on Σ divided by Γ then there is an isotopy Ψs : Σ → N = νΣ(s ∈ [0, 1]

), Ψ0 = idΣ and Ψs|Γ = idΓ such that Ψs(Σ) is convex for all s

and FΨ1(Σ) = Ψ1(F).


Therefore, by fixing Γ, any foliation divided by Γ can be thought of asthe characteristic foliation; in conclusion Γ determines the germ of thecontact structure along Σ. The next lemma shows a connection betweenconvex surfaces and Legendrian knots on them. First we need the followingdefinition.

Definition 5.1.17. A properly embedded 1-submanifold C of a convexsurface Σ is nonisolating if C is transverse to Γ and the closure of everycomponent 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., thereexists an isotopy ψs : Σ → N = νΣ

(s ∈ [0, 1]

), ψ0 = idΣ such that ψs(Σ)

is convex for all s ∈ [0, 1], ψ1(ΓΣ) = Γψ1(Σ), and ψ1(C) is Legendrian.

Remark 5.1.19. The nonisolating condition guarantees that C can beextended to a singular foliation divided by Γ. Then by Theorem 5.1.16we can realize this foliation on Σ as the characteristic foliation and henceC 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 convexsurface (possibly with Legendrian boundary) and Σ 6= S2. Then Σ hasa tight neighborhood if and only if Γ contains no homotopically trivialcomponent. If Σ = S2 then νΣ is tight if and only if Γ consists of a singlecomponent.

Proof (sketch). We give a proof for the “only if” direction. Suppose thatΓ contains at least two components one of which is homotopically trivial.Let γ denote the homotopically trivial curve which bounds a disk D. Letγ′ be a curve parallel to γ such that γ′ ∩ Γ = ∅. Then γ′ is nonisolatingon Σ and hence by the Legendrian realization principle we can make γ′

Legendrian (so that it stays disjoint from γ). This implies that the surfaceframing of γ′ agrees with its contact framing by Exercise 5.1.14. Thus thedisk bounded by γ′ on Σ is an overtwisted disk by definition. Now supposethat Γ has only one component γ which is homotopically trivial. Take ahomotopically essential non-separating simple closed curve δ ⊂ Σ− ΓΣ anduse the “folding” method of Honda [76] to introduce a pair of dividing curvesparallel to δ. Then repeat the previous argument to find an overtwisteddisk.


Exercises 5.1.21. (a) Use Theorem 4.3.8 to show that if the dividing seton a closed convex surface Σ 6= S2 in a contact 3-manifold (Y, ξ) consistsof 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 givenΣ in a tight contact 3-manifold (Y, ξ) into convex position. It is clear thatχ(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 whenΣ 6= S2. If Σ = S2 then Σ − ΓΣ 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 thedividing set Γ on T 2 consists of 2n parallel circles (n ≥ 1).

By fixing an identification of T 2 with R2/Z2, the slope of these parallelcurves is called the slope of the torus at hand. Of course, there is nocanonical choice of identification of T 2 with R2/Z2 in general. In particularcases, however, there are natural directions to choose — for example ifT 2 = ∂(S1 × D2) or T 2 = ∂νK is the boundary of the neighborhood of aLegendrian knot then the meridian µ provides an obvious direction.

Example 5.1.23. Consider N = R2 × (R/Z) ≃ R2 × S1 with the 1-form

α = cos(2πz) dx − sin(2πz) dy

in the coordinates (x, y) for R2 and z for R/Z. First we check that α is acontact form on N : since

dα = 2π sin(2πz) dx ∧ dz + 2π cos(2πz) dy ∧ dz,

we have

α ∧ dα = 2π cos2(2πz) dx ∧ dy ∧ dz + 2π sin2(2πz) dx ∧ dy ∧ dz

= 2π dx ∧ dy ∧ dz.

The contact form α on N induces a contact form on the solid torus Nd =(x, y, z) | x2 + y2 ≤ d2

for d > 0. We claim that ∂Nd is a convex torus.

To this end, consider the vector field v = x ∂∂x + y ∂

∂y . It is clear that v is


transverse to ∂Nd. To show that v is a contact vector field, we check thatLvα = α (see Lemma 5.1.2). For the given α we calculate

ιvα = ι(x ∂

∂x+y ∂

∂y

)α = ιx ∂∂xα+ ιy ∂

∂yα = x cos(2πz) − y sin(2πz),

dιvα = cos(2πz) dx − 2πx sin(2πz) dz − sin(2πz) dy − 2πy cos(2πz) dz,

and

ιvdα = ιx ∂∂xdα+ ιy ∂

∂ydα = 2πx sin(2πz) dz + 2πy cos(2πz) dz.

Then it follows by Cartan’s formula Lvα = ιvdα+ dιvα that Lvα = α. Thecentral circle

C =

(x, y, z) ∈ Nd | x = y = 0

acquires a canonical contact framing. This framing can be given by thelongitude λ that is obtained by pushing C along a vector field which istransverse to C and stays inside the contact planes ζ = kerα. If wechoose the vector field that is orthogonal to C, it is easy to calculate thatλ = (d sin(2πz), d cos(2πz), z

). Since ∂Nd is a convex torus in a contact3-manifold, there are dividing curves on ∂Nd induced by the contact vectorfield v. By definition, the dividing curves consist of the points x ∈ ∂Nd suchthat v(x) ∈ ζ(x), i.e, when αx

(v(x)

)= 0. The solution of the equation

(cos(2πz) dx − sin(2πz) dy

) (x∂

∂x+ y

∂

∂y

)= x cos(2πz) − y sin(2πz) = 0

or equivalently the equation

z =1

2πtan−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, withthe trivialization of ∂Nd by λ and the meridian µ the slope of the dividingcurves comes out to be equal to ∞. Here µ will correspond to the x- and λto the y-axis; hence the slope being p

q means that Γ is parallel to the curvepλ+ qµ. In fact, we can visualize the contact planes as follows: The planesare horizontal at z = 0 and start twisting as z is increasing and they becomehorizontal again when z = 1. So the characteristic foliation consists of twosingular lines of slope = ∞ and parallel nonsingular leaves of slope 6= ∞.(Notice that this characteristic foliation is not generic.)


Remark 5.1.24. In general, on S1 ×D2 only the meridian µ is canonical,hence the slope of ∂(S1×D2) is well-defined only up to an action of SL2(Z)

leaving µ fixed, i.e., of the action

(1 m0 1

)— the Dehn twists changing

the framing. It is not hard to see that using this equivalence any nonzeroslope 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 abovematrix the slopes p

q and ppm+q are equivalent.

Exercise 5.1.25. Find slopes equivalent to 23 and 1.

For topologically simple 3-manifolds the dividing curves may determine theentire contact structure. The following is a fundamental result which isessential 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 ∂D3 which makes ∂D3 convex with connecteddividing set. Then there exists a unique extension of ξ to a tight contactstructure on the 3-disk D3 up to an isotopy which fixes the boundary.

Exercise 5.1.27. Using Theorem 5.1.26 show that the 3-sphere S3 admits(up to isotopy) a unique tight contact structure.

The exercise above can be solved by a simple-minded approach to findan 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 alongconvex surfaces until we end up with a disjoint union of 3-disks. At eachstep we keep track of all possible configurations of dividing curves on thesesurfaces along which we cut our 3-manifold. We will apply this strategybelow to find an upper bound for the number of tight contact structureson the solid torus for the case when the boundary slope of the dividingcurves is equal to 1

n . We will first state a basic lemma called the “edgerounding” which is frequently used to transfer dividing sets between twoconvex surfaces meeting along a Legendrian curve.

Exercise 5.1.28. Let Σi be a convex surface with dividing set Γi for i = 1, 2.Assume that ∂Σ2 is a Legendrian curve in Σ1. Let A = Γ1 ∩ ∂Σ2 andB = Γ2∩∂Σ2. Then between two adjacent points of A there is a point in Band between two adjacent points of B there is a point in A. (Hint: Considerthe unique geometric model of contact structures in a neighborhood of theLegendrian curve.)


Lemma 5.1.29 (Edge rounding, [76]). Let Σi be a convex surface withthe dividing set Γi for i = 1, 2 and assume that ∂Σ1 = ∂Σ2 is Legendrian.Then using the standard local model around ∂Σ1 we can glue Σ1 to Σ2 byrounding the edge ∂Σ1 = ∂Σ2 to get a smooth surface Σ so that the dividingcurves Γi connect up as shown in Figure 5.1.29 to form a dividing set Γ onΣ.

Σ1

Σ

Σ

2

Figure 5.2. Connecting up the dividing curves while rounding an edge

Theorem 5.1.30. Suppose that ξ1 and ξ2 are two tight contact structureson S1 × D2 with two parallel dividing curves on the convex boundary∂(S1 × D2) having slope equal to 1

n for some n ∈ Z. Then ξ1 and ξ2are isotopic.

Proof (sketch). Notice first that 1n and 1

m are equivalent boundary slopesfor any m,n ∈ Z and −1 = 1

−1 also represents this class. Hence it sufficesto classify the tight contact structures for any one of these slopes. It isclear that a meridian on the convex surface ∂(S1 × D2) is nonisolatingand therefore we can isotope this meridian into Legendrian position by theLegendrian Realization Principle. Notice that the twisting tw(∂D,D) of thecontact planes along ∂D with respect to a spanning diskD of the meridian isnegative. Thus D can be isotoped to a convex disk by Remark 5.1.7. Thentightness of the contact structures at hand implies by Giroux’s criterion


(Theorem 5.1.20) that the dividing set ΓD on the disk D contains no closedcomponents, hence ΓD is a single arc connecting two points a1 and a2 on∂D. Let b1, b2 ∈ ∂D denote the points of the intersection of D with thedividing set on the convex boundary of the solid torus. Now we have aconvex torus intersecting a convex disk along a Legendrian curve and weknow the dividing sets on these surfaces. Hence by Exercise 5.1.28, b1 ispositioned between a1 and a2 while b2 is positioned between a2 and a1 on the(oriented) circle ∂D. Next we cut S1 ×D2 along D and smooth the cornersby rounding the edges using Lemma 5.1.29. Notice that when we removea neighborhood νD of D from S1 × D2 we get a 3-disk D3 such that thedividing set on its boundary is connected. Now Eliashberg’s Theorem 5.1.26concludes the proof: near the boundary and near the spanning disk D thecontact structures ξ1 and ξ2 are isotopic (shown by the dividing curves),and the complement of νD in S1 × D2 is D3 with connected dividing seton its boundary. Therefore Theorem 5.1.26 extends the above isotopy toS1 ×D2, 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 configurationsfor the dividing curves on it, since the dividing curves on ∂(S1 ×D2) willintersect ∂

(pt.×D2

)in more points: if the slope is r = p

q then ∂D inter-

sects the dividing set of ∂(S1 ×D2) in 2p points. Every configuration givesa potential tight contact structure, and so this argument gives a (poten-tially weak) upper bound for the number of tight structures. In fact, manyof the different configurations correspond to isotopic tight structures. Inorder to get the classification, Honda followed a slightly different path, andmanipulated 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 onS1 × D2 with boundary slope zero: in this case ∂

(pt. × D2

)is disjoint

from the dividing curves of the boundary, therefore pt.×D2 (after havingbeen isotoped to have Legendrian boundary) provides an overtwisted disk.


5.2. Contact structures and Heegaard decompositions

In this section we review a construction of Torisu [168] associating a uniquecontact structure to an open book decomposition of a 3-manifold. Torisu’sresult is based on the work of Giroux on convex contact structures. Wefollow 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 thegiven open book decomposition in the sense of Giroux. (See Chapter 9 forrelevant definitions regarding open book decompositions and their relationto 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, whileπ : Y − L → S1 denotes the fibration of the open book decomposition.)Then by presenting the circle S1 as the union of two closed (connected) arcsS1 = A1 ∪ A2 intersecting each other in two points, the open book decom-position (L, π) naturally induces a Heegaard decomposition Y = U1 ∪Σ U2

of the 3-manifold Y : one only needs to verify the simple observation thatUi = π−1(Ai) ∪ L are solid handlebodies. The surface Σ along which thesehandlebodies are glued is simply the union of two pages π−1(A1 ∩ A2) 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 ξ1, ξ2 are contact structureson Y satisfying:

(i) ξi|Uj(i = 1, 2; j = 1, 2) are tight, and

(ii) Σ is convex in (Y, ξi) and L is the dividing set for both contact structures.

Then ξ1 and ξ2 are isotopic. In addition, the set of such contact structuresis nonempty.

Proof. Suppose that a page F of the given open book is a genus-g surfacewith r boundary components. Then ∂U1 = Σ is a closed surface of genush = 2g + r − 1. First we would like to argue that there is at most onetight contact structure on the handlebody U1 such that L is the dividingset on Σ. Since U1 is a genus-h handlebody, it is clear that we can find hhomologically linearly independent curves α1, α2, . . . , αh on Σ which boundh disjoint disks in U1 so that when we cut along these disks we get the 3-diskD3. The key point of our construction is that we can choose α1, α2, . . . , αhin such a way that each αk intersects the dividing set L ⊂ Σ twice fork = 1, 2, . . . , h. We depicted a choice of such curves for r = 3, g = 2 inFigure 5.4. The disk Dk spanned by αk can be visualized as the disk whichis swept out in U1 (see Figure 5.3) by swinging the left-half of the curve αkuntil it coincides with its right-half.

α

α

αα

α α

L

L

L

6

5

43

1

2

Figure 5.4. The α-curves.

Now we proceed exactly as in the proof of Theorem 5.1.30. First we putthe curves α1, α2, . . . , αh into Legendrian position and make the spanningdisks D1,D2, . . . ,Dh convex. The diving set on each Dk will be an arcconnecting two points on the boundary, for k = 1, 2, . . . , h. Then we cut


along these disks and round the edges to get a connected dividing set onthe remaining D3 and use Eliashberg’s theorem to show the uniqueness of atight contact structure on U1 with the assumed boundary condition. Clearlywe can prove the same result for the handlebody U2. To finish this part ofthe proof of the theorem we need to show the existence of a tight contactstructure η1 on U1 (and η2 on U2) which has L as its dividing set on Σ. Theidea is to embed U1 into an open book whose compatible contact structure(see Chapter 9) is Stein fillable, and hence tight. Such an embedding ofthe genus-g handlebody into a Stein fillable contact structure will be shownin Exercise 11.3.5(c). Suppose now that ηj is a tight contact structure onUj whose dividing set is equal to the binding L ⊂ ∂Uj for j = 1, 2. Letφ : ∂U1 → ∂U2 be the diffeomorphism defining the Heegaard decompositionY = U1 ∪Σ U2. The tight contact structure ηj on Uj induces a foliation Fjon Σ. Now φ(F1) (as well as F2) is a singular foliation on ∂U2 divided by L(since φ is the identity on L). Then by Giroux’s flexibility Theorem 5.1.16we can isotope Σ in Y so that φ(F1) and F2 agree by this isotopy and hencewe can glue the tight contact structure η1 and η2 to get a contact structureξ on Y . The uniqueness of such a contact structure ξ on Y follows from theuniqueness of η1 and η2. Notice that the tightness of η1 and η2 does notimply tightness for the glued up contact structure ξ.

Example 5.2.2. Consider the open book decomposition of S3 induced bythe positive (resp. negative) Hopf link. The associated contact structure ξis tight (resp. overtwisted).

Let (Yi, ξi) be a contact 3-manifold for i = 1, 2. To define the contactconnected sum (Y = Y1#Y2, ξ = ξ1#ξ2) just delete a Darboux ball Di

from Yi (i = 1, 2) and glue Y1 − int D1 to Y2 − int D2 by a diffeomorphismf : ∂(Y1 − int D1) → ∂(Y2 − int D2) which takes the dividing set Γ1 on∂(Y1 − int D1) to the dividing set Γ2 on ∂(Y2 − int D2). This operation iswell 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 contactconnected sum of the corresponding contact structures. (See [68] for analternative proof.)

Exercise 5.2.4. Show that the contact structure associated to the openbook of S3 induced by a (p, q)-torus knot is tight. (Hint: See Example 9.1.4.)

6. Spinc structures on 3- and 4-manifolds

Spinc structures turn out to be very useful tools in understanding homotopicproperties of contact structures. In addition, gauge theoretic invariants —such as Seiberg–Witten and Ozsvath–Szabo invariants — are defined forspinc 3- and 4-manifolds. This chapter is devoted to the review of spinc

structures — with a special emphasis on the 3- and 4-dimensional case.Throughout this chapter we will assume that the reader is familiar withthe basics of the theory of characteristic classes. (For an excellent referencesee [116].) For a more complete treatment of spinc structures the reader isadvised to turn to [113].

6.1. Generalities on spin and spinc structures

We begin our discussion by recalling the related and much more standardsubject 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 connectedLie group with a map ρ : Spin(n) → SO(n) which is the principal Z2-bundle of the unique nontrivial real line bundle on SO(n) (n ≥ 3). (Recallthat π1

(SO(n)

)= H1

(SO(n); Z

)= Z2.) Let X be a given oriented

Riemannian n-manifold and let p : PSO(n) → X denote the principal SO(n)-bundle of orthonormal frames in TX. A spin structure on X is a principalSpin(n)-bundle s : PSpin(n) → X with a map π : PSpin(n) → PSO(n) suchthat pπ = s and fiberwise π is just the double cover ρ : Spin(n) → SO(n).

100 6. Spinc structures on 3- and 4-manifolds

In other words, the associated principal SO(n)-bundle PSpin(n) ×ρ SO(n) isisomorphic to PSO(n). Two spin structures P1 and P2 are equivalent if thereis a bundle isomorphism ϕ : P1 → P2 such that s1 = s2ϕ where si : Pi → Xare the bundle maps of the principal Spin(n)-bundles for i = 1, 2. The setof 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 asimilar definition provides spin structures on E.

Theorem 6.1.2. An oriented Riemannian n-manifold X admits a spinstructure if and only if w2(X) = 0. In that case the number of inequivalentspin structures is equal to

∣∣H1(X; Z2)∣∣ . In fact, H1(X; Z2) admits a free

and transitive action on Spin(X).

In a similar fashion, it can be shown that a principal SO(n)-bundle E → Xadmits a spin structure if and only if w2(E) = 0, and the number ofinequivalent spin structures is again equal to

∣∣H1(X; Z2)∣∣ .

Spinc structures

The group Spinc(n) is defined as S1 × Spin(n)/Z2 where Z2 =± (1, 1) ∈

S1 × Spinc(n)

, and ±1 is defined as ker ρ ⊂ Spin(n). It followsthat Spinc(n) admits an S1-fibration over SO(n) (n ≥ 3); this mapρ : Spinc(n) → SO(n) can be characterized as the principal S1-bundle ofthe unique nontrivial complex line bundle on SO(n) (n ≥ 3). Notice thatH2(SO(n); Z) = Z2 for n ≥ 3. Again, a spinc structure on an n-dimensionalmanifold X is a principal Spinc(n)-bundle s : PSpinc(n) → X with a mapπ : PSpinc(n) → PSO(n) such that s = p π and fiberwise π is just ρ; equiva-lently, in bundle theoretic terms PSpinc(n)×ρSO(n) ∼= PSO(n). As in the spincase, spinc structures can be defined for any principal SO(n)-bundle. Themap α : Spinc(n) → S1 we get by the formula

±(z,A)

7→ z2 enables us to

associate a line bundle — the determinant line bundle — L = PSpinc(n)×αC

to a given spinc structure PSpinc(n). In an equivalent way, a spinc structure

on X can be regarded as an element u ∈ H2(PSO(n); Z) whose restriction toevery fiber of PSO(n) → X is the unique nontrivial element of H2(SO(n); Z):by considering the S1-bundle su : L→ PSO(n) corresponding to u, the com-position su p provides a principal Spinc(n)-bundle structure on L andhence a spinc structure in the above sense. Similarly to the spin case, wesay that spinc structures P1, P2 → X are equivalent if there is a bundle

6.1. Generalities on spin and spinc structures 101

isomorphism h : P1 → P2 satisfying s1 = s2 h, where si : Pi → X are thebundle projections. The set of equivalence classes of spinc structures on afixed manifold X will be denoted by Spinc(X). As the above reformulationshows, we can regard Spinc(X) as subset of H2(PSO(n); Z).

Theorem 6.1.3. Let PSpinc(n) be a given spinc structure with determinantline bundle detPSpinc(n) = L. Then c1(L) ≡ w2(X) (mod 2). In addition,if c ∈ H2(X; Z) satisfies c ≡ w2(X) (mod 2) then there is a spinc structurewith determinant line bundle L satisfying c1(L) = c.

Proof (sketch). The natural map α× ρ : Spinc(n) → SO(2) × SO(n) canbe shown to be the unique double cover of SO(2) × SO(n) which extendsto a double cover of SO(n + 2), hence TX admits a spinc structure if andonly if there is a line bundle L such that TX ⊕ L admits a spin structure.This latter is equivalent to w2(TX ⊕ L) = w2(TX) + c1(L)

∣∣2

= 0, provingthe claim.

The group H2(X; Z) admits a free and transitive action on Spinc(X) (ifthe latter is nonempty) as follows: for s ∈ Spinc(X) ⊂ H2(PSO(n); Z) anda ∈ H2(X; Z) the action of a on s is given by

s 7→ s + p∗(a)

where p : PSO(n) → X is the bundle map of the frame bundle. The naturalgroup homomorphism Spin(n) → Spinc(n) shows that a spin structureinduces a spinc structure. It follows that such an induced spinc structurehas trivial determinant line bundle. Conversely, if det (PSpinc(n)) is trivialfor a spinc structure then it can be induced by a spin structure, since thetriviality of the determinant line bundle shows that the cocycle structure ofPSpinc(n) can be homotoped into the kernel kerα = Spin(n).

The collar neighborhood theorem for the embedding ∂X → X providesa splitting of TX|∂ = T (∂X)⊕R near the boundary, implying in particular,that a spin (or spinc) structure on X naturally induces a similar structureon ∂X. (As always, R denotes the trivial real line bundle.) After havingdispensed with the above general discussion, in the rest of this chapter wefocus on the 3- and 4-dimensional case and relate spinc structures to othergeometric objects on such manifolds.


6.2. Spinc structures and oriented 2-plane fields

We start with the 3-dimensional case. It is fairly easy to see that Spin(3) =SU(2) ∼= S3 and Spinc(3) = U(2). Notice also that from the theory ofcharacteristic classes it follows that for a 3-manifold X we have w2(X) =w2

1(X), therefore any oriented 3-manifold admits a spin (and so spinc)structure. A spinc structure on an oriented 3-dimensional Euclidean vectorspace V can be given by specifying a complex hermitian plane W with amap γ : V → HomC(W,W ) satisfying γ(v)∗γ(v) = −|v|2 idW . Globally, aspinc structure on an oriented Riemannian 3-manifold is simply a continuousfamily of spinc structures on the tangent spaces, i.e., a pair (W,ρ) whereW → Y is a hermitian C2-bundle (a U(2)-bundle) on the 3-manifold Y andρ : TCY → HomC(W,W ) is a bundle homomorphism satisfying ρ(v)∗ρ(v) =−|v|2 idW . The equivalence with the definition given in Section 6.1 is clear:PSpinc(3) = PU(2) corresponds to the principal U(2)-bundle of W whileρ : TCY → HomC(W,W ) and the map π : PSpinc(3) → PSO(3) determine eachother. Next we discuss a more geometric presentation of spinc structureson 3-manifolds; this presentation will be more suitable for our purposes inour subsequent discussions. Let Ξ(Y ) denote the space of oriented 2-planefields 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-planefield we get a bijection Ξ(Y ) → V ect(Y ).

Definition 6.2.1. Two nowhere vanishing vector fields v1 and v2 are saidto be homologous if v1 is homotopic to v2 outside a disk D3 ⊂ Y (throughnowhere vanishing vector fields).

This equivalence relation — together with the identification given above —induces an equivalence relation on Ξ(Y ) and hence on π0

(Ξ(Y )

). For the

proof of the following statement see [169].

Proposition 6.2.2 (Turaev, [169]). Spinc(Y ) can be identified with theset of equivalence classes of elements of π0

(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 anatural lift to U(2).

6.2. Spinc structures and oriented 2-plane fields 103

(b) Verify that a C2-bundle W → Y admits a nowhere vanishing section.Show that a spinc structure (W,ρ) induces a nowhere vanishing section ofTY → 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) wheret ∈ Spinc(Y ) and φ ∈ Γ(W ) is of unit length.

An oriented 2-plane field, or more specifically a contact structure ξnaturally induces a spinc structure which will be denoted by tξ. Letp : π0

(Ξ(Y )

)→ Spinc(Y ) denote the map associating tξ to ξ. In the rest of

this section we briefly recall the classification of oriented 2-plane fields (upto 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 → S2 can beassociated to ξ ∈ Ξ(Y ). In particular, on Y = S3 the oriented 2-plane fieldsare in one-to-one correspondence with elements of [S3, S2] = π3(S

2) ∼= Z.Using the Pontrjagin-Thom construction, the space [Y, S2] can be identifiedwith the framed cobordism classes of framed 1-manifolds in Y . Homotopiesoutside of a disk (i.e., spinc structures) can be parameterized by the 1-manifolds in Y up to cobordism, i.e., with H1(Y ; Z) ∼= H2(Y ; Z). A fiberp−1(t) for a spinc structure t admits an [S3, S2] ∼= Z-action: for any n we can“twist” the given framing of the framed link corresponding to the oriented2-plane field by n. Viewing this action from another point of view, oriented2-plane fields (or, equivalently, the orthogonal vector fields correspondingto them) inducing a fixed spinc structure t ∈ Spinc(Y ) can be assumed tobe identical outside of a disk D3 ⊂ Y . Then Z acts on p−1(t) by connectsumming a given (Y, v) (where [v] ∈ p−1(t)) with the elements of

(S3, w) |w is a nowhere zero vector field on S3

.

By pulling back the generator of H2(S2; Z) with the map fξ : Y → S2

associated to ξ ∈ Ξ(Y ) we get a second cohomology class Γξ ∈ H2(Y ; Z).This class will depend on the chosen trivialization of TY , but for ξ1, ξ2 ∈Ξ(Y ) the difference Γξ1 − Γξ2 is independent of this choice, since it can beidentified with the obstruction of fξ1 being homotopic to fξ2 on Y − D3.This observation again shows the existence of a natural free and transitiveH2(Y ; Z)-action on Spinc(Y ). It is not hard to see that c1(ξ) ∈ H2(Y ; Z)


(where we regard ξ as an oriented R2-, hence a complex line bundle) is equalto 2Γξ: by definition Γξ is the pull-back of [S2] = PD[point] ∈ H2(S2; Z)while ξ is the pull-back of the tangent bundle TS2, hence

c1(ξ) = f∗ξ(c1(TS

2))

= f∗ξ (PD[two points]).

Consequently, if H2(Y ; Z) ∼= H1(Y ; Z) has no 2-torsion, then c1(ξ) deter-mines the spinc structure tξ induced by ξ. Notice that c1(ξ) = c1(tξ) forthe induced spinc structure tξ, since a second cohomology class uniquely ex-tends from Y −D3 to Y on a 3-dimensional manifold. Recall that Z admitsa transitive action on p−1(t) for any spinc structure t. In the statementbelow, Z0 is understood to be equal to Z.

Proposition 6.2.5 (Gompf, [64]). Let t ∈ Spinc(Y ) be a given spinc

structure. The fiber p−1(t) can be identified with Zd(t) where d(t) denotesthe divisibility of c1(t), and is zero if c1(tξ) is a torsion class.

In conclusion, the homotopy type of a 2-plane field ξ is uniquely specifiedby the induced spinc structure tξ and the framing of the corresponding 1-manifold in Y . This latter invariant is an element of Zd(t) in general, and itis hard to work with, except in the case of torsion first Chern class c1(tξ).In this case the set of framings (an affine set for Z) can be lifted to a subsetof Q as follows: Suppose that ξ ∈ Ξ(Y ) has torsion first Chern class c1(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 complextangencies along ∂X, i.e., to TY ∩ JTY .

Lemma 6.2.6 (Gompf, [64]). The expression

d3(ξ) =1

4

(c21(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 aclosed almost-complex 4-manifold (X,J) we have c21(X,J) = 3σ(X) +2χ(X).

Exercises 6.2.8. (a) Verify that if c1(ξ) is torsion then c21(X,J) ∈ Q iswell-defined. (Hint: Cf. Remark 3.1.11.)

(b) Show thatd3(ξ) | ξ ∈ Ξ(S3)

⊂ Z + 1

2 . (Hint: Use the fact that fora 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. Spinc 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 ishomotopic to the oriented 2-plane field of complex tangencies along ∂X.The rational number d3(ξ) defined for those ξ ∈ Ξ(Y ) which have torsionc1(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 fieldsξ1, ξ2 induce the same spinc structure t and that c1(t) is torsion. Then[ξ1] = [ξ2] if and only if d3(ξ1) = d3(ξ2).

Later we will show explicit computations for d3(ξ) of some contact struc-tures ξ.

6.3. Spinc structures and almost-complex structures

Next we consider the geometric interpretation of spinc structures on 4-manifolds. Recall that Spin(4) = SU(2) × SU(2) and so

Spinc(4) = S1 × SU(2) × SU(2)/± (1, id, id);

alternatively, Spinc(4) =

(A,B) ∈ U(2) × U(2) | detA = detB. The

isomorphism between the above groups is given by the map

(A,B) 7→(α,A

(α−1 00 α−1

), B

(α−1 00 α−1

))

with α2 = detA = detB. Spin and spinc structures in dimension 4 can alsobe defined as follows. First let V be a 4-dimensional oriented Euclideanvector space. A spin structure on V is a pair (V +, V −) of 1-dimensionalquaternionic vector spaces with hermitian metrics together with an isomor-phism γ : V → HomH(V +, V −) compatible with the metrics. (Note thatthe 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± → Xare quaternionic line bundles with hermitian metrics (i.e., SU(2)-bundles)and ρ : TX → HomH(S+, S−) is a bundle isomorphism compatible with thechosen metrics. Using the cocycle structures of the bundles S± → X it isfairly easy to see that this definition coincides with the general one given inSection 6.1.


A spinc structure on a vector space V is a pair (V +, V −) of com-plex planes with hermitian metrics such that detC V

+ ∼= detC V− to-

gether with an isomorphism γ : V ⊗ C → HomC(V +, V −) which satisfiesγ(v)∗γ(v) = −|v|2 idV + . (It is not hard to verify that the symmetry groupof (V ±, γ) is isomorphic to Spinc(4).) Once again, by globalizing the aboveconstruction, we can define a spinc structure on X by a triple (W+,W−, ρ)where W± → X are hermitian C2-bundles with detW+ ∼= detW− andρ : TX ⊗ C → HomC(W+,W−) is a bundle isomorphism which satisfiesρ(v)∗ρ(v) = −|v|2 idW+. The proof of equivalence is again an easy exercise.As a simple homological argument shows (see e.g. [66]) the set of spinc

structures on an oriented 4-manifold X is always nonempty, and hence itis (noncanonically) isomorphic to H2(X; Z). An almost-complex structureJ naturally induces a spinc structure sJ : the almost-complex structure re-duces the structure group of TX from SO(4) to U(2), and the map

A 7→((

detA 00 1

), A

)∈ U(2) × U(2)

provides the desired lift from U(2) to Spinc(4). Alternatively, since J givesrise to the bundles Λp,qJ (X), we can take W+ to be equal to Λ0,0

J (X)⊕Λ0,2J (X)

and W− = Λ0,1J (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 inducedspinc structure sJ ∈ Spinc(X) since the above construction provides a spinc

structure on X − x1, . . . , xn where J is defined and (since both S3 andD4 admit unique spinc structures) it extends uniquely to X. It can beshown that J1, J2 induces the same spinc structure if J1 is homotopic to J2

outside of a 1-dimensional submanifold (containing all points where Ji areundefined). Hence

Proposition 6.3.1. The set of spinc structures Spinc(X) on X can beidentified with

J almost-complex structure on X − x1, . . . , xn

for some x1, . . . , xn ∈ X/ ∼,

where ∼ is the equivalence relation described above.


If X is a compact 4-manifold with J as above, then the complex tangenciesalong ∂X provide an oriented 2-plane field, giving a geometric interpretationof the restriction map Spinc(X) → Spinc(∂X) of spinc structures.

Exercises 6.3.2. (a) Show that s = (W±, γ) ∈ Spinc(X) is induced by analmost-complex structure if and only if c2(W

+) = 0.

(b) Verify that for a closed 4-manifold X the identity

c2(W+) =

1

4

(c21(W

+) − 3σ(X) − 2χ(X))

holds.

If the second cohomology group contains torsion elements (for example, if Yis a rational homology 3-sphere) it is quite complicated to work with spinc

structures directly. In such cases c1 might not determine the spinc structure,and we cannot work with torsion second cohomology classes through theirvalues on embedded surfaces. The underlying smooth 3-manifold can alwaysbe presented as the oriented boundary of a smooth 4-manifold built from a0-handle and some 2-handles only. Studying spinc structures on Y throughtheir extensions to simply connected 4-manifolds (i.e., to manifolds where wedo not have torsion (co)homologies) turns out to be very useful in numeroussituations.

Exercise 6.3.3. Show that ifX is simply connected then the restriction mapSpinc(X) → Spinc(∂X) is onto. (Hint: Apply the long exact cohomologysequence of the pair (X,∂X) and use the fact that H1(X; Z) = 0.)

Therefore, instead of studying t ∈ Spinc(∂X) we can focus on some s ∈Spinc(X) with s|∂X = t. Since π1(X) = 1, the spinc structure s ∈ Spinc(X)is uniquely determined by c1(s), and this class is specified by its values onthe second homologies of X. So suppose that X is a compact 4-manifoldwith boundary, given by a Kirby diagram involving a unique 0-handle andt 2-handles, attached along the knots Ki (i = 1, . . . , t) with framings ni(i = 1, . . . , t). The corresponding basis of H2(X; Z) is denoted by α1, . . . , αt.Suppose furthermore that J is an almost-complex structure on X with firstChern class c1(X,J) satisfying

⟨c1(X,J), αi

⟩= mi

for i = 1, . . . , t. Recall that c1(X,J) is a characteristic cohomology element,that is, ⟨

c1(X,J), αi⟩≡ QX(αi, αi) (mod 2).


Denote the induced oriented 2-plane field of complex tangencies on Y = ∂Xby ξ. Using the notation of Section 2.3 (see text preceding Exercises 2.3.4)we get that the Poincare dual of c1(X,J) is equal to

∑ti=1mi[Di]. This

element maps to∑t

i=1miµi ∈ H1(Y ; Z). From the relations among thehomology classes µi we get a presentation of H1(Y ; Z), and we can easilyidentify c1(ξ) and decide whether it is torsion or not. If c1(ξ) is torsion thenfor some n ∈ N the class nc1(ξ) = 0, implying that nPD

(c1(X,J)

)maps

to zero under the map

H2(X,Y ; Z)ϕ2−→ H1(Y ; Z),

hence it is in the image of ϕ1 : H2(X; Z) → H2(X,Y ; Z). Since ϕ1 isexplicitly described in Section 2.3, it is a simple matter of solving a linearsystem of equations to find c ∈ H2(X; Z) with the property that ϕ1(c) =nPD

(c1(X,J)

). The linking matrix of the Kirby diagram defining X

enables us to determine c·c = QX(c, c), leading to a computation of c21(X,J)since this latter term is equal to 1

n2 c · c ∈ 1nZ. Having this quantity at hand

now it is an easy exercise to determine d3(ξ) since the linking matrix of theKirby diagram provides χ(X) and σ(X). Of course, in general it is ratherhard to find an appropriate (X,J) for a given (Y, ξ). As we will see, for acontact 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. Supposethat (Y, t) is a given spinc 3-manifold and the cobordism is defined byattaching a (4-dimensional) 2-handle along K ⊂ Y . Fix a 4-manifold Xwith ∂X = Y which admits a handle decomposition with 0- and 2-handlesonly, and let s ∈ Spinc(X) be chosen in such a way that s|∂X = t. Notethat since X is simply connected, the spinc structure s is determined bythe values of c1(s) on a generating system of the second homology groupH2(X; Z). A spinc structure s1 on X ∪W extending s ∈ Spinc(X), thatis, a spinc cobordism (W, s1) from (Y, t) can be specified now by the valuem of c1(s1) on the 2-homology defined by the 2-handle giving rise to W .Since H2(W,∂W ; Z) ∼= Z〈g〉 with g a generator, the value

⟨c1(s1), g

⟩= n

specifies the extension s1 of t. The computation of the self-intersection of gand so of c21(s1) follows the same line as it is discussed in Section 2.3.

Exercises 6.3.4. (a) Suppose that [K] = 0 in H1(Y ; Z). Determine thenumber of possible extensions of a given spinc structure t to the cobor-dism given by the handle attachment along the knot K ⊂ Y with surgerycoefficient being equal to 0 (with respect to the Seifert framing).


(b) Suppose that (Y, t) is a spin 3-manifold. Show that s1 ∈ Spinc(W ) is aspin extension if and only if, with the above notations,

⟨c1(s1), g

⟩= 0.

(c) Find an example of a spinc cobordism (W, s) such that⟨c1(s1), g

⟩= 0

but s1 ∈ Spinc(W ) is not spin. (Hint: Start with a nonspin structuret ∈ Spinc(Y ) and extend it.)

(d) Consider the cobordismW given by Figure 2.12. Let the spinc structures ∈ Spinc(W ) satisfy

⟨c1(s), g

⟩= n. Determine c21(s) and 1

4

(c21(s) −

3σ(W ) − 2χ(W ))∈ Q.

7. Symplectic surgery

After these preparatory chapters now we are ready to describe the surgeryscheme in the symplectic category. First we will deal with the general cut-and-paste operation and then examine the handle attachment procedure indetail. The chapter concludes with the description of a version of surgerywhich 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 asymplectic dilation or Liouville vector field if Lvω = ω. Notice that sincedω = 0 we have that Lvω = dιvω + ιvdω = dιvω, therefore the aboveequation translates to dιvω = ω. A codimension-1 submanifold Y ⊂ (X,ω)is of contact type if there is a vector field v defined on some neighborhoodνY of Y which is a symplectic dilation and is transverse to Y .

Remark 7.1.2. Notice the similarity with the definition of convex surfacesin contact manifolds. The important difference is that now a symplecticdilation v has a direction: −v is not a dilation anymore, since L−vω = −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 outof (into) U .

Let LY = 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 LY is a line field on Y . Considerthe 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 LY is “equal to” the vector field vH , where vH isspecified by the equation dH = ιvH

ω — that is, the vector field vH is in theline field LY . Recall that since ω is nondegenerate, the formula dH = ιvH

ωuniquely determines vH . Therefore the fact that vH ∈ LY easily followsfrom the fact that for any u ∈ TY we have ω(u, vH) = dH(u) = 0 since Hdoes 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∗ωand α is nonzero on LY .

Proof. We prove the theorem in the special case when Y = H−1(a) forsome function H : X → R and regular value a. Suppose that Y is of contacttype with vector field v. Consider α′ = ιvω = ω(., v) and take α = i∗α′.Now ω = Lvω = dιvω = dα′ implies i∗ω = dα. In order to evaluate α onLY notice that α(vH) = (ιvω)(vH) = ω(vH , v) = −dH(v) 6= 0 since v istransverse to Y = H−1(a). Since vH spans LY , the second property followsfor α. For the converse direction extend the given α to α′ defined on theneighborhood νY in such a way that dα′ = ω. (This can be done since νYretracts to Y .) If v is defined by the equation ιvω = α′ we get that Lvω =dιvω = dα′ = ω, moreover −dH(v) = ω(vH , v) = ιvω(vH) = α′(vH) 6= 0implies 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 His not very restrictive. For a codimension-1 submanifold Y ⊂ X we canalways find H such that Y ⊂ H−1(a) for some regular value a — and thisdescription is enough for our purposes. Equality cannot always be achieved,since the complement of Y might be connected, preventing the existence ofan appropriate H — this is the case, for example, if X = T 2 and Y is ahomologically essential circle on it.

Proposition 7.1.6. If Y ⊂ (X,ω) is of contact type then the 1-form αprovided by Theorem 7.1.4 is a contact form on Y .

Proof. We need to examine dα on kerα. Notice that dα = i∗ω andkerα ∼= TY/LY , on which i∗ω is obviously a symplectic form, proving thatα is a contact form. (For kerα ∼= TY/LY notice that kerα∩LY = 0, hencethe map sending u ∈ kerα ⊂ TY to [u] ∈ TY/LY is injective, and so anisomorphism by dimension reasons.) Notice that the contact structure ξ onY induced by α is cooriented by α.

7.1. Symplectic cut-and-paste 113

Remark 7.1.7. Alternatively, using α = ιvω we can compute α ∧ dα =ιvω ∧ d(ιvω) = ιvω ∧ Lvω = ιvω ∧ ω = 1

2 ιv(ω ∧ ω), so α ∧ dα is nowherezero on a hypersurface Y transverse to v, therefore ξ = kerα is a contactstructure on Y (after using the appropriate orientation).

Informally, the Liouville vector field v in the definition of a hypersurface ofcontact type helps us to determine the symplectic structure near Y . Thismeans that if we “know” ω on Y (through, for example, the induced contactform α) then we “know” ω near Y . To make this picture more rigorous, weprove the following

Proposition 7.1.8. Suppose that Y ⊂ (X,ω) is a hypersurface of contacttype (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 = ιvω

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 neighborhoodsνY ⊂ 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 nowthe diffeomorphism can be isotoped to a symplectomorphism.

Now we are in the position to prove the theorem which allows us toperform symplectic cut-and-paste.

Theorem 7.1.9 ([38]). Suppose that Ui ⊂ Xi are codimension-0 submani-folds and ωi symplectic forms on Xi (i = 1, 2) such that Ui are ωi-convex andthe boundaries Yi = ∂Ui with the induced contact structures are contacto-morphic. Then the surgered manifold (X1 − U1) ∪ U2 admits a symplecticstructure.

Proof. Consider the contact forms αi = ιviωi (i = 1, 2) and take the

symplectization Symp(Y, ξ) with contact structure ξ = kerα1 induced byα1. Now α1(Y ) = 1 × Y ⊂ Symp(Y, ξ) = (0,∞) × Y . For the con-tactomorphism Ψ: (Y, kerα1) → (Y, kerα2) there is a function f : Y → R

such that Ψ∗α2 = fα1; the graph of f : Y → R in Symp(Y, ξ) will be de-noted by α2(Y ) ⊂ Symp(Y, ξ = kerα1). Fix neighborhoods Ni ⊂ Xi andN ′i ⊂ Symp(Y, ξ) of Y and αi(Y ), respectively, which are pairwise sym-

plectomorphic (such neighborhoods are provided by Proposition 7.1.8). By


rescaling ω2 we can achieve that f < 1, hence N ′1 and N ′

2 can be chosen tobe disjoint. Considering V ⊂ Symp(Y, ξ) bounded by N ′

1 and N ′2 we can

form [X1 − (U1 −N1)]∪ V ∪ (U2 ∪N2). (Notice that topologically V is triv-ial, it serves as an interpolation between the symplectic structures on thetwo pieces.) By applying the above symplectomorphisms on the overlappingregions we can glue the symplectic forms together, producing a symplecticstructure on the smooth 4-manifold (X − U1) ∪ U2.

As a special case of the above construction we outline a proof of atheorem of Gompf:

Theorem 7.1.10 (Gompf, [64]). Suppose that for i = 1, 2 the closed sym-plectic 4-manifolds (Xi, ωi) contain closed symplectic 2-dimensional sub-manifolds Σi ⊂ Xi satisfying g(Σ1) = g(Σ2) and [Σ1]

2 + [Σ2]2 = 0. Fix

an identification f : Σ1 → Σ2 and consider an orientation reversing liftF : ∂νΣ1 → ∂νΣ2 of f . Then the normal connected sum X1#FX2 =(X1 − νΣ1) ∪F (X2 − νΣ2) admits a symplectic structure.

Proof (sketch). We assume first that [Σ1]2 < 0. In that case Σ1 admits

an ω1-convex, and Σ2 (with [Σ2]2 > 0) an ω2-concave neighborhood — as

their local models show. Then all we need to do is to show that the contactstructures on the boundaries are contactomorphic. Let α1 be the contactform on ∂νΣ1 and α2 the pull-back of the contact form of ∂νΣ2 by F . Letαt = tα1 + (1− t)α2

(t ∈ [0, 1]

)be a path connecting them. Since dα1 and

dα2 are both positive multiples of π∗ω1|Σ1for π : ∂νΣ1 → Σ1, we conclude

that the αt are all contact forms on ∂νΣ1: notice that dαt = t dα1 + (1 −t) dα2 is also a positive multiple of π∗ω|Σ1

, and kerαt is always transverseto the fibers of ∂νΣ1 → Σ1, hence dαt is nondegenerate on kerαt. Thisshows that α1 and α2 are isotopic, therefore Gray’s Theorem 4.1.16 showsthat they are contactomorphic, hence the previous construction proves thetheorem. If [Σ1]

2 = [Σ2]2 = 0 then just use a function which fiberwise turns

the punctured unit disk in R2 symplectically inside out.

Remark 7.1.11. The original proof of Gompf for the above theorem restson the symplectic neighborhood theorem, for details see [64]. Notice thatthis construction — applied for CP1 ⊂ CP2 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 fieldexists). 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 aStein manifold), and contactomorphism can be proved by relying on someform 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) ofthe above gluing procedure — when we glue a 4-dimensional 2-handle to asymplectic 4-manifold (X,ω) with ω-convex boundary. For a more generaldiscussion of gluing handles see [173]. Let us take the standard 2-handle Has the closure of the component of

R4 −

x21 + x2

2 −1

2(y2

1 + y22) = −1

∪x2

1 + x22 −

ε

6(y2

1 + y22) =

ε

2

which contains the origin, see the shaded region in Figure 7.1, cf. [38]. Itinherits the symplectic structure ω0 = dx1∧dy1+dx2∧dy2 from the standardstructure on R4. Consider the vector field

v = 2x1∂

∂x1− y1

∂

∂y1+ 2x2

∂

∂x2− y2

∂

∂y2.

Exercises 7.2.1. (a) Show that v is equal to ∇f for the function f =x2

1 + x22 − 1

2y21 − 1

2y22 with respect to the standard Euclidean metric.

(b) Check that α = ιv(ω0) = 2x1 dy1 + y1 dx1 + 2x2 dy2 + y2 dx2.

(c) Show that Lvω0 = ω0 and that v is transverse to the boundary ofthe standard 2-handle. (Hint: Use the fact that Lvω0 = dιvω0. Fortransversality compute df(v) and show that it is equal to 4x2

1+4x22+y2

1+y22.)

(d) Show that the attaching circle S = x1 = x2 = 0, y21 + y2

2 = 2 ⊂ ∂H isLegendrian with respect to the contact structure ξ = kerα generated by von the boundary of the 2-handle. Similarly, the belt circle B = y1 = y2 =

0, x21 + x2

2 = ε2 ⊂ ∂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 isopposite to the orientation ∂H inherits from H, while the two orientationscoincide near B.

Now applying the previous construction we get


, yy1 2

, xx1 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 tothe induced contact structure). Then a 2-handle H can be attached to Xalong L in such a way that ω extends to X ∪ H as ω′ and ∂(X ∪ H) isω′-convex.

Proof. According to the Legendrian neighborhood theorem, L ⊂ ∂X andS ⊂ ∂H admit contactomorphic neighborhoods. Choose ε in the definitionof H so small that the attaching region of H becomes a subset of thisneighborhood. The contactomorphism between the neighborhoods of L ⊂∂X and S ⊂ ∂H will provide a suitable gluing map. Notice that since thismap is dictated by Theorem 4.1.15, the framing of the handle attachmentis also given. The last statement follows from patching the vector fieldstogether.

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 #mS

1×D3 admits a symplectic structure with ω-convex boundary

7.2. Weinstein handles 117

— just repeat the handle attachment for m 1-handles starting with thestandard 4-disk

(D4, ωst|D4

)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 weneed to identify the framing of the 2-handle H we have to use in the aboveconstruction. Recall that L admits a canonical framing (as a Legendrianknot), hence we need to understand the framing of the gluing relative tothis canonical one. We think of a framing as a vector field in the tangentspace along the knot transverse to its tangent vector field. Therefore weneed to identify two vector fields along L: v1 is the vector field in ξ which istransverse to the tangent of L (providing the contact framing), while v2 isthe image of the direction we push-off the attaching circle when measuringframings. Notice that for this computation we can work in the standardhandle H: the vector field v2 is defined as an image of a vector fieldin H, while v1 is the image of the corresponding vector field along theLegendrian knot S ⊂ H, since the gluing map is a contactomorphism.For this computation, fix a parametrization of the attaching circle S =x1 = x2 = 0, y2

1 + y22 = 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). Restrictingthe contact form α to the tangent to S we get

√2 cos t dx1 +

√2 sin t dx2,

therefore the contact framing (i.e., the vector field along S(t) which isorthogonal to s′(t) and is in the kernel of α) can be represented by thevector field V (t) = (sin t,− cos t, 0, 0). Now the framing of the gluing ismeasured by pushing off S in the (x1, x2)-direction in the handle. Thecorresponding vector field can be chosen, for example as (1, 0, 0, 0). Sincethe two unit length vector fields intersect each other once, the differenceof the two framings can be clearly represented by a meridian of the knot.Taking the orientations into account, we see that the contact framing makesone positive full turn around the origin, therefore the framing we get bypushing the knot slightly in the (x1, x2)-direction is (−1) when comparedto the contact framing. In conclusion

Theorem 7.2.4 (Weinstein). Suppose that (X,ω), ∂X and L are as inProposition 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 ωextends to X ∪H as in Proposition 7.2.2.

Corollary 7.2.5. Let L ⊂ (S3, ξst) be a given Legendrian link. Then L

equips the 4-manifold X defined by handle attachment along the smoothlink underlying L (with framings tb(Li)− 1) with a symplectic structure ωsuch that ∂X is ω-convex.


Proof. Apply the above theorem for D4 with the symplectic structureit inherits from (R4, ω0) and for the outward pointing radial vector fieldv = x ∂

∂x + y ∂∂y + z ∂

∂z + t ∂∂t .

Exercises 7.2.6. (a) Let the 3-manifold Y be given as 0-surgery on theright-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 andcompare framings.)

(b) Find a symplectic 4-manifold with ω-convex boundary diffeomorphic tothe 3-manifold given by the surgery diagram of Figure 7.2. (Hint: Convertthe 0-framed circle into a 1-handle, see Figure 7.3.)

04

Figure 7.2. Stein fillable 3-manifold

0

4

44

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 boundaryexists. In the gluing construction discussed above, however, we only needthe existence of this vectorfield near the Legendrian knot L ⊂ ∂X. As ittruns out, the necessary vector field exists near the given knot under muchweaker assumptions: it exists if (X,ω) is only a weak filling of the contact3-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 thevector field on the handle points in the wrong direction. Therefore the re-sulting 4-manifold carries no natural symplectic structure. Notice also thatthe framing coefficient of this latter operation is +1 with respect to thecontact framing of the knot in ∂X. This operation will have interesting in-terpretation in the realm of contact surgery, see Chapter 11. Viewing thislatter construction from another point of view, we see that a symplectic2-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 lyingin 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 thehandle along ω-convex boundary with framing (+1) (relative to the contactframing).

Lemma 7.3.1. Take a Legendrian curve L ⊂ (∂X, ξ) and push it off alongits 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) andanother 2-handle along L′ with framing (+1) (again, measured with respectto the contact framing) then the resulting 4-manifold will have boundarydiffeomorphic to ∂X.

Proof. This fact is quite obvious from the smooth point of view, since bysliding L′ over L we will get a 0-framed circle which is just the boundary ofa 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 geta cancelling pair of handles. This means that we can erase them withoutchanging the 4-manifold. Since the surgery along the sphere does not changethe boundary, we conclude that the boundary after the handle attachmentsis diffeomorphic to ∂X.

As we will see, the new 3-manifold we get after the handle attachmentwith framing (+1) carries a natural contact structure. In Section 11.2 wewill sharpen the above lemma to prove contactomorphism for the resultingstructure after a (−1)- and a (+1)-surgery on L and its contact push offL′. In another context we will see that although the symplectic structuredoes not extend through the handle when glued along the belt circle B, an


appropriate almost-complex structure does extend to H −pt., providing,for example, a spinc structure on the manifold X ∪H. We will discuss thisaspect of the gluing theorem in the next section.

8. Stein manifolds

In this chapter we interpret the Weinstein handle attachment in the Steincategory, leading us to Eliashberg’s celebrated theorem. To put this resultin 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)

∣∣ ≤ supy∈K

∣∣f(y)∣∣ for all f holomorphic on X

.

The manifold X is holomorphically convex if for all K ⊂ X compact theholomorphic convex hull K is compact. This property is equivalent to therequirement that for any infinite discrete set D ⊂ X there is a holomorphicfunction f : X → C which is unbounded on D. Yet another equivalentproperty for a domain Ω ⊂ Cn is the existence of a holomorphic functionf : 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 6= y ∈ X there is a holomorphic functionf : X → C such that f(x) 6= f(y) and for every x ∈ X there are holomor-phic functions f1, . . . , fn and a neighborhood U of x such that zi = fi|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, Cn and every closed analytic submanifold of Cn is Stein. Thenext 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-dimensionalStein manifold biholomorphically and properly embeds into Cq.

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 surfaceif it admits a proper biholomorphic embedding S → C4. That is, S is asmooth affine 2-dimensional complex analytic submanifold in C4.

Another, technically more involved equivalent way of defining Stein man-ifolds is to require the vanishing of the sheaf cohomology groups Hq(X,S)for q ≥ 1 and any coherent sheaf S (see [70], for example). For our purposesyet another, more topological definition will be the most suitable. First afew related definitions are in place:

Definition 8.1.5. A smooth function ϕ : X → R on a complex manifoldX is (strictly) plurisubharmonic if ϕ is (strictly) subharmonic on everyholomorphic curve C ⊂ X. Recall that ϕ is subharmonic if for r smallenough ϕ(x0) ≤ 1

2πr

∫B(x0,r)

ϕ(x) dx; or alternatively ∆ϕ ≥ 0 for the Laplace

operator ∆. A function ϕ : X → R is an exhausting function ifx ∈ X |

ϕ(x) < c

is relatively compact in X for all c ∈ R. Recall that a mapϕ : 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 zizi is a plurisubharmonicexhausting function on Cn. Moreover, if f is holomorphic (and not identi-cally 0 on any component) then log |f | is plurisubharmonic; e.g. log |z|2 isplurisubharmonic.

Let now Y ⊂ X be a codimension-1 submanifold of X. The complextangencies along Y form a complex hyperplane distribution in TY , whichcan be (at least locally) given as kerα for some 1-form α. The Levi formLY (x, y) is defined as LY (x, y) = dα(x, Jy) (where J is multiplication by√−1). Taking the orientation of Y into account, LY 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 LY (x, y) =∑n

i,j=1∂2ϕ∂zi∂zj

xiyj .

Definition 8.1.6. The hypersurface Y ⊂ X is strictly pseudoconvex (orJ-convex ) if its Levi form is positive definite.

In this case the complex hyperplane distribution can be proved to give acontact structure, since the definition requires dα to be nondegenerate onkerα.

Remark 8.1.7. It can be proved that if Y is strictly pseudoconvex then itcannot be touched by a holomorphic curve from inside. More precisely,if Y = ϕ−1(0) is pseudoconvex and Y is oriented as the boundary ofX = ϕ−1

((−∞, 0]

)then any holomorphic curve C ⊂ X with boundary

is transverse to Y , in particular intC ∩ Y = ∅. This property explains“convexity” in the definition.

A function ϕ : X → R turns out to be strictly plurisubharmonic if the

associated Levi form ( ∂2ϕ∂zi∂zj

) 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 = hϕ isplurisubharmonic. A more invariant reformulation of the above fact can begiven as follows. Suppose that a smooth function ϕ : X → R is given on thecomplex manifold X. Consider the associated 2-form ωϕ = −dJ∗dϕ. (HereJ∗ : T ∗X → T ∗X is the dual of J . The operator J∗d is frequently denotedby dC, hence ωϕ = −ddCϕ.) This 2-form gives rise to a symmetric tensorgϕ(x, y) = ωϕ(x, Jy).

Proposition 8.1.8. The smooth function ϕ : X → R on the complexmanifold X is strictly plurisubharmonic if and only if gϕ is a Riemannianmetric. In particular, this property implies that the exact 2-form ωϕ isnondegenerate, hence is an exact symplectic form, while gϕ is a Kahlermetric on X.

Note that if X is complex 2- (hence real 4-) dimensional then the definite-ness of LY on the (real) 2-dimensional distribution on a 3-manifold Y ⊂ Xis in fact equivalent to requiring that the distribution is a contact structure.Therefore in this dimension ϕ : X → R is strictly plurisubharmonic if andonly if the complex tangencies provide a contact structure on the smoothpoints of the level sets ϕ−1(a). The gradient vector field ∇ϕ (with respect tothe Riemannian metric gϕ) is a symplectic dilation since L∇ϕωϕ = d(ι∇ϕωϕ)


and (ι∇ϕωϕ)(v) = ωϕ(∇ϕ, v) = gϕ(∇ϕ, Jv) = −dϕ(Jv) = −J∗dϕ(v), henced(ι∇ϕωϕ) = −dJ∗dϕ = ωϕ. Recall that on Cn there are many plurisubhar-monic functions, and so Stein manifolds (being complex submanifolds of Cn

for some n) admit many plurisubharmonic functions as well. In fact, theconverse of this statement also holds:

Theorem 8.1.9 (Grauert, [69]). If a complex manifold X admits a properplurisubharmonic function ϕ : X → [0,∞) then X is Stein.

For Xn ⊂ Cm the distance function f(z) = |z − p|2 from a generic pointp ∈ Cm defines a proper Morse function f : Xn → [0,∞) with critical pointsof index at most n [114]. In conclusion

Theorem 8.1.10 (Grauert, [69]). The complex surface S is Stein if andonly if it admits a proper Morse function f : S → [0,∞) such that away fromthe critical points f−1(t) is a contact 3-manifold (with complex tangent linesas ξ) for all t.

Remarks 8.1.11. (a) It is not very hard to see that a plurisubharmonicfunction 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 withboundary then for a plurisubharmonic function ϕ the restriction ϕ|C has nolocal maximum in the interior intC. This property is closely related to thealternative reformulation of pseudoconvexity described in Remark 8.1.7.

(b) According to a result of Eliashberg and Gromov, two plurisubharmonicfunctions ϕ and ψ with complete gradient flows define symplectomorphicsymplectic structures (X,ωϕ) and (X,ωψ) on a Stein manifold X.

A compact manifold W with boundary will be called a Stein domain if thereis a Stein manifold X with plurisubharmonic function ϕ : X → [0,∞) suchthat 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 domainif it admits a proper plurisubharmonic function which is constant on theboundary. More generally, a cobordism W (with boundary −Y1 ∪ Y2) isa Stein cobordism if W is a complex cobordism with a plurisubharmonicfunction f : W → R such that f−1(ti) = Yi, t1 < t2.

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 givenin the previous chapter to the case of Stein surfaces. Recall that in thissetting we assume that ∂W is the level set of a plurisubharmonic functionand we consider the contact structure on ∂W provided by the distribution ofcomplex 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 attachinga Weinstein handle H to W along L, the Stein structure can be extendedto W ∪H.

Proof (sketch). The idea of the proof is the following: first we approximatethe Legendrian knot L with a C∞-close real algebraic Legendrian knot. Inthis way the attaching map can be chosen to be complex analytic, providingus a complex structure on W ∪H. Notice that by the framing assumptioncomplex lines will match up under the gluing. Therefore the proof reducesto 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 tothe 2-handle now proceeds by turning it into a standard model and thenextending. For details see [25]; for an explicit description of the shape ofthe 2-handle see [50].

Corollary 8.2.2. A Legendrian link L ⊂ (S3, ξst) determines a Stein mani-fold XL. Topologically XL is given by 2-handle attachments along the linkL with framings tb(Li) − 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 structurealways extends.

(b) The product Y × I of a contact 3-manifold (Y, ξ) and I = [0, 1] can beequipped with a Stein structure. In addition, if (Y, ξ) is overtwisted then theframing condition yields no restriction in the gluing. Therefore a cobordism(involving only 2-handles) on an overtwisted 3-manifold always admits Steinstructure. (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.


The theory of Stein manifolds from the point of view of handle calculuswas carefully developed by Gompf in [65], see also Chapter 11 of [66]. Herewe highlight only one result of [65], which will be important in our laterconsiderations: the identification of the first Chern class of a Stein structuregiven by handle attachments. Suppose that the Stein surface (X,J) is givenby attaching Weinstein 2-handles to (D4, Jst) along the Legendrian linkL = (L1, . . . , Lt) ⊂ (S3, ξst), and suppose that ΣLi

⊂ X denotes the surfacecorresponding to the knot Li (see Section 2.3). The value of c1(X,J) of theresulting complex structure on the homology defined by the knot is givenby the following

Proposition 8.2.4 (Gompf, [65]). The first Chern class of the resultingcomplex structure is given by

⟨c1(X,J), [ΣLi

]⟩

= rot(Li).

Proof. By definition, the value of the first Chern class c1(X,J) is equalto the obstruction of extending a complex trivialization of TD4 to thecomplex 4-manifold X we get after the handle attachment. To determinethis obstruction, we fix trivializations on D4 and on the handle and computethe obstruction for splicing them together. Along the boundary S3 wecan take the vector field ∂

∂x (spanning the standard contact structure ξst,regarded as ker(dz+ x dy) on the finite part of S3), and an inward pointingnormal v — these two vector fields span TD4 over C —, and extend themover the 4-disk. In the handle consider the tangent vector field τ and theoutward pointing normal w along the attaching circle. These two vectorfields extend to a complex trivialization of the tangent bundle of the handlewhere the spanning disk of the attaching circle is viewed as part of iR2 ⊂ C2.Now under the handle attachment we map w to v and τ into ξ. Theobstruction for extending the trivialization given on D4 is now simply therotation number of τ with respect to the chosen vector field ∂

∂x , which is bydefinition the rotation number of the Legendrian knot K ⊂ (S3, ξst).

Remark 8.2.5. Similar statement holds when we glue Stein 1-handles firstto D4 and then add the 2-handles; for details see [65].

Exercises 8.2.6. (a) Equip the handlebodies given by Figure 2.14 withStein structures.

(b) Equip RP3 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 certainembedded surfaces in complex 4-manifolds. Let us fix a complex manifold(X4, J) and an embedded oriented surface Σ2 ⊂ X4. As always, J : TX →TX denotes the almost-complex structure induced by the complex structureof X; all dimensions are understood to be real. The theory presentedbelow resembles to the discussion about Bennequin’s inequality given inSection 4.3.

Definition 8.3.1. A point p ∈ Σ is a complex point if TpΣ = JTpΣ, i.e.,the tangent space of Σ at p is a complex line. The noncomplex points of theembedding are called real points. For a generic embedding complex pointsare isolated.

Saying the above condition in another way, p ∈ Σ is an isolated complexpoint if there are complex coordinates (z1, z2) in X such that Σ locally canbe given by

z2 = f(z1)

with p = (0, 0) and ∂f

∂z (0, 0) = 0. This can beseen by noting that the vectors

X(z1) = (1,∂f

∂x(z1)) and Y (z1) = (1,

∂f

∂y(z1))

(with z1 = x + iy) form a real basis of T(z1,f(z1))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(z1)

1 ∂f∂y (z1)

)= −2i

∂f

∂z(z1) = 0.

Yet another way to see the picture is the following: If Gr2(4) denotesthe Grassmannian of oriented 2-planes in R4 then consider the associatedbundle Gr2(X) = PX ×Gl2(R) Gr2(4) for the principal frame bundle PX →X. By taking the tangent planes of Σ we get a lift of the embeddingΣ → X to F : Σ → Gr2(X). The complex tangent lines define a subsetC Gr(X) ⊂ Gr2(X) and for the projection π : Gr2 → X we have thatπ(C Gr(X)∩F (Σ)

)⊂ Σ is precisely the set of complex tangencies. The fact

that complex tangencies are isolated for a generic embedding now followsfrom a general transversality result of Thom.

Definition 8.3.2. The index Ip ∈ Z of an isolated complex point p ∈ Σ is

defined as the winding number of ∂f∂z around a small circle around p.


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 bewritten either as z2 = zk1z1 with k ≥ 0 or as z2 = z−k+1 with k < 0, and soIp = k. The index of the embedding Σ ⊂ X is defined as I(Σ) =

∑p∈Σ Ip.

Definition 8.3.3. The complex point p ∈ Σ is elliptic if Ip = 1 andhyperbolic if Ip = −1.

In the Grassmannian picture the index Ip can be interpreted as anintersection multiplicity: if π(P ) = p for P ∈ C Gr(X) ∩ F (Σ) ⊂ Gr2(X)then Ip is the multiplicity of the intersection of C Gr(X) with F (Σ) atP . For a generic embedding Σ ⊂ X all complex points are isolated andeither elliptic or hyperbolic. Now the orientation of Σ makes us able toassign a sign to each complex point p: this sign is positive if the complexorientation of TpΣ coincides with the orientation of it inherited from Σ andit is negative otherwise. Said another way, C Gr(X) falls into the disjointunion C Gr (X)+∪C Gr (X)− according to whether the orientation of the 2-plane is the complex one or its opposite. The complex point p ∈ Σ is positiveif p = π(P ) for P ∈ C Gr (X)+∩F (Σ) and negative if P ∈ C Gr (X)−∩F (Σ).We define I+(Σ) (I−(Σ)) as the sum of Ip for all positive (resp. negative)complex points of Σ ⊂ X. Obviously I+(Σ) + I−(Σ) = I(Σ). It turns outthat 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) =

⟨c1(X), [Σ]

⟩. In conclusion

I±(Σ) = 12(χ(Σ) + [Σ]2 ±

⟨c1(X), [Σ]

⟩).

Proof (sketch). By choosing a metric on X fix a projection of TX|Σ tothe normal bundle νΣ. For proving the first identity, choose a vector fieldv in the tangent bundle TΣ. Applying J to it and projecting the result toνΣ we get a section σ of the normal bundle νΣ. The projection providesa zero of σ at p ∈ Σ if and only if Jvp is in the tangent plane TpΣ, whichhappens if and only if either p is a complex point or vp = 0. Checking thesigns of the zeros the first formula follows.

For the second formula consider the section s = ω(v1, v2)−1v1 ∧C v2 of

the complex line bundle Λ2C(TX)

∣∣Σ, where ω is a symplectic form on Σ and

v1, v2 is a local frame for TΣ. Now the zeros of s correspond to thosepoints of Σ where v1 and v2 are not independent over C, i.e., in the complexpoints of Σ. A careful checking of the signs expresses the (signed) sum ofzeros of s as I+(Σ) − I−(Σ). On the other hand the sum of zeros of s isequal to c1(ΛC(TX)

([Σ])

=⟨c1(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] 6= 0 or Σ is not a sphere.

Theorem 8.3.5. If there is an open subset U ⊂ X such that Σ ⊂ U and Uadmits a Stein structure then I±(Σ) ≤ 0.

Proof. By the adjunction inequality of Theorem 1.2.1 we have

[Σ]2 + |⟨c1(U), [Σ]⟩ | ≤ −χ(Σ)

once U is Stein (and Σ is not a null-homologous sphere). Since c1(U) =c1(X), the formulae for I±(Σ) imply the result.

According to a beautiful result of Forstneric, the converse of the abovetheorem also holds:

Theorem 8.3.6 (Forstneric, [49]). If I±(Σ) ≤ 0 for a generic embeddingΣ ⊂ X then there is a Stein domain U ⊂ X containing an isotopic copyof Σ.

Proof (sketch). The proof of this theorem involves two major steps. Firstwe use a cancellation theorem due to Eliashberg and Kharlamov: if p, q ∈ Σare complex points with equal sign and Ip + Iq = 0 then Σ can be isotopedto cancel p and q without changing the other complex points or introducingnew complex tangencies. Therefore the assumption guarantees that we canisotope Σ to have only hyperbolic complex point. Then a local construc-tion near real and hyperbolic points together with a patching argumentprovides a neighborhood U of Σ with a plurisubharmonic function show-ing its Stein property: In a complex chart around a complex hyperbolicpoint pj take the nonnegative function ρj(z1, z2) = |z2 − z2

1|. Now define ρ0

on a tubular neighborhood of Σ as h(v) where we implicitly identified thetubular neighborhood with the normal bundle of Σ and h is the quadraticform of a Riemannian metric on the normal bundle νΣ. For the complexpoints p1, . . . , pm choose χj smooth cut-off functions (j = 1, . . . ,m) sup-ported by the (disjoint) complex coordinate neighborhoods which are con-stant 1 near pj. The function ρ =

∑mj=1 χjρj + (1 − χj)ρ0 can be shown to

be plurisubharmonic, proving the fact that Σ admits a neighborhood withStein structure.

In fact, this argument provides a Stein neighborhood basis for Σ, thatis, Stein neighborhoods Uαα∈(0,1) with Uβ = ∪α<βUα and Uβ = ∩α>βUαand all Uα retract to Σ. It is known that the presence of an elliptic point on


Σ obstructs the existence of such a basis: the existing holomorphic Bishopdisks [27] around the elliptic point (which cannot fit into all Uα) wouldprovide holomorphic extensions of functions. Notice the similarity betweenthese 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 contact3-manifolds. He showed that there is a one-to-one correspondence betweencontact structures (up to isotopy) and open book decompositions (up topositive stabilization/destabilization) on a closed oriented 3-manifold. Thischapter is devoted to the introduction of relevant notions and also someparts 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 → S1 such that the fibers are interiorsof 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 thebinding of the open book decomposition. The monodromy of the fibrationπ is called the monodromy of the open book decomposition.

Any locally trivial bundle with fiber F over an oriented circle is canonicallyisomorphic to the fibration I × F/(1, x) ∼

(0, h(x)

)→ I/∂I ≈ S1 for

some self-diffeomorphism h of F . In fact, the map h is determined bythe fibration up to isotopy and conjugation by an orientation preservingself-diffeomorphism of F . The isotopy class represented by the map h iscalled the monodromy of the fibration. Conversely given a compact orientedsurface F with nonempty boundary and h ∈ ΓF (the mapping class group ofF ) 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) canbe canonically identified with r copies of T 2 = S1 × S1, where the first S1

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 D2 × S1 to F (h) so that ∂D2 isidentified with S1 = I/∂I and the S1 factor in D2 × S1 is identified with aboundary component of ∂F , F (h) can be completed to a closed 3-manifoldY equipped with an open book decomposition. In conclusion, an elementh ∈ ΓF determines a 3-manifold together with an “abstract” open bookdecomposition on it. Notice that by conjugating the monodromy h of anopen book on a 3-manifold Y by an element in ΓF we get an equivalent openbook 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 bookconcretely 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 complementY − L fibers over S1, in such a way that each fiber intersects a tubularneighborhood of L in a curve isotopic to L. It is clear that the binding ofan open book decomposition is a fibered link and conversely a fibered linknaturally induces an open book decomposition with our definition.

Theorem 9.1.3 (Alexander, [9]). Every closed and oriented 3-manifoldadmits an open book decomposition.

Proof (sketch). There are several different proofs of this classical theoremof Alexander. We first outline a proof using branched covers. Every closedand oriented 3-manifold Y is a 3-fold branched cover Y → S3. The proofof this fact rests on the following. Choose a Heegaard decomposition of Yas U1 ∪f U2 (here U1 and U2 are solid genus-g handlebodies and f ∈ Γg is amapping class) and represent U1, U2 as triple branched covers of B3. Gluingthe two B3 together we can realize any mapping class for the gluing of thetwo handlebodies [75, 117], hence the result follows. We can assume thatthe branch locus of this cover is transverse to the pages of an open bookdecomposition of S3. Notice that S3 admits several open book decomposi-tions (see examples below). Thus we get an open book decomposition of Yby lifting an open book decomposition of S3 to the cover.

Next we describe a proof given in [148]. It is well-known that everyclosed and oriented 3-manifold is obtained from S3 by a (±1)-surgery alonga link L. Moreover we can assume that there is an unknot K ⊂ S3 suchthat each component Li of L links K exactly once. Consider the trivial openbook of S3 whose binding is K and whose pages are the spanning disks forK with trivial monodromy (cf. Example 9.1.4(a)). It is clear that when weremove a neighborhood Vi of Li to perform surgery along Li we punctureonce every page of this open book on S3. Observe that the boundary of

9.1. Open book decompositions of 3-manifolds 133

a puncture (which is a meridional curve to Li in S3) becomes longitudinalby a (±1)-surgery along Li. By performing surgery along Li we glue in anannulus (bounded by Li and this longitudinal curve on ∂Vi) to a punctureon each page of the trivial open book of S3 to obtain a page of an openbook on Y . Notice that a page of this open book on Y is planar, i.e., it is adisk with holes since to each puncture we glue in an annulus. The bindingis given by K ∪ L and the monodromy is given by an (appropriate) Dehntwist along a curve parallel to each boundary component Li and identitynear K. (Yet another proof of this theorem using Lefschetz fibrations wasgiven by Harer, see Section 10.2.)

Examples 9.1.4. (a) Taking L to be the z-axis and considering thehalf-planes with boundary L we get an open book decomposition on R3.Alternatively, take π : R3 −

(0, 0, z)

→ S1 ⊂ R2 given by

π(x, y, z) =1√

x2 + y2(x, y).

This open book decomposition extends to the one point compactificationS3 as a genus-0 open book decomposition with binding the unknot andmonodromy equal the identity idD2 of the disk. The resulting open bookdecomposition is called the standard open book decomposition on S3. Thispicture is another manifestation of the fact that S3 is the union of two solidtori, one is the neighborhood of the binding and the other one is the unionof the pages.

(b) Let h be the right-handed Dehn twist along the middle circle S1 ×12

in S1 × [0, 1]. Using h as monodromy, it defines a 3-manifold together withan (abstract) open book decomposition. We denote the corresponding openbook decomposition by ob+. Using methods we will discuss in the nextchapter, we can see that the 3-manifold given by h is the 3-sphere S3: Justconsider the Lefschetz fibration given by h and realize that it can be builtusing a single 1-handle and a 2-handle cancelling it; see Figure 9.1 for aKirby diagram of this 4-manifold. In fact, the binding of the resulting openbook decomposition can be identified with the positive Hopf link and thepages are just the obvious Seifert surfaces. To see this, slide the circlesrepresenting ∂

(S1 × [0, 1]

)over the (−1)-framed 2-handle and cancel the

1-handle/2-handle pair. Then the circles a = S1 × 0 and b = S1 × 1will be linked once in the new S3. This is depicted by the fine lines inFigure 9.1. Taking h−1 corresponds to reversing the orientation on theLefschetz fibration and hence on its boundary S3. Therefore the resulting


−1

aa

b

b

Figure 9.1. The circles a and b after handle cancellation

open book decomposition has the negative Hopf link as its binding — thisopen book decomposition will be denoted by ob−. An alternative way togive ob± is by considering H± = r1r2 = 0 ⊂ S3 equipped with polarcoordinates (r1, θ1, r2, θ2) coming from S3 ⊂ C2 and π±(r1, θ1, r2, θ2) =θ1 ± θ2. The standard open book decomposition becomes L = r2 = 0 andπ(r1, θ1, r2, θ2) = θ2 in these coordinates.

(c) Let p and q be relatively prime integers such that p, q ≥ 2. It iswell-known that a (p, q) torus knot T(p,q) is a fibered knot. This givesan open book decomposition of S3, where the fiber is a surface of genus12(p − 1)(q − 1) with one boundary component and the monodromy is aproduct 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 specialcase of the Murasugi sum) and explain how to construct the fibered surfaceof a torus knot (i.e., a page of the open book of S3 induced by the torusknot) 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 positiveHopf band) in S3 as it is shown in Figure 9.2. Suppose that (L,F ) is anotherfibered link with its fibered surface in an arbitrary 3-manifold Y . Choosean arc α in F connecting two points on the boundary L = ∂F . Take aneighborhood να of α in F and thicken this into a 3-ball. Now apply the


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 takea connected sum of Y with S3 along these 3-balls such that να = α× I andνβ = β × I are glued in a way that α is identified with I ⊂ νβ and β isidentified with I ⊂ να. In fact plumbing a Hopf band is a special case of amore general operation called the Murasugi sum which is defined similarlyfor 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 fiberedlink whose monodromy is the product of the monodromies of these fiberedlinks in the following sense: First extend the monodromies of each fiberedsurface onto the glued up surface (obtained by plumbing) by identity andthen take the product of the resulting diffeomorphisms.

For example, we can plumb two positive Hopf links to get a (2, 3) torusknot (the right-handed trefoil) with its fibered surface. Simply identify aneighborhood of the arc α in one Hopf band with a neighborhood of the arcβ in the other Hopf band, transversely as shown in Figure 9.3. The resultingmonodromy will be the product of two right-handed Dehn twists along thecurves also drawn in Figure 9.3(c). Note that the two curves (one of whichis drawn thicker) intersect each other only once and they stay parallel whenthey go through the left twist on the surface. We can iterate this plumbingoperation to express the monodromy of a (2, q) torus knot as a product of(q − 1) right-handed Dehn twists.


(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 fiberedsurface 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 viewthis figure as two rows of “gates”. First construct the row of gates in theback as described above and then plumb a Hopf band in the front row andproceed as above to obtain a second row of gates. It should be clear that we


can iterate this process to build as many rows of gates as we wish. Hencethe monodromy of the (p, q) torus knot is a product of right-handed Dehntwists.

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 Fis specified by h ∈ ΓF . Attach a 1-handle to the surface F connecting twopoints on ∂F to obtain a new surface F ′. Let α be a closed curve in F ′ goingover the new 1-handle exactly once. Define a new open book decompositionwith h tα ∈ ΓF ′ , where tα denotes the right-handed Dehn twist along α.The resulting open book decomposition is called a positive stabilization ofthe one defined by h. If we use a left-handed Dehn twist instead then we callthe result a negative stabilization. The inverse of the above process is calledpositive (negative) destabilization. Notice that the resulting monodromydepends on the chosen curve α.

We can view the stabilization/destabilization as plumbing/deplumbing Hopfbands. Since plumbing a Hopf band on the 3-manifold level is equal to the


connected sum with S3, by definition we do not change the topology of theunderlying 3-manifold.

There is another technique, called twisting, for constructing new openbook decompositions of 3-manifolds. Suppose that C is a curve embedded ina page F of a given open book decomposition in S3. Twisting is defined asperforming a (±1)-surgery along C with respect to the framing C acquiresby the page. It is easy to see that by this operation we add a Dehn twistalong C to the monodromy of the original open book decomposition. Inparticular if C is unknotted in S3 and the surgery coefficient of C in S3

turns out to be ±1 (with respect to its Seifert framing) then the resultingmanifold is again S3 and hence we get a new open book of S3.

Theorem 9.1.9 (Harer, [73]). Every open book decomposition in S3 isrelated to the trivial one by a sequence of plumbings and twistings.

In fact Harer conjectured that one can entirely omit the twisting operation inthe theorem above. Harer’s conjecture was recently verified by Giroux (andalso independently by Goodman [68]). It was showed that any two openbook decompositions of an arbitrary integral homology sphere are relatedby 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 anda (cooriented) contact structure ξ on Y are called compatible if ξ can berepresented 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 transversebinding induced by α agrees with the boundary orientation of the pages.

This definition of compatibility is natural in the sense that the conditionsthat α > 0 on the binding and dα > 0 on the pages is a strengthening ofthe 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 formon every page is equivalent to the condition that the Reeb vector field ofα is transverse to the pages. (Hint: Recall that the Reeb vector field Rαis determined as the unique direction where dα degenerates, and a volumeform is nondegenerate.)

9.2. Compatible contact structures 139

(b) Show that an open book decomposition and a contact structure arecompatible if and only if the Reeb vector field of α is transverse to thepages (in their interiors) and tangent to the binding.

Intuitively an open book is compatible with a contact structure if we canpush the contact planes arbitrarily close to the tangents of the pages (excepton the binding) of the open book. Recall that the Reeb vector field Rα forα is transverse to the contact planes.

Next we would like to look at the simplest example of a compatibleopen book decomposition and a contact structure. Recall that the standardcontact structure on R3 can be given as the kernel of the form α1 = dz+x dyand the pages of the standard open book decomposition on R3 is given bythe half-planes around the z-axis (see Example 9.1.4). Notice that dα1 isdegenerate when restricted to a page, and therefore clearly is not a volumeform. In fact ∂

∂z is the Reeb vector field for α1 and it is tangent to allthe pages of the open book decomposition. However, if we multiply α1 bythe positive function f(x, y, z) = e−x

2+ e−y

2we get a contact form which

represents the same contact structure and using this form, rather than thestandard one, we show that that the standard contact structure and thestandard open book decomposition on R3 are compatible. It is clear thatthe binding (the z-axis) is transverse to the contact planes and ∂

∂z orients

the binding. Let α = (e−x2+ e−y

2)(dz + x dy). Then

dα = −2xe−x2dx ∧ dz − 2ye−y

2dy ∧ dz +

((1 − 2x2)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 betangent to the pages. So we checked two conditions in the definition but westill have to verify the condition about the orientations. We want to showthat the orientation on the binding induced from a page (which is orientedby 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 = (x2 + y2 + 1)(dz+ x dy) onR3. Show that dβ is a volume from on the pages of the standard open bookdecomposition 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 thestandard tight contact structure ξst on

S3 =

(z1, z2) ∈ C2 | |z1|2 + |z2|2 = 1.


The contact structure ξst can be given by the kernel of the 1-form

α = r21dθ1 + r22dθ2,

where zj = rjeiθj , for j = 1, 2. The simplest open book decomposition of S3

is given by the binding L = r2 = 0 and the fibration π(r1, θ1, r2, θ2) = θ2.Then, for a fixed θ2, π

−1(θ2) is the interior of a disk bounded by the bindingL. This is the trivial open book decomposition of S3 where the binding isan unknot, the pages are disks and the monodromy is the identity. We cansee that the trivial open book decomposition of S3 is compatible with ξstas follows: The tangent to the binding L is given by ∂

∂θ1and the contact

form is dθ1 when restricted to r2 = 0. Therefore the binding is transverseto the contact structure ξst. The contact form restricted to a page is r21dθ1and thus d(r21dθ1) = 2r1 dr1 ∧ dθ1 is a volume form. This also shows thatthe orientation induced on the binding as the boundary of a page coincideswith the orientation induced by the contact form α.

The roots of Giroux’s result invoked at the beginning of this chapter goback 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 compatiblecontact 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 ×D2),

where F is an oriented surface with boundary, h is a self-diffeomorphism of Fpreserving ∂F pointwise and F (h) =

(F × [0, 1]

)/((x, 1) ∼

(h(x), 0

)) is the

relative mapping torus of the element h ∈ ΓF . To simplify notation, belowwe assume that the boundary ∂F has only one component. To find a contactform on Y , we find a contact form on F × [0, 1] first, which descends to thequotient F (h) and then extend it over the solid torus S1×D2 ≃ ∂F×D2. 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 S of 1-forms η satisfying

(1) dη is a volume form on F , and

(2) η = tdθ near ∂F ,


is nonempty and convex. For proving the claim, choose a volume form Ωon F with ∫

FΩ = 1 and Ω|C = dt ∧ dθ.

Let η1 be any 1-form on F which equals tdθ near ∂F . Then by Stokes’Theorem we obtain

∫

F(Ω − dη1) =

∫

FΩ −

∫

Fdη1 = 1 −

∫

∂Fη1 = 1 −

∫

∂Fdθ = 0.

Hence the closed 2-form Ω − dη1 represents the trivial class in cohomologyand vanishes near ∂F . By deRham’s theorem there is a 1-form γ on F with

dγ = Ω − dη1

and γ vanishes near ∂F . Define η2 = η1 + γ. Then dη2 = Ω is a volumeform on F and η2 = tdθ near ∂F , showing that S 6= ∅. Let ϕ1 and ϕ2 betwo 1-forms in S. Then

d(τϕ1 + (1 − τ)ϕ2

)= τdϕ1 + (1 − τ)dϕ2 > 0

on F andτϕ1 + (1 − τ)ϕ2 = tdθ

near ∂F , which shows the convexity of the set S.

Let η be any 1-form in S. Then h∗η also belongs to the set S: dh∗η =h∗dη is a volume form on F and h∗η = η = tdθ near ∂F . By convexity, the1-form

η(x,τ) = τηx + (1 − τ)(h∗η)x

is in S for each τ and descends to the quotient F (h) where x is in the fiberand τ is in the base circle. Thus dη induces a volume form when restrictedto a page of our open book decomposition. Notice that when we glue thetwo 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 forall (x, τ) =

((t, θ), τ

)near ∂F (h) = ∂F × S1. Let dτ be a volume form on

S1. We claim thatα1 = η + κπ∗dτ

is a contact form on F (h) for some sufficiently large constant κ > 0, whereπ denotes the projection of F (h) onto the circle S1 = [0, 1]/ ∼. To provethe claim we fix a point (x, τ) ∈ F (h) and choose an oriented basis u, v,wof T(x,τ)F (h) such that dη(x,τ)(u, v) > 0 and π∗(u) = π∗(v) = 0. This means


that the vectors u and v are tangent to the fiber and w is transverse to thefibration. Thus we get

(α1 ∧ dα1)(x,τ)(u, v,w)

= (η ∧ dη)(x,τ)(u, v,w) + κ(x,τ)

(dτ(π∗w)dη(x,τ)(u, v)

).

Hence we conclude that

(α1 ∧ dα1)(x,τ)(u, v,w) > 0

for κ(x,τ) sufficiently large since dτ(π∗w)dη(x,τ)(u, v) is positive by the choiceof the oriented basis (u, v,w). By compactness of F (h), there is a sufficientlylarge κ > 0 such that α1 ∧ dα1 > 0 on F (h).

Now we would like to identify a collar neighborhood of ∂F (h) with atubular 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 identifycoordinates as (θ, r, φ) ≈ (θ, 2 − t, τ) where 1 ≤ r ≤ 1.5. Then the 1-formα1 defined above is given by

α1 = (2 − r)dθ + κdφ

on ∂F × (D(1.5) − D(1)) since η = tdθ near the boundary and π∗dτ isidentified 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 fromr = 0 but it does not extend across r = 0. Consider the 1-form

α2 = (2 − r2)dθ + r2dφ

instead, which is a contact form near r = 0 since α2 ∧ dα2 = 4rdθ∧ dr∧ dφ.Now we claim that we can “connect” α1 and α2 by contact 1-forms, i.e., wecan find smooth functions

f1, f2 : [0, 1.5] → R

so that the 1-form α = f1(r)dθ + f2(r)dφ is a contact form on ∂F ×D(1.5)where α equals α2 near r = 0 and equals α1 for 1 ≤ r ≤ 1.5. Note that thenecessary and sufficient condition for α to be a positive contact form is thatα ∧ dα > 0 which is equivalent to the condition

f1f′2 − f2f

′1 > 0

(away from r = 0) as the following simple calculation shows:


α ∧ dα = (f1dθ + f2dφ) ∧ (f ′1 dr ∧ dθ + f ′2 dr ∧ dφ)

= (f1f′2 − f2f

′1) dθ ∧ dr ∧ dφ.

To guarantee the condition f1f′2 − f2f

′1 > 0 we choose smooth functions

f1(r) and f2(r) such that

f1(r) =

2 − r2 if 0 ≤ r ≤ 0.52 − r if 1 ≤ r ≤ 1.5

f2(r) =

r2 if 0 ≤ r ≤ 0.5κ if 1 ≤ r ≤ 1.5

satisfying f ′1(r) < 0 (0.5 ≤ r ≤ 1), and f ′2(r) > 0 (0.5 ≤ r ≤ 1). Itis clear that we can find such smooth functions. We claim now that thesmooth 1-form α is compatible with the given open book decomposition.Note that by construction dα = dα1 = dη is a volume form on the fibersof F (h). On the other hand dα = f ′1 dr ∧ dθ + f ′2 dr ∧ dφ is a volume formon φ = constant, 0 ≤ r ≤ 1.5 in ∂F ×D(1.5). Note that the orientationdt∧dθ on the collar of a fiber F in F (h) and the orientation −dr∧dθ on thesurface φ = constant, 0 < r ≤ 1.5 in the solid torus ∂F ×D(1.5) matchup (via the identification t = 2 − r ) to orient the pages of the given openbook decomposition. So we conclude that dα induces a volume form whenrestricted to a page of our open book decomposition. Moreover, the tangentto the binding is given by ∂

∂θ and it is clearly transverse to (2−r2)dθ+r2dφ.Finally notice that the orientation induced on the binding by the volumeform dα agrees with ∂

∂θ .

Exercise 9.2.6. We proved above that the 1-form α1 = η + κπ∗dτ is acontact 1-form for sufficiently large κ. Show that the contact planes kerα1

approach the tangents of the pages in the open book decomposition asκ→ ∞.

Theorem 9.2.5 was substantially refined by Giroux [63]. He proved

Proposition 9.2.7. Any two contact structures compatible with a givenopen book decomposition are isotopic.


Proof. Suppose that ξ0 and ξ1 are two contact structures compatible witha given open book decomposition of a closed oriented 3-manifold Y . Thenthere are contact forms α0 and α1 such that ξi = kerαi, where dαi isa positive volume form on the pages and αi is transverse to the bindingL, for i = 0, 1. Choose coordinates (θ, r, φ) near L in which the bindingand the pages are given by r = 0 and φ = const, respectively. Letα = f(r)dφ, where f(r) is a nondecreasing function which equals 0 forsmall r and which equals 1 for r ≥ r0. Extend α to Y as π∗dτ (which agreeswith dφ in a tubular neighborhood of the binding), where π : Y − L → S1

is the fibration and dτ is a volume form on S1. Notice that in this way wedefine a global 1-form α on Y which vanishes near the binding. Then the1-forms αi,t = αi + tα, t ≥ 0 are all contact. It is easy to see that αi,t is acontact form away from the binding: αi,t∧dαi,t = αi∧dαi+ tπ∗dτ ∧dαi > 0since αi is a contact form and dαi is a volume form on the pages. Moreoverfor t large enough, the forms αs,t = (1 − s)α0,t + sα1,t (0 ≤ s ≤ 1) arealso contact. Again, when we consider αs,t ∧ dαs,t away from the binding,the only terms which are not necessarily positive are s(1 − s)α1 ∧ dα2 ands(1 − s)α2 ∧ dα1. But the rest of the terms are positive and some of themare multiplied with the parameter t. This shows that for large enough t,αs,t is contact for all 0 ≤ s ≤ 1 and hence α0,t and α1,t are isotopic, whichin turn implies that α0 and α1 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 inGoodman’s thesis [68] which is a slight modification of Giroux’s originalconstruction. 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 planestwist 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. Inorder to find such a cell decomposition first cover (Y, ξ) by a finite numberof Darboux balls. Then take a cell decomposition of Y such that each 3-cell


lies in the interior of one of these Darboux balls. Isotope the 1-skeleton tobe 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 thentake a refinement of the cell decomposition by appropriately subdividingD.

Definition 9.2.10. Let G be the 1-skeleton of a contact cell decompositionof (Y, ξ). The ribbon R of G is a (smoothly embedded) surface in Y suchthat the surface R retracts onto G, TpR = ξp for p ∈ G and TpR 6= ξp forp /∈ 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 Ycompatible 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 twoopen book decompositions compatible with a fixed contact structure, ina similar fashion as Proposition 9.2.7 does in the converse direction. Thisresult says that two open book decompositions are compatible with the samecontact structure if and only if they admit common positive stabilization.We will not deal with the proof of this statement in these notes, althoughthe 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 decompositionof Y there is a compatible contact structure ξ on Y . Contact structurescompatible 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 fixedcontact structure admit common positive stabilization.

Remark 9.2.12. When we stabilize a compatible open book obξ on (Y, ξ)we take a connected sum of Y with S3 at the topological level, so that theresulting manifold is diffeomorphic to Y . In the case of positive stabilizationthe resulting open book is obtained by plumbing a positive Hopf bandF+ to a page of obξ. Recall that the contact structure on S3 compatiblewith the open book ob+ (induced by the positive Hopf link H+ ) is thestandard tight contact structure. Hence by Remark 5.2.3, the contactstructure compatible with the resulting open book is a contact connectedsum ξ#ξst, which is isotopic to ξ. In conclusion, positive stabilization doesnot change the (compatible) contact structure. On the other hand, negative


stabilization does change the contact structure since the contact structureon S3 compatible with the open book ob− (induced by the negative Hopflink H− ) is an overtwisted contact structure. Consequently, the contactstructure compatible with an open book obtained by negative stabilizationis necessarily overtwisted.

The next natural question is how to read off contact topological propertiesof 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 tightnessand fillability properties of the contact structure translate to factorizabil-ity of the monodromy of some compatible open book decomposition. Thedifficulty in using such characterizations lies in the fact that the open bookdecomposition is not uniquely defined for a contact structure, rather it isgiven up to a complicated equivalence relation between various mappingclass groups. The relation between properties of mapping classes and thecorresponding contact structures is still not completely understood. By theclassification of overtwisted contact structures one can prove

Corollary 9.2.13 (Giroux). A contact 3-manifold (Y, ξ) is overtwisted ifand only if it admits a compatible open book decomposition which is anegative stabilization of another open book decomposition.

Proof. We already showed in Remark 9.2.12 that the contact structurecompatible with an open book obtained by a negative stabilization is over-twisted. It is not hard to see that negative stabilization of an open bookchanges the oriented 2-plane field induced by the compatible contact struc-ture by adding one to its 3-dimensional invariant. This follows from theconnected sum formula for the 3-dimensional invariant (cf. Chapter 11).The classification of overtwisted contact structures shows that any oriented2-plane field v0 is homotopic to an overtwisted contact structure ξ0 and weknow that there is an open book obξ0 compatible with ξ0. Let (Y, ξ) be anovertwisted contact 3-manifold. Now take the oriented 2-plane field v0 forwhich the contact structure compatible with the negative stabilization ofthe corresponding open book obξ0 is homotopic to the oriented 2-plane fieldinduced by ξ. Consider the open book ob which is obtained by a negativestabilization of obξ0. By construction, the contact structure ξob compatiblewith 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 ξob are isotopic, hence ξ and ob are compatible, proving the result.


Corollary 9.2.14 (Giroux). In an integral homology 3-sphere, any twoopen book decompositions can be related by a sequence of plumbing anddeplumbing positive and negative Hopf bands.

Proof (sketch). We plumb sufficiently many negative Hopf bands to oneof the given open book decompositions so that the 3-dimensional invariantsof the resulting contact structures become equal. (We might have to plumbextra negative Hopf bands to guarantee that both contact structures areovertwisted (cf. Corollary 9.2.13)). Thus these overtwisted contact struc-tures are homotopic and therefore isotopic. Consequently, the resultingopen book decompositions have a common positive stabilization by Theo-rem 9.2.11. Notice that by the assumption it follows that the 3-manifoldsupports a unique spinc structure, hence oriented 2-plane fields are classi-fied by their 3-dimensional invariants d3 ∈ Z up to homotopy.

Another corollary makes use of Legendrian surgery:

Corollary 9.2.15 (Giroux, Matveyev). The contact 3-manifold (Y, ξ) isStein fillable if and only if Y admits an open book decomposition compatiblewith ξ whose monodromy admits a factorization into right-handed Dehntwists only.

A proof of this theorem is given in Theorem 10.3.4. For various notionsof fillability of contact structures see Chapter 12, and for factorizations ofmapping 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. Noticethat the contact structures compatible with a given open book decompo-sition are all tight or all overtwisted by Proposition 9.2.7. Hence we willcall an open book decomposition overtwisted if a contact structure com-patible with this open book decomposition is overtwisted. Let α, β ⊂ F beproperly embedded oriented arcs which intersect transversely on an orientedsurface F . The algebraic intersection number ialg(α, β) is the oriented sumover interior intersections. The geometric intersection number igeom(α, β) isthe unassigned count of interior intersections, minimized over all boundaryfixing isotopies of α and β. The boundary intersection number i∂(α, β) isone-half the oriented sum over intersections at the boundaries of the arcs,after the arcs have been isotoped, fixing boundary, to minimize geometricintersection. See Figure 9.5 for sign conventions. In particular, given anarc α on a page F of an open book decomposition with monodromy h, we


−

α

β

α αββ

+

+

Figure 9.5. Sign convention for intersection numbers on a surface

can consider ialg(α, h(α)

), igeom

(α, h(α)

)and i∂

(α, h(α)

). Here h(α) is

oriented by reversing the orientation on h(α) which is obtained by pushingforward the orientation of α by the monodromy h.

Definition 9.2.16. A properly embedded arc α is sobering for a monodromyh if

ialg(α, h(α)

)+ igeom

(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 forh then the given open book decomposition is overtwisted.

In order to prove this result, Goodman constructs a surface with Legendrianboundary which violates Eliasberg’s inequality given by Theorem 4.3.7. Asan example, we consider the simplest case: the open book decompositionob− of S3 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 9.6is 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 theannulus. (Hint: Observe that i∂

(α, h(α)

)= −1, while ialg

(α, h(α)

)=

igeom(α, h(α)

)= 0.)

In the light of Proposition 9.2.17 this implies that the induced open bookdecomposition is overtwisted. The arc α, however, is not a sobering arc


α

( )αh

Figure 9.6. A sobering arc

for H+. In fact, since the monodromy of the open book decompositionob+ is a right-handed Dehn twist, the compatible contact structure is Steinfillable by Corollary 9.2.15 and therefore it gives the standard tight contactstructure by the classification of contact structures on S3.

Proposition 9.2.19 (Goodman, [68]). If an arc α ⊂ F satisfies

ialg(α, h(α)

)+ igeom

(α, h(α)

)+ i∂

(α, h(α)

)= −1,

then the open book decomposition with page F and monodromy hn isovertwisted for any n > 0.

For an application of Proposition 9.2.19 consider a genus-g surface F withonly one boundary component and let δ be a curve parallel to the boundary.Then the open book decomposition with page F and monodromy t−nδ isovertwisted for n > 0, where tδ denotes the right-handed Dehn twist alongδ. To see this, we first observe that

tδ = (ta0ta1 · · · ta2g−1ta2g )4g+2

where the curves a0, a1, . . . , a2g are depicted in Figure 15.5. Now the arc αshown in Figure 9.7 will be a sobering arc for h = t−1

a2gt−1a2g−1

· · · t−1a1 t

−1a0 , since

ialg(α, h(α)

)+ igeom

(α, h(α)

)+ i∂

(α, h(α)

)= −1.

We depict α and h(α) in Figure 9.7. Then since t−nδ = hn(4g+2), theopen book decomposition with monodromy t−nδ is overtwisted for n > 0by Proposition 9.2.19. On the other hand, the open book decompositionwith page F and monodromy tnδ (n ≥ 0) is Stein fillable and hence tight.


(α)h

2g ααa

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 openbook decomposition which has a sobering arc.

Proof. If the final open book decomposition has a sobering arc then it isovertwisted by Theorem 9.2.17, and positive stabilization/destabilizationdoes not change the contact structure (up to isotopy). Conversely, if thecontact structure is overtwisted then it has a positive stabilization whichis a negative stabilization of some other open book decomposition. On theother hand, negative stabilization can be realized by plumbing a negativeHopf band. Now the solution of Exercise 9.2.18 shows the existence of asobering 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 → D2 is called a simpled-fold cover if there is a finite set Q in the interior of D2, called the branchset and each p ∈ D2 has a neighborhood U over which π : π−1(U) → Ubehaves 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 diskprojecting to U as a double cover branched over p, i.e., can be modeledby the complex map z 7→ z2 around the origin, while the others are disksprojecting diffeomorphically.

9.3. Branched covers and contact structures 151

Definition 9.3.2. A smooth map h : Y → S3 is called a simple d-fold coverwith branch set B ⊂ S3 if it is locally diffeomorphic to the product ofan interval with a simple d-fold cover of a disk D2 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 S3 branched over a closed braid.

Proof. Let F be the page of a given open book decomposition on Y withconnected binding. Choose a 3-fold simple branched cover π : F → D2.Then we can realize the monodromy of the open book decomposition as thelift of a braid (viewed as a diffeomorphism of the disk D2 fixing ∂D2 and thebranch set in D2) under the branched covering map π. Now consider theclosure of this braid (viewed as a geometric object in the usual sense) in S3

with respect to an axis A. Note that S3 has an open book decompositionwith disks Dt as pages, A = ∂Dt as binding, and the identity map asmonodromy. This construction gives a branched covering map h : Y → S3

such that each page of the given open book decomposition is realized ash−1(Dt) and the binding is simply equal to h−1(A).

Corollary 9.3.4 (Giroux). Every closed contact 3-manifold (Y, ξ) is asimple 3-fold cover of (S3, ξst) branched along a transverse link.

Proof. For a given contact 3-manifold (Y, ξ) Theorem 9.2.8 provides an openbook decomposition of Y (with connected binding) which is compatible withthe contact structure ξ. By Theorem 9.3.3, on the other hand, this openbook decomposition is a simple 3-fold cover h : Y → S3 branched over aclosed braid B. Let α be the standard contact form on S3. Note that h∗αis not a contact form on Y . We denote by C the set of points in Y where hfails to be a local diffeomorphism. Next we follow the discussion in [67] toconstruct a contact form on Y using the branched covering map h. Thereis a tubular neighborhood U1 of B with coordinates (θ, x, y) where

gα = dθ + x dy − y dx

for some positive function g on U1. Extend g to a positive function on S3

such that g = 1 outside a tube U2 slightly larger than U1 and then defineα′ = gα. Using polar coordinates, α′ can be given by

dθ + r2dφ

on U1 − B. Then we deform h by an isotopy of S3 supported in U1 to anonsmooth covering map H which is a local diffeomorphism on Y −C. Since


H is a local diffeomorphism, β = H∗α′ is a contact form on Y −C. Moreover,we can find local coordinates (θ0, r0, φ0) in a tubular neighborhood U of Csuch that the form β is given by

(1 + nr20)dθ0 + r20dφ0

for some integer n. We can extend this form to a smooth contact formon Y , again denoted by β. To see this we write the form β in Cartesiancoordinates as

β =(1 + n(x2

0 + y20))dθ0 + x0 dy0 − y0 dx0

and check thatβ ∧ dβ = 2 dθ0 ∧ dx0 ∧ dy0 > 0.

Let A denote the axis of the closed braid B which is the branch set ofh : Y → S3. We can assume that both the braid B and the axis A aretransverse to the standard contact structure ξst on S3. Consider the trivialopen book decomposition of S3, whose pages are the disks Dt, the bindingis the axis A = ∂Dt and the monodromy is the identity map. The pagesof the open book decomposition on Y are then given by h−1(Dt) and thebinding is equal to h−1(A). To show that the contact form β is compatiblewith the open book decomposition on Y , we need to check the conditionsgiven in Definition 9.2.1. It is easy to see that the binding h−1(A) of ouropen book decomposition is transverse to the contact structure β on Ysince A is transverse to α and the covering map is a local diffeomorphismon the points of the binding. We also need to check that dβ induces avolume form on every page of the open book decomposition on Y . Sinceβ = H∗α′ = h∗α away from the set C, the contact form β is simply the pullback of α by the covering map h. Now dβ = dh∗α = h∗dα is a volume formon h−1(Dt) − (h−1(Dt) ∩ U), because dα induces a volume form on a pageDt of the trivial open book decomposition on S3 and the covering map h isa local diffeomorphism. Near the set C, the contact form β is given by

(1 + nr20)dθ0 + r20dφ0

in local coordinates. Note that

dβ = 2r0 dr0 ∧ dφ0 = 2 dx0 ∧ dy0

on a small disk obtained by fixing θ0 and clearly it is a volume form on thissmall disk which doubly covers a disk in S3 with a branching point at r0 = 0.

9.3. Branched covers and contact structures 153

This proves that dβ is a volume form on every page. The condition aboutthe orientation is clear since we use the same orientation preserving map topull back the contact form and to construct the branched cover. Thus, byProposition 9.2.7 the contact structure ker β is isotopic to ξ, which finishesthe 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 termsof Lefschetz fibrations and Lefschetz pencils. In this chapter we give thenecessary definitions and sketch this topological descriptions of symplecticand Stein manifolds. In the discussion we include achiral Lefschetz fibra-tions as well; these more general objects are useful in viewing open bookdecompositions as boundaries of certain achiral Lefschetz fibrations. Thechapter concludes with some applications of Lefschetz fibrations in variouslow 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 : X4 → Σ2 is an achiral Lefschetzfibration if df is onto with finitely many exceptions p1, . . . , pk = C ⊂ intX(called the set of critical points), the map f is a locally trivial surface bundleover Σ − f(C) and around pi ∈ C and qi = f(pi) ∈ f(C) there are complexcharts Ui, Vi on which f is of the form z2

1 + z22 . We call the fibers f−1(qi)

(qi ∈ f(C)) singular, while the other fibers are regular. A fibration isrelatively 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 finiteset B ⊂ X (called the base point set) together with a map f : X−B → CP1

such that each point b ∈ B has a coordinate chart on which f can be givenby the projectivization C2 − 0 → CP1 and around its critical points fbehaves as in (a).

156 10. Lefschetz fibrations on 4-manifolds

(c) An achiral Lefschetz fibration/pencil is called a Lefschetz fibration/pencilif the complex charts Ui, Vi around the critical and base points pi andqi = f(pi) appearing in the above definition respect the given orientationsof X and Σ.

Remark 10.1.2. Notice that in Definition 10.1.1(a) the manifolds mighthave boundaries. If the typical fiber f−1(t) is a closed surface thenf−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 ofachiral Lefschetz fibrations/pencils is more general than the one withoutthe adjective achiral ; although the terminology might suggest the contrary.We did not take the courage for changing this unfortunate phenomenon, wewill 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′ suchthat 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(z1, z2) equals z2

1 +z22 , so a nearby

regular fiber is given by z21 + z2

2 = t, and after multiplying f by a unitcomplex number we can assume t > 0. For the discussion below, assumethat the chart does respect the orientation of X around the critical point.If we intersect the fiber with R2 ⊂ C2, we obtain the circle x2

1 + x22 = t

in R2 (where zj = xj + iyj and R2 is spanned by x1 and x2). This cir-cle bounds a disk Dt ⊂ R2 and as t → 0 the disk Dt shrinks to a pointin R2. By definition ∂Dt = Ft ∩ R2 is the vanishing cycle of the criticalpoint, and we explicitly see the singular fiber F0 being created from Ft bythe collapse of ∂Dt. A regular neighborhood νF0 of the singular fiber isobtained from the neighborhood νFt by adding a regular neighborhood ofDt. This latter neighborhood is clearly a 2-handle H attached to νFt, withcore disk equal to Dt. (In fact, a corresponding Morse function can begiven locally by g = −Re f , or g(z1, z2) = y2

1 + y22 − x2

1 − x22. This Morse

function provides a handlebody decomposition of the relative handlebodybuilt on ∂νFt for a regular fiber Ft.) Suppose that ∂νFt contains a fiberFs, 0 < s < t. Then the core of the 2-handle H is Ds and the attach-ing circle is the vanishing cycle ∂Ds ⊂ Fs. We describe the framing of thehandle attachment by comparing it with the framing on ∂Ds ⊂ ∂νFt deter-mined by the surface Fs. At a point (

√s cos θ,

√s sin θ, 0, 0) ∈ ∂Ds ⊂

R2 ⊂ R2 × iR2 ∼= C2 the vector (− sin θ, cos θ, 0, 0) is tangent to ∂Ds.

10.1. Lefschetz pencils and fibrations 157

Since ∂Ds lies in Fs, which is holomorphic in the given local coordi-nates, the vector field v(θ) = (0, 0,−i sin θ, i cos θ) on ∂Ds is also tan-gent to Fs. This can be seen explicitly by taking, for example, the curve(√s− t2 cos θ,

√s− t2 sin θ,−it sin θ, 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 thenormal to ∂Ds inside the surface Fs. This framing has to be compared withthe 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 correspondingvector 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 orientationinto 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 tobe 1. On the other hand, we would like to specify the framing coefficientof the 2-handle with respect to the surface framing, therefore the aboveargument translates to −1. For the case of a critical point admitting anorientation reversing coordinate chart the above argument passes throughwith an orientation reversal at the last moment, implying that the framingcoefficient in that case is +1 (with respect to the fiber framing). The abovereasoning can be obviously inverted in the following sense: Suppose thatf : X → Σ is an achiral Lefschetz fibration with ∂Σ 6= ∅ and γ ⊂ ∂X is agiven knot which lies in a fiber. Let X ′ be given by attaching a 2-handleto X along γ with framing ±1 relative to the surface framing of γ. Then fextends as an achiral Lefschetz fibration to X ′ → Σ.

Remark 10.1.4. By adding the standard shaped 2-handle, the map willnot 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 wholefiber. On the other hand, f can be extended to a manifold diffeomorphicto X ′ (similarly to the procedure of “smoothing corners”, encountered inRemark 2.1.1), and this is the content of the above argument.

Next we determine the monodromy around a critical value. For anybundle with fiber F over an oriented circle, the monodromy is determined bya single diffeomorphism ψ representing the image of the canonical generatorof π1(S

1) in ΓF . The bundle is then canonically isomorphic to the fibrationI × F/((1, x) ∼

(0, ψ(x)

)→ I/∂I ≈ S1. Given a Lefschetz fibration

f : X → Σ and a disk D ⊂ Σ inheriting the orientation of Σ, we can considerthe monodromy of the bundle f |∂D provided that the oriented circle ∂D


avoids the critical values of f . If D contains no critical values then f |D istrivial, as is the monodromy (i.e., ψ is isotopic to idF ). If D contains aunique critical value, however, the monodromy is nontrivial provided f |D isrelatively minimal. A local computation shows

Proposition 10.1.5. If f−1(c) contains a unique critical point then themonodromy around c is a Dehn twist along a simple closed curve. If theorientation of the chart containing c ∈ C respects the orientation of Xthen the Dehn twist is right-handed, otherwise it is left-handed. The simpleclosed curve is isotopic to the vanishing cycle of the singular fiber underexamination.

(For the definition of a Dehn twist see Appendix 15.) Notice the assumptionabout the number of critical points in a fiber. It is not hard to see that anyLefschetz fibration admits a perturbation such that f is injective on C.Therefore by fixing a natural generating system of π1(Σ − f(C)) we get aword in Γg: if gi in the generating system is defined as gi = [γi] with eitherγi = ∂Di for disks Di satisfying |f(C) ∩Di| = 1 and [hi] (i = 1, . . . , 2g(Σ))is a natural generating system of π1(Σ) then the fibration can be encodedby the word t1 . . . tnΠ[αi, βi] where the ti are the monodromies around γiand αi (resp. βi) are the monodromies around hi (resp. hi+g(Σ)). The worduniquely determines the fibration since a Dehn twist determines its definingvanishing cycle up to isotopy, and from this information the fibration can berecovered by adding 2-handles along the vanishing cycles with appropriateframings. In fact, if the fiber of the fibration f : X → Σ is a manifoldwith r boundary components then the resulting word naturally lives in Γg,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 geometriccounterparts of certain special words in various mapping class groups. Thisrelation will be discussed in more details in Section 15.2.

Suppose that f : X → D2 is an achiral Lefschetz fibration, such thateach singular fiber contains a unique critical point. We will describe anelementary handlebody decomposition of X using essentially the definitionof an achiral Lefschetz fibration and Proposition 10.1.5. We select a regularvalue q0 of the map f in the interior of D2, an identification of the fiberf−1(q0) ∼= F (a compact surface with possibly nonempty boundary), anda collection of arcs si in the interior of D2 with each si connecting q0to qi, and otherwise disjoint from each other. We also assume that thecritical values are indexed so that the arcs s1, . . . , sm appear in order as we


travel counterclockwise in a small circle about q0. Let V0, . . . , Vm denote acollection of small disjoint open disks with qi ∈ Vi for each i, see Figure 10.1.Since an achiral Lefschetz fibration is a locally trivial F -bundle away from

0

1

3

ms

V

2V

3q

2q

mq

4q

VV 1q

s1

2s

4V

Vm

3

4

s

s

q0

Figure 10.1. Fibration over the disk

the critical points, we have f−1(V0) ∼= D2 × F with ∂(f−1(V0)

) ∼= S1 × F .Let ν(si) be a regular neighborhood of the arc si. Now the discussionfollowing Remark 10.1.2 shows that f−1(V0 ∪ ν(s1)∪V1) is diffeomorphic toD2 × F with a 2-handle H1 attached along a circle γ1 contained in a fiberpt. × F ⊂ S1 × F . Moreover, the 2-handle H1 is attached with framing(±1) relative to the natural framing on γ1 inherited from the the fiber. (Thecurve γ1 was called the vanishing cycle.) In addition, ∂

((D2 × F ) ∪ H1

)

is diffeomorphic to an F -bundle over S1 whose monodromy is equal to theDehn twist tγ1 along γ1. Continuing counterclockwise around q0, we addthe remaining critical values to our description, yielding that

X0∼= f−1

(V0 ∪

( m⋃

i=1

ν(si)

)∪( m⋃

i=1

Vi

))

is diffeomorphic to (D2 × F ) ∪ (⋃mi=1Hi), where each Hi is a 2-handle

attached along a vanishing cycle γi in a fiber of S1 × F → S1 with relative


framing (±1). Furthermore, the part of

∂X0∼= ∂

((D2 × F ) ∪

( m⋃

i=1

Hi

))

which maps to ∂D is an F -bundle over S1, whose monodromy is the productof Dehn twists along the vanishing cycles. We will refer to this product asthe global monodromy of the fibration. Suppose that an achiral Lefschetzfibration f : X → Σ admits k singular fibers. The Euler characteristic of Xcan be easily computed as χ(F )χ(Σ)+k since we add k 2-handles to a surfacebundle. The computation of σ(X), however, turns out be a nontrivial issue.There is a signature formula for hyperelliptic Lefschetz fibrations [35] andthere is an algorithm to compute the signature for Lefschetz fibrations overS2 given in [130].

After these topological preparations we begin our discussion about therelation between Lefschetz fibrations and symplectic/Stein manifolds. Forthe rest of this chapter we assume that all Lefschetz fibrations are of the typegiven by Definition 10.1.1(c), i.e., the complex coordinate charts respectthe orientation fixed on X. We start with the case of closed symplecticmanifolds; Stein surfaces will be discussed in the next section. The mostimportant 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] ∈ H2(X; R) is integral then X admits a Lefschetz pencilsuch that the generic fiber is a smooth symplectic submanifold.

Exercise 10.1.7. Prove that every symplectic manifold (X,ω) admits asymplectic form ω′ such that [ω′] ∈ H2(X; R) lifts to an integral cohomologyclass, i.e., it is in the image of the map H2(X; Z) → H2(X; R) induced bythe inclusion Z → R.

Remark 10.1.8. The proof of this theorem is rather involved, here werestrict ourselves merely to a quick indication of the main idea. Let L→ Xbe the complex line bundle with c1(L) = h ∈ H2(X; Z) (where h mapsto [ω] under the map H2(X; Z) → H2(X; R)). To prove Theorem 10.1.6,Donaldson showed that if k is large enough, then L⊗k → X admits a sections such that s−1(0) ⊂ X is a symplectic submanifold. Using the same basicidea, he also showed that for k even larger there are linearly independentsections s0, s1 ∈ Γ(L⊗k) such that the submanifolds

(t0s0 + t1s1)

−1(0) ⊂ X | [t0 : t1] ∈ CP1


are symplectic and form a Lefschetz pencil on X. The proof is based on atechnique of Kodaira for embedding Kahler manifolds in CPN , although theanalytical details are much more subtle in the symplectic case. Specifically,it was proved that the map x 7→

[s0(x) : s1(x)

]∈ CP1 (defined on

X −s−10 (0) ∩ s−1

1 (0)

) provides a Lefschetz fibration on some blow-upof 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-manifoldthen it decomposes as W ∪D where W is a Stein domain and D is a D2-bundle over a surface Σg.

Proof (sketch). Take a section σ ∈ Γ(L⊗k) as above and consider thefunction 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 D2-bundle over the surface Σg = s−1(0), the conclusionfollows.

Donaldson’s theorem admits a converse (which is considerably simpler toprove):

Theorem 10.1.10 (Gompf, [66]). If the smooth, closed 4-manifold Xadmits a Lefschetz fibration such that the homology class of the fiber isnonzero in H2(X; R) then X admits a symplectic structure with the fibersbeing 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 fiberscan be taken care of by implementing the existing local models around thecritical points. For details see [66, Chapter 10]. The main idea in the proofis 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 ofa symplectic structure from the base to it. This leaves the fiber directionsintact and takes care for the orthogonal directions. In fact, by taking evenlarger multiples we can arrange that finitely many (fixed) sections of thefibration become symplectic as well. This leads us to the following:

Corollary 10.1.12 (Gompf, [66]). If a smooth, closed 4-manifold X admitsa Lefschetz pencil then it carries a symplectic structure such that the genericfiber is a smooth symplectic submanifold.


Proof. By blowing up X we get X#nCP2 equipped with a Lefschetz fi-bration, moreover the n exceptional curves (being sections) can be chosento 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 10.1.10about the homology class of the fiber is not very restrictive: if the fibergenus is not equal to one or the fibration has at least one singular fiber thenit is fulfilled, see [66]. (For torus fibrations the statement does not neces-sarily hold, as the obvious torus fibration S1 × S3 → S2 coming from theHopf map S3 → S2 shows.)

10.2. Lefschetz fibrations on Stein domains

In [73] Harer proved that if a smooth 4-manifold X is obtained by attaching1- and 2-handles to D4 then it admits an achiral Lefschetz fibration overD2. Notice that the boundary of an achiral Lefschetz fibration f : X → D2

acquires a canonical open book decomposition induced from the fibration:compose the map f with the radial projection π : D2 − 0 → ∂D2 to getπ f :

(∂X − f−1(0)

)→ S1, providing an open book decomposition on

∂X with binding ∂f−1(0). An alternative proof of Theorem 9.1.3 followsfrom this fact since every closed oriented 3-manifold Y is the boundaryof a smooth 4-manifold obtained by attaching 2-handles to D4. Loi andPiergallini [104] (and later Akbulut and the first author [7]) showed that aStein domain always admits a Lefschetz fibration structure:

Theorem 10.2.1 (Loi–Piergallini, [104]). If W is a Stein domain then itadmits a Lefschetz fibration structure. In addition, we can assume that thevanishing cycles in the resulting fibration are homologically essential.

In fact, Loi and Piergallini proved that any Stein domain can be given asan analytic branched cover of D4 along a holomorphic curve, or of D2 ×D2

along a positive braided surface. The theorem above follows from this result.

Proof. We describe the proof of this theorem given by Akbulut and thefirst author [7]. The proof explicitly constructs the vanishing cycles of theLefschetz fibration, and associates to every Stein domain infinitely manypairwise 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 homologicallynontrivial in the fiber F . Note that a simple closed curve on a surface withat most one boundary component is homologically trivial if and only if itseparates the surface. Sometimes we will refer to a homologically trivial(resp. nontrivial) curve as a separating (resp. nonseparating) curve. Apositive allowable Lefschetz fibration over D2 with bounded fibers will beabbreviated as PALF. (Here the adjective “positive” just emphasizes thatwe are working with Lefschetz fibrations, that is, all singular fibers give riseto right-handed Dehn twists in the monodromy.)

In the following we digress to give the details of a construction which isdue to Lyon [106]. We say that a link in R2 is in a square bridge positionwith respect to the plane x = 0 if the projection onto the plane is regularand each segment above the plane projects to a horizontal segment andeach one below to a vertical segment. Clearly any link can be put in asquare bridge position. (Notice that we require the horizontal segment topass over the vertical; therefore in putting a projection in square bridgeposition we have to pay special attention to possible illegal crossings. Forthese 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 arrangedby isotopy so that each horizontal segment is a subset of

0 × [0, 1] × zi


z

y

Figure 10.3. Trefoil knot in a square bridge position

for some 0 < z1 < z2 < · · · < zp < 1 and each vertical segment is a subsetof

0 × yj × [0, 1]

for some 0 < y1 < y2 < · · · < yq < 1. Now consider the 2-disk

Di = [ε, 1] × [0, 1] × zi

for each i = 1, 2, . . . , p and the 2-disk

Ej = [−1,−ε] × yj × [0, 1]

for each j = 1, 2, . . . , q, where ε is a small positive number. Attach thesedisks by small bands (see Figure 10.4) corresponding to each point (0, yi, zj)for i = 1, . . . , p and j = 1, . . . , q. If p and q are relatively prime then theresult is the minimal Seifert surface F for a (p, q) torus knot K such thatK ∩ L = ∅ and L ⊂ F . It is easy to see that each component of the link Lis a nonseparating curve on the surface F . Moreover we can choose p andq arbitrarily large by adding more disks of either type D or type E. Thisconcludes our digression.


x

Di

Ej

L

z

y

Figure 10.4. Attaching disks

Let K be a torus knot in S3. It is well-known that K is a fibered knotand the corresponding fibration induces an open book decomposition ofS3 whose monodromy is a product of nonseparating positive Dehn twists,cf. Example 9.1.4. This factorization of the monodromy defines a PALFX → D2 such that the induced open book decomposition on S3 = ∂X isequal to the open book decomposition given by the torus knot.

Exercise 10.2.2. Verify that for any torus knotK the 4-manifold (PALF )Kunderlying the corresponding Lefschetz fibration is diffeomorphic to D4.(Hint: Consider the handlebody decomposition of (PALF )K and use Kirbycalculus; in particular, locate cancelling 1-handle/2-handle pairs.)

Returning to the proof of Theorem 10.2.1 suppose first that W is a Steindomain built by 2-handles only. This means that we attach Weinstein 2-handles to D4 along the components Li of a Legendrian link L in S3 = ∂D4

with framing tb(Li) − 1 to get the Stein domain W . Hence our startingpoint is a Legendrian link diagram in (R3, ξst) ⊂ (S3, ξst). First we smoothall 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 thatclockwise rotation results in a diagram with vertical segments passing overhorizontal ones, contradicting our convention for square bridge position.)Then the construction of Lyon described above allows us to find a torus


z

y

Figure 10.5. Rotation of a Legendrian knot into square bridge position

knot K with its Seifert surface F such that each Li is an embedded circleon F for i = 1, 2, . . . , n. In Figure 10.6 we depicted the embedding of theright-handed trefoil knot into the Seifert surface of the (5, 6) torus knot.Let L+

i be a parallel copy of Li on the surface F , and let lk(Li, L+i ) be

the linking number of Li and L+i computed with parallel orientations. This

linking number is called the surface framing of Li. We will denote it bysf(Li). Then we observe that the surface framing of Li will pick up a −1at each left corner of the link in square bridge position and will change bythe amount of writhe at each under/over-crossing. To see this, imagine aparallel copy L+

i of Li on the surface F then cut out and straighten thenarrow band on the surface bounded by Li and L+

i . Notice, however, thatthis is exactly the recipe how the Thurston–Bennequin invariant of Li iscalculated in its Legendrian position (before rotating and smoothing itscorners): −1 for each left kink plus the writhe of the knot. Thus we get

tb(Li) = sf(Li).

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 Hi

to D4 along Li with framing tb(Li) − 1 = sf(Li) − 1 for i = 1, 2, . . . , n.By our discussion of the handle decomposition of a Lefschetz fibration


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 D4 ∼= (PALF )K over the 2-handles to get a new PALF sinceL = L1, . . . , Ln is embedded in a fiber F of ∂(PALF )K

∼= S3. Thus weshowed that W ∼= D4 ∪H1 ∪ · · · ∪Hn

∼= (PALF )K ∪H1 ∪ · · · ∪Hn admitsa PALF and the global monodromy of this PALF is the monodromy of thetorus knot K composed with positive Dehn twists along the Li’s. Noticethat the Dehn twists along the Li’s commute since they are pairwise disjointembedded curves on the surface F .

Now we turn to the general case. Suppose thatW is a Stein domain obtainedby attaching 1- and 2-handles to D4. First of all, we would like to extend(PALF )K on D4 to a PALF on D4 union 1-handles. Recall that attachinga 1-handle to D4 (with the dotted-circle notation) is the same as pushingthe interior of the obvious disk that is spanned by the dotted circle into theinterior ofD4 and scooping out a tubular neighborhood of its image fromD4.To reach our goal, we represent the 1-handles with dotted-circles stackedover the front projection of the Legendrian tangle which is in standard formas it is described in [65]. Then we modify the handle decomposition bytwisting the strands going through each 1-handle negatively once. In thenew diagram the Legendrian framing will be the blackboard framing withone left-twist added for each left cusp. This is illustrated by the seconddiagram in Figure 10.7.


tangleLegendrian

tangleLegendrian

tangleLegendrian

......

.

......

.

......

.

......

.

......

....

....

......

.

......

.

Figure 10.7. Legendrian link diagram in square bridge position

Exercise 10.2.3. Verify that the twisting operation does not change thetopology of the underlying 4-manifold. (Hint: See Figure 10.8. In (b) weintroduce a cancelling 1-handle/2-handle pair, in order to get (c) we slidethe new dotted circle over the old one, then in (d) we slide the strands over


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 thelink 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 thewhole diagram as a link in S3. We put this link diagram in a square bridgeposition as in the previous case (see Figure 10.7) and find a torus knot Ksuch that all link components lie on the Seifert surface F of K. Beforeattaching the 1-handles we isotope each dotted circle in the complementof the rest of the link such that it becomes transverse to the fibers of thefibration S3 −K → S1, meeting each fiber only once, see [106] for details.Now for each 1-handle we push the interior of the disk spanned by thedotted circle into D4 and this becomes a section of (PALF )K . Thus byattaching a 1-handle to D4 we actually remove a small 2-disk D2 from


each fiber of (PALF )K , and hence obtain a new PALF on D4 union a 1-handle. In other words, we extend the open book decomposition on S3

induced by the torus knot K to an open book decomposition on S1 × S2.The boundary of the disk we remove from the fiber becomes a componentof the binding of the open book decomposition on S1 × S2. Notice thatthis circle becomes a longitudional curve after the surgery on the boundary(induced by attaching the 1-handle) since we swap meridian and longitudeby a 0-surgery. After attaching all the 1-handles to D4 we get an open bookdecomposition on the connected sum of k copies of S1 × S2 for some k ∈ N

and a new PALF on kS1 × D3 such that the regular fiber is obtained by

removing disjoint small disks from F . Then as in the previous case we canextend our PALF on D4 ∪ 1-handles to a PALF on D4 ∪ 1-handles ∪ 2-handles. The global monodromy of the constructed PALF is the productof the monodromy of the torus knot K and right-handed Dehn twists alongvanishing cycles corresponding to the 2-handles. Finally, we note thatthe (p, q) torus knot can be constructed using arbitrarily large p and q.Therefore our construction yields infinitely many pairwise nonequivalentPALF’s, since for chosen p and q the genus of the regular fiber will be equalto 1

2(p − 1)(q − 1).

In the proof of Theorem 10.2.1 we constructed an explicit Lefschetzfibration on a Stein domain which is given by its handle decomposition.The boundary of this PALF has an open book decomposition induced fromthe fibration and it also acquires a contact structure induced from the Steindomain. It turns out that the induced open book and the contact structureare compatible. In the following we will outline a proof of this fact dueto Plamenevskaya [146] which is obtained by a slight modification of theproof of Theorem 10.2.1. Recall that in the proof of Theorem 10.2.1 wesmoothly isotoped the Legendrian link into a square bridge position in orderto put it on the Seifert surface of a torus knot and we forgot about thecontact structure. One can modify this construction as follows: For a givenLegendrian link L in (R3, ξst) there exists a surface F containing L suchthat dα is an area form on F (where α = dz+x dy), ∂F = K is a torus knotwhich is transverse to ξst and the components of L do not separate F . Theconstruction of the surface F is identical to the one in the proof above exceptthat we first isotope L by a Legendrian isotopy so that in the front projectionall the segments have slope (±1) away from the points where L intersects theyz-plane, see Figure 10.9. Then we use narrow strips around these segmentsas 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


of a torus knot K = ∂F . By further isotopies we can ensure that L lies inF , dα is an area form on F and ∂F = K is transverse to ξst. Now thicken

Figure 10.9. Legendrian link diagram

this one page F (which carries the Legendrian link L) into a handlebodyU1 which is the union of an interval worth of pages (see Section 5.2) so thatdα is an area form on every page. Now we can fiber the complementaryhandlebody U2 in S3 with binding K and pages diffeomorphic to F sinceK is a fibered knot. So far we obtained an open book decomposition ofS3 which is expressed as a union of two “half” open books, one of whichis compatible with ξst. We would like to extend the contact structure ξstto the fibered handlebody U2 (as some contact structure ξ) so that it iscompatible with the open book decomposition on U2. This can be achieved(see [146]) by an explicit construction of a contact form on U2 similar tothe one we described in Theorem 9.2.5. Hence we get a contact structureξ on S3 which is compatible with our open book decomposition. Since themonodromy of this open book decomposition is a product of right-handedDehn twists, ξ is Stein fillable and therefore isotopic to ξst. Moreover, byconstruction ξ and ξst coincide on U1 so that the isotopy between ξ andξst can be assumed to be the identity on U1. Now we need to show thatthe contact structures ξ and ξst are isotopic on U2 relative to ∂U2. Noticethat ∂U2 can be made convex and one can check that the binding K is thedividing set on ∂U2. Uniqueness (up to isotopy) of a tight contact structurewith such boundary conditions was shown in the proof of Theorem 5.2.1.In summary we proved


Proposition 10.2.4 (Plamenevskaya, [146]). For a given Legendrian linkL in (S3, ξst) there exists an open book decomposition of S3 satisfying thefollowing conditions:

(1) the contact structure ξ compatible with this open book decompositionis isotopic to ξst,

(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 ξst,

(5) the surface framing of L (induced by the page F ) is the same as itscontact framing induced by ξ (or ξst).

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 compatibleopen book obξ in a contact 3-manifold (Y, ξ). Then the surface framing ofC (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 bindingand dα > 0 on the pages of obξ. Then the Reeb vector field Rα is transverseto the pages (by Exercise 9.2.2) as well as to the contact planes. Hence Rαdefines 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, wehave an algorithm which constructs an explicit PALF on a Stein domainX. This algorithm also yields an open book decomposition on ∂X which iscompatible with the contact structure induced from the Stein domain.

Exercise 10.2.6. Find a PALF on D2×T 2 using the given algorithm. Alsofind an open book decomposition of T 3 which is compatible with the contactstructure induced from the Stein domain D2 × T 2. (Hint: See Figure 12.8for a Stein structure on D2 × T 2.)

The converse of the above theorem also holds, namely

Theorem 10.2.7 (Loi–Piergallini, [104]). Every PALF admits a Steinstructure.

10.3. Some applications 173

Proof. We describe the proof given in [7]. LetX be a PALF. We can assumethat the boundary of a regular fiber is connected by plumbing Hopf bands ifnecessary. It is clear that X is obtained by a sequence of steps of attaching2-handles X0 = D2 × F → X1 → X2 → · · · → Xn = X, where eachXi−1 is a PALF and Xi is obtained from Xi−1 by attaching a 2-handle to anonseparating curve Ci lying on a fiber F ⊂ ∂Xi−1 with framing sf(Ci)−1.Notice that D2 × F has a Stein structure since it is obtained from D4 byattaching 1-handles only. Inductively, we assume that Xi−1 has a Steinstructure and thus ∂Xi−1 has an induced compatible contact structure. In[53] it was shown that this induced contact structure agrees with Torisu’scontact structure given in Section 5.2. Let Σ denote the double of a page Fof the open book (as in Section 5.2) induced on ∂Xi−1 by the PALF. We canassume that Σ is a convex surface which is divided by the binding ∂F . Thesimple closed curve Ci on the convex surface Σ is nonisolating with respectto the dividing curve ∂F since we assumed that Ci is a nonseparating curve.Then we apply the Legendrian Realization Principle (cf. Lemma 5.1.18) tomake Ci Legendrian such that the surface framing sf(Ci) of Ci is equal toits Thurston–Bennequin framing (see Exercise 5.1.14). The result followsby 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 sothat the nonisolating condition is quaranteed even for homologically trivialcurve.

10.3. Some applications

In this section we use ideas developed above to solve certain low dimensionalproblems.

Theorem 10.3.1. If W is a Stein domain then we can embed it into aminimal, closed, symplectic 4-manifold X.

Proof. We know that a Stein domain W admits a PALF. By plumbing Hopfbands if necessary, we may also assume that the boundary of the regularfiber F is connected. The fibration induces an open book decomposition of∂W with connected binding ∂F . First we enlarge W to W by attaching a


2-handle along the binding ∂F with framing 0 (with respect to the surfaceframing) to get a Lefschetz fibration over D2 with closed fibers. Hence ∂Wis an F -bundle over S1, where F denotes the closed surface obtained bycapping off the surface F by gluing a 2-disk along its boundary. Let ΓFdenote the mapping class group of the closed surface F . Now we can easilyextend W into a Lefschetz fibration X over S2 with regular fiber F . Lettc1tc2 · · · tck be the global monodromy of the PALF on W , where ci denotesa simple closed curve on F for i = 1, 2, . . . , n. Then this product (aftercapping off the boundary component) can be viewed as a product in ΓF .We clearly have

tc1tc2 · · · tckt−1ckt−1ck−1

· · · t−1c1 = 1.

By Lemma 15.1.16 we can replace every left-handed Dehn twist by a productof right-handed Dehn twists to obtain a factorization of the identity into aproduct of right-handed Dehn twists. This factorization gives a Lefschetzfibration X over S2 (with closed fibers) which admits a symplectic structureby Theorem 10.1.10. We can assume that the genus of F (and therefore of F )is at least two so that the hypothesis [F ] 6= 0 ∈ H2(X; R) in Theorem 10.1.10is automatically satisfied. Consequently, the Stein domain W is embeddedinto a closed symplectic 4-manifold X. As we will see, by taking fiber sumif 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-manifoldwith some extra structure was first noticed by Lisca and Matic [97]. Theyproved 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 Kahler embeddingf : W → X. This observation was used to distinguish homotopic but non-isotopic (Stein fillable) contact structures. The above embedding of a Steindomain into a minimal, closed, symplectic 4-manifold is due to Akbulut andthe first author [8]. In another direction, we will show that any symplecticfilling embeds into a closed symplectic 4-manifold, see Theorem 12.1.7.

The following converse of Theorem 10.3.1 easily follows from Theorem 10.2.7and Remark 10.2.8; compare also with Corollary 10.1.9.

Corollary 10.3.3. Let f : X → S2 be a Lefschetz fibration which admitsa section. Then by removing a neighborhood of the section union a regularfiber 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


a contact structure and the monodromy of a compatible open book decom-position, cf. Corollary 9.2.15. For definitions of various fillability notionssee Section 12.1.

Theorem 10.3.4 (Giroux). A contact 3-manifold (Y, ξ) is Stein fillableif and only if it admits a compatible open book decomposition with mon-odromy h ∈ Γg,r such that h = ta1 . . . tan with tai

right-handed Dehn twistsalong 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 toanother.

Proof. Suppose that h = ta1 . . . tan for some right-handed Dehn twists alonghomotopically nontrivial simple closed curves. Then the (positive) Lefschetzfibration with total monodromy h is a Stein filling of Y . Conversely, anyStein 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 needto 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 ofcertain contact 3-manifold. Furthermore, these examples show that fibersums of holomorphic Lefschetz fibrations do not necessarily admit complexstructures.

Theorem 10.3.5 ([131]). There are infinitely many (pairwise nonhomeo-morphic) 4-manifolds which admit genus-2 Lefschetz fibrations but do notcarry complex structure with either orientation.

Proof. Matsumoto [108] showed that S2 × T 2#4CP2 admits a genus-2Lefschetz fibration over S2 with global monodromy (tβ1 · · · tβ4)

2, whereβ1, . . . , β4 are the curves depicted by Figure 10.10. Let Bn denote thesmooth 4-manifold obtained by the twisted fiber sum of the Lefschetz fi-bration S2 × T 2#4CP2 → S2 with itself, using the diffeomorphism hn ofthe fiber Σ2, where h denotes the right-handed Dehn twist about the curveα which is depicted in Figure 10.11. Then Bn admits a genus-2 Lefschetz

fibration over S2 with global monodromy (tβ1 · · · tβ4)2(thn(β1) · · · thn(β4))

2.

Standard theory of Lefschetz fibrations gives that

π1(Bn) = π1(Σ2)/⟨β1, . . . , β4, h

n(β1), . . . , hn(β4)

⟩,

showing that π1(Bn) = Z ⊕ Zn.


β2β1

β β43

Figure 10.10. Vanishing cycles

α

Figure 10.11. The twisting curve α

Exercise 10.3.6. Show that the Lefschetz fibration Bn → S2 admits asection. (Hint: Verify the statement for S2 × T 2#4CP2 → S2 and splicethe sections together.)

The definition of Bn provides a handlebody decomposition for it and shows,in particular, that the Euler characteristic χ(Bn) is equal to 12. Since Bnis the fiber sum of two copies of S2 × T 2#4CP2, we get that the signatureσ(Bn) = −8, consequently b2(Bn) = 12, b+2 (Bn) = 2 and b−2 (Bn) = 10. LetMn denote the n-fold cover of Bn with π1(Mn) ∼= Z. Easy computationshows that b+2 (Mn) = 2n and b−2 (Mn) = 10n. This allows us to show thatBn does not admit a complex structure (see [131] for details).

Next we study the problem of finding the minimal number of singularfibers 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 thereare almost-complex structures J and j on X and Σ resp., such that f ispseudoholomorphic, that is, df J = j df .


Using Seiberg–Witten theory, in particular Taubes’ results on Seiberg–Witten invariants of closed symplectic 4-manifolds, this observation quicklyleads us to the proofs of the following two results:

Proposition 10.3.8 ([155]). Suppose that f : X → Σ is a given Lefschetzfibration with g(Σ) > 0. Then the fibration X → Σ is relatively minimal ifand 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 relativelyminimal. For the converse direction suppose that X is not minimal. UsingTaubes’ result, a (−1)-sphere S can be displaced to be a J-holomorphicsubmanifold, hence f |S : S → Σ is a holomorphic map. By the assumptionon 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 → S2 is relativelyminimal if and only if X#fX is minimal.

Here are a few corollaries of the above propositions:

Corollary 10.3.10. If X → Σ is a relatively minimal Lefschetz fibrationand g(Σ) > 0 then c21(X) ≥ 0. If X → S2 is relatively minimal thenc21(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#fX) = 2σ(X)and χ(X#fX) = 2χ(X) + 4g − 4, therefore by Proposition 10.3.9 we havethat

0 ≤ c21(X#fX) = 3σ(X#fX) + 2χ(X#fX)

= 6σ(X) + 4χ(X) + 8g − 8 = 2c21(X) + 8g − 8,

which implies the result.

Corollary 10.3.11. A genus-g Lefschetz fibration X → S2 has at least 45g

singular fibers.


Proof (sketch). We can assume that X is relatively minimal. Let n ands denote the number of homologically nontrivial and homologically trivialvanishing cycles, respectively. Then 4 − 4g ≤ c21(X) = 3σ(X) + 2χ(X) ≤3(n − s) + 2(4 − 4g + n + s) = 5n − s + 8 − 8g, implying n ≥ 4

5g. Theinequality σ(X) ≤ n− s (see [130] for example) follows from the fact that a2-handle attachment can change the signature of the 4-manifold by at most1, and if the vanishing cycle is homologically trivial, such an attachmentreduces the signature.

If we allow the base space to have higher genus then the problem of findingthe minimal number of singular fibers in a relatively minimal Lefschetzfibration is almost completely solved, see Chapter 15.

11. Contact Dehn surgery

Now we are in the position to describe the contact version of the smoothsurgery scheme we started our notes with. This method provides a richand yet to be explored source of all kinds of contact 3-manifolds. Theapproach to 3-dimensional contact topology we outline here was initiated byDing and Geiges [16, 17], see also [18, 19]. Using contact surgery diagrams— and applying achiral Lefschetz fibrations — we will make connectionto Giroux’s theory on open book decompositions, and we will also showa way to determine homotopic properties of the contact structures underexamination. We begin by reviewing the classification of tight structures onS1×D2 due to Honda — this is the result which allows us to define contactsurgery diagrams.

11.1. Contact structures on S1 × D2

The idea of Honda in the classification of tight contact structures on thesolid torus S1 ×D2 is roughly the following: there is a strong relationshipbetween contact structures on the solid torus S1×D2 with certain boundarycondition, the thickened torus T 2 × [0, 1] with certain related boundaryconditions, and on the lens space L(p, q) where p and q depend on the aboveboundary conditions. Legendrian surgery provides many tight, in fact, Steinfillable contact structures on lens spaces — this gives a lower bound for thenumber of structures on the solid torus. Using convex surface theory thenHonda gives an upper bound for that number, which matches with thelower bound given by the surgeries. This concludes the proof. Here we willshow the lower bound in the general case by using Legendrian surgery, andproduce a (generally much weaker) upper bound for the number of tightstructures on S1 × D2. 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 caseour contact surgery construction will rely on. Let us start by stating theresult of Honda. For this, let us assume that for p ≥ q ≥ 1 the rationalnumber −p

q is equal to [r0, . . . , rk] where [r0, . . . , rk] denotes the continued

fraction expansion of the rational number −pq , i.e.,

−pq

= r0 −1

r1 − 1r2−...−

1rk

with ri ≤ −2 (i = 0, . . . , k) in case p > q. (For p = q = 1 we have k = 0 andr0 = −1.) Notice that any nonzero slope on the boundary of the solid toruscan be transformed into the form −p

q with p ≥ q ≥ 1 and (p, q) = 1 by a self-

equivalence S1 ×D2 → S1 ×D2. The case p = 0 needs different treatment,see the concluding remark of this section. Now fix relative primes (p, q) withp ≥ q ≥ 1.

Theorem 11.1.1 (Honda, [76]). The solid torus S1 × D2 has exactly∣∣(r0 + 1)(r1 + 1) · · · (rk−1 + 1)rk∣∣ nonisotopic tight contact structures with

convex boundary having two dividing curves of slope −pq . Consequently,

any nonzero boundary slope can be given as the boundary of a tight contactstructure on S1 ×D2.

Remark 11.1.2. Recall that the dividing set on a convex torus in a tightcontact structure consists of 2n parallel circles of some common slope r.The above theorem provides the classification of tight contact structures onS1 ×D2 with convex boundary having dividing set of two components. Forresults regarding the general (i.e., n > 1) case see [76] — those results willnot be used in this volume.

Example 11.1.3. If the boundary slope is 1n for some n ∈ Z then by a self-

diffeomorphism we can transform it to −11 = −1, hence k = 0 and r0 = −1,

consequently (up to isotopy) there is a unique tight contact structure onS1×D2 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 thesolid torus with fixed boundary condition. As we already saw, L(p, q) canbe given as −p

q -surgery on the unknot K ⊂ S3 — and this is equivalent to

attaching 4-dimensional 2-handles to D4 along a chain of (k + 1) unknotswith framings r0, . . . , rk, cf. Exercise 2.2.7(c). In fact, the unknots can beput in Legendrian position, and since ri < −1, by adding zig-zags we can

11.1. Contact structures on S1× D2 181

arrange ri = tb(Ki) − 1 to hold. Note that there is a certain freedom inadding the zig-zags to the Legendrian unknot shown by Figure 11.1: intotal the zig-zags can be positioned in

∣∣ (r0 + 1) · · · (rk + 1)∣∣ different ways.

All these choices produce diffeomorphic Stein domains with some inducedcontact structures on the boundary. It is not hard to determine the spinc

structures induced by these contact structures, and a direct computationeasily shows that the structures are not isotopic. Recall that a fixed diagraminduces a Stein structure on the underlying smooth 4-manifold X withcomplex structure J satisfying

(11.1.1)⟨c1(X,J), [Σi]

⟩= rot(Ki)

with [Σi] ∈ H2(X; Z) denoting the homology element corresponding to theknot Ki, cf. Chapter 2. Now c1(X,J) determines a spinc structure on X andits restriction to ∂X is the spinc structure induced by the contact structureof the surgery diagram.

Exercise 11.1.4. Show that the contact structures given by the aboveLegendrian surgery diagrams on L(p, q) all have different spinc structures.

There is another way, involving much deeper theory, to distinguish thesestructures. According to Proposition 8.2.4 (or Equation 11.1.1) the c1-invariants of these Stein domains are all different, hence Theorem 11.1.5 ofLisca and Matic applies:

−1

−3

−1 −1

Figure 11.1. Adjusting contact framing by stabilization


Theorem 11.1.5 (Lisca–Matic, [97]). Suppose that J1, J2 are two Steinstructures on a fixed smooth 4-manifold X and ξ1, ξ2 are the induced contactstructures on ∂X. If c1(X,J1) 6= c1(X,J2) then ξ1 and ξ2 are not isotopic.

Remark 11.1.6. The proof of this statement rests on the fact that aStein domain can be embedded into a minimal surface of general type, andthis 4-manifold has only two Seiberg–Witten basic classes. Two isotopiccontact structures with different c1-invariants would produce more basicclasses. 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∣∣ (r0 + 1) · · · (rk + 1)

∣∣ tight contact structures on L(p, q)are all distinct. In fact, the above mentioned relation between contactstructures on L(p, q) and on the solid torus with some fixed boundarycondition, together with Theorem 11.1.1 finishes the classification of contactstructures on lens spaces:

Theorem 11.1.7 (Honda, [76]). Any tight contact structure on L(p, q) isisotopic to one of the structures given as Stein boundaries above. Conse-quently L(p, q) carries

∣∣ (r0 +1) · · · (rk +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 contactstructures covered by the theorem of Lisca are the ones for which all thezig-zags in the diagram are either on the left or on the right — these twostructures are actually contactomorphic and universally tight.

The link description of the contact structures shows that all these structurescontain a Legendrian knot K, the Legendrian realization of the normal circleto, say, the left-most surgery curve in the chain, such that L(p, q) − int νKis 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 seethat the slope of the dividing curves on this torus is −p′

q′ with pq′ − qp′ = 1when viewed from the complementary solid torus. From this equation weget that −p′

q′ has continued fraction representation [r0, . . . , rk−1, rk + 1] if

−pq = [r0, . . . , rk]. Since the neighborhood of K is standard, in this way

we found∣∣(r0 + 1) · · · (rk + 1)

∣∣ isotopy classes of tight contact structures

on S1 × D2 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 S1× D2 183

contact structures stated in Theorem 11.1.1 (where the statement was given

for pq rather than for p′

q′ ), and is equal to 1 for slopes of the form 1n .

Now we turn to the derivation of the upper bound for the number oftight structures on S1 ×D2 in general. In doing so we will follow the proofof Theorem 5.1.30. To this end fix a tight contact structure ξ on S1 ×D2

with convex boundary and the fixed boundary slope of the dividing setΓ∂(S1×D2) on it. Again, we only deal with the case when Γ∂(S1×D2) has twocomponents. Consider the meridional simple closed curve µ ⊂ ∂(S1 ×D2)which becomes homotopically trivial when viewed in S1 ×D2. Put it intoLegendrian position, consider the spanning disk D ⊂ S1 ×D2 with ∂D = µand isotope this disk into convex position. Since ξ is tight, the dividing setΓD on D contains no closed components, hence ΓD is equal to a collectionof arcs with boundary on ∂D = µ. From the fact that the contact planesrotate in the same direction when travelling around µ it follows that theintersection points ΓD ∩ ∂D and Γ∂(S1×D2) ∩ ∂D follow each other in analternating manner, that is, for consecutive intersections x, y ∈ ΓD ∩ ∂Dthere is a unique z ∈ Γ∂(S1×D2) ∩ ∂D between x and y and vice versa,cf. Figure 11.2. Since Γ∂(S1×D2), and so Γ∂(S1×D2) ∩ ∂D is given by theboundary condition, the number |ΓD∩∂D| and so the number of arcs in ΓDis also fixed. Since there is an upper bound for the possible configurationsof the embedded arcs of ΓD with these boundary conditions, this argumentprovides an upper bound for the tight contact structures near D in termsof the boundary slope −p

q . Since by Eliashberg’s Theorem there is a unique

(up to isotopy) tight contact structure on S1 ×D2 − νD = D3, the abovereasoning provides an upper bound for the number of tight structures onS1 ×D2 with the given boundary condition encoded by the slope −p

q of thedividing set Γ∂(S1×D2) on the boundary. This upper bound is in generalfar from being sharp. Isotoping the disk D in a fixed contact structure ξwe might get different configurations for ΓD, although the contact structurehas not been changed. Honda’s method of manipulating the dividing setswith bypasses yields an equivalence relation among possible configurationsof dividing sets on D and concludes in a sharp upper bound for the numberof 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.


x y z

T2Γ

DΓ

Figure 11.2. Dividing sets Γ∂(S1×D2) and ΓD

Remark 11.1.9. Throughout the argument above we assumed that theboundary slope is different from zero. The reason is that there is notight contact structure on S1 ×D2 with boundary slope zero: in this case∂(pt. × D2

)can be isotoped to be disjoint from the dividing curves of

the boundary, therefore pt.×D2 (after being isotoped to have Legendrianboundary) 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-manifoldsadapted to the contact category. The discussion presented here rests on thework of Ding and Geiges [16, 17]. Suppose that K ⊂ (Y, ξ) is a Legendrianknot in a given contact 3-manifold. As we already saw, K comes with acanonical 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 Yr(K) also admits a contactstructure, we have to describe the surgery procedure a little more carefully.As the Legendrian neighborhood theorem shows, for some positive δ thereis a contact embedding f : (Nδ , ζ) → (Y, ξ) with f(C) = K where

Nδ =

(φ, x, y) | x2 + y2 ≤ δ⊂ S1 × R2,

ζ = cos(2πnφ) dx − sin(2πnφ) dy

andC =

(φ, x, y) | x = y = 0

,

see Example 5.1.23. Let

N2δ =

(φ, x, y) | x2 + y2 ≤ 2δ⊂ S1 × R2.

Now we will cut out f(Nδ) ⊂ Y and reglue N2δ by a diffeomorphism

g : (N2δ − intNδ) → (N2δ − intNδ)

which maps boundary to boundary and on N2δ − intNδ∼= T 2 × I it maps

the meridian µ to pµ + qλ. Such a map obviously exists on T 2, and thiscan be trivially extended to N2δ− intNδ. Considering the contact structureζ1 = (g∗)

−1(ζ) on N2δ − intNδ we need the following

Proposition 11.2.1. For p 6= 0 the contact structure ζ1 extends to a tightcontact structure ζ ′ on N2δ.

Proof. Using the identification given by

(p p′

q q′

)

with pq′ − p′q = 1 to glue S1 ×D2 back in, we need to choose the slope onthe solid torus to match up with the old longitude, which is isotopic to the


dividing curve. Recall that the slope of the boundary of the neighborhoodof a Legendrian knot can be assumed to be equal to ∞ by choosing thelongitude given by the contact framing. The meridian of S1 ×D2 will mapto pµ+ qλ. Computing the inverse of the above matrix, the inverse imageof the longitude turns out to be −p′µ + pλ, hence the slope of the tightcontact structure on S1×D2 we need should be equal to − p

p′ . According toTheorem 11.1.1 this boundary condition can be fulfilled by a tight contactstructure on S1 ×D2 once p 6= 0.

Now identifying w ∈ N2δ− intNδ with f(g(w)

)∈ Y −f(Nδ) we glue N2δ to

Y − f(Nδ) and get a manifold Y ′ with glued up contact structure ξ′. Fromthe construction it is clear that

Y ′ = Y pq(K).

The contact structure ξ′ depends on the choice of the extension of ζ1 to N2δ.In general this extension is not unique, but — as the classification given inthe previous section shows — uniqueness holds for p = 1. (For p = 1 we canchoose p′ = 0 and q′ = 1, hence we have to understand tight structures onS1×D2 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 orderto have a well-defined construction one needs to check that the resultingcontact structure ξ′ is (up to isotopy) independent of all the choices madethroughout the above gluing process. This is the content of

Theorem 11.2.2 (Ding–Geiges, [17]). If the extension ζ1 inN2 is fixed thenthe 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 toisotopy by the Legendrian knot K and q ∈ Z.

The construction above allows us to prove numerous classical results incontact topology. For example

Corollary 11.2.3 (Martinet, [107]). Every 3-manifold admits a contactstructure.

Proof. Every closed 3-manifold can be given as rational surgery on a linkin S3. Put the link into Legendrian position in (S3, ξst) and recomputethe framing coefficients with respect to the contact framing. The previousprocedure provides a contact structure on the desired 3-manifold. (Byadding zig-zags if necessary we can always avoid contact 0-framings.)


Remark 11.2.4. A refined version of this theorem will be discussed inProposition 11.3.15.

From the 4-dimensional point of view, integral surgeries are especiallyimportant, since these correspond to 4-dimensional 2-handle attachments.In this sense, contact (±1)-surgery produces both a 4-manifold and a uniquecontact structure on its boundary. Recall that for (−1)-surgery the result-ing cobordism admits a Stein structure as well. We have already considereda 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 bepresented as contact (±1)-surgery along a Legendrian link in (S3, ξst).

Theorem 11.2.5 (Ding–Geiges, [17]). For any closed contact 3-manifold(Y, ξ) there is a Legendrian link L = L+ ∪ L− ⊂ (S3, ξst) such that contactsurgery 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 ourobservation of Lemma 7.3.1. Therefore suppose that (Y, ξ) is a given contactmanifold, 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 thecontact manifold (Y ′, ξ′).

Lemma 11.2.6 (The Cancellation Lemma, Ding–Geiges [16]). The contact3-manifolds (Y, ξ) and (Y ′, ξ′) are contactomorphic. The contactomorphismcan be chosen to be the identity outside of a small tubular neighborhood ofthe Legendrian knot L.

Proof. The complete proof of this useful lemma relies on the solution ofthe following two exercises.

Exercises 11.2.7. (a) Computing contact Ozsvath–Szabo invariants verifythat 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 unknotand (−1)-surgery on its Legendrian push-off gives a tight contact structureξ on S3. (Hint: From (a) deduce that the result of the (+1)-surgery is Steinfillable, 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 aneighborhoodN of L containing L′. It is easy to see that the two surgeries do


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 alongL, verifying the statement. By the Legendrian neighborhood theorem thistightness can be checked on a model case, for example if L is the Legendrianunknot in (S3, ξst). In this case, however, Exercise 11.2.7 shows that theresult of (−1)-surgery on L and (+1)-surgery on L′ embeds into a tightcontact S3, hence it is tight, completing the proof.

Corollary 11.2.8. Suppose that L, L′ are Legendrian knots in a surgerydiagram for (Y, ξ) such that L′ is the contact push-off of L and there isa neighborhood of L disjoint from the rest of the diagram and containingL and L′ only. If we do (−1)-surgery on L and (+1)-surgery on L′ thenthe diagram given by the same link after deleting L and L′ yields the samecontact 3-manifold (Y, ξ).

Now we can begin the proof of Theorem 11.2.5. We will prove thistheorem in two steps. First we reduce the problem to the case of anovertwisted 3-sphere, and then in the next section we finish the proof byexplicit diagrams for those contact 3-manifolds.

Proof. (Reduction of Theorem 11.2.5 to an overtwisted S3.) Perform con-tact (+1)-surgery on the Legendrian knot L ⊂ (S3, ξst) pictured by Fig-ure 11.3. It is not hard to see that the result is an overtwisted structure ξ1

+1

Figure 11.3. Overtwisted contact structure on S3

on S3, see Exercise 11.2.10(a). Now consider (Y, ξ) and take the connectedsum with (S3, ξ1). The result is an overtwisted (Y, ξ2) which can be givenas 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 S3 by a topologi-cal surgery along a link, and since the complement of each knot in (Y, ξ2)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


the topological surgery. In conclusion, a sequence of contact (+1)-surgeriesturns (Y, ξ) into (S3, ξ′) with some contact structure ξ′. Adding one morecopy of (S3, ξ1) to the whole process, the resulting (S3, ξ′) can be assumedto be overtwisted. Now reversing the surgeries we get that contact (−1)-surgery on a Legendrian link in some overtwisted contact 3-sphere (S3, ξ′)yields (Y, ξ). In conclusion, once we have a surgery presentation for (S3, 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 Etnyreand Honda we can actually assume that the Legendrian link L ⊂ (S3, ξst)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) = (S3, ξ1) given by (+1)-surgery along the knot L of Figure 11.3 wehave the above sharpening of Theorem 11.2.5.

Exercises 11.2.10. (a) Show that the contact structure ξ1 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 framingon 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 slightlydifferent lines. In [17] they worked out a way for turning contact rationalsurgeries into contact (±1)-surgeries. Since this method is very useful in


applications, below we describe the algorithm — for the proof the readeris advised to turn to [17, 18]. Let us first assume that we want to performcontact r-surgery on the Legendrian knot L with r < 0. In this casethe surgery can be replaced by a sequence of contact (−1)-surgeries alongLegendrian knots associated to L as follows: suppose that r = −p

q and the

continued fraction coefficients of −pq are equal to [r0 + 1, r1, . . . , rk], with

ri ≤ −2 (i = 0, . . . , k). Consider a Legendrian push-off of L, add |r0 + 2|zig-zags to it and get K0. Push this knot off along the contact framing andadd |r1 + 2| zig-zags to it to get K1. Do contact (−1)-surgery on K0 andrepeat the process with K1. After (k + 1) steps we end up with a diagraminvolving only contact (−1)-surgeries. According to [17, 18] the result of thesequence 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 S1×D2 with the correct boundaryslope. This non-uniqueness is present in the sequence of (−1)-surgeries aswell: we have a freedom in adding the zig-zags in each step either on theright or on the left. It is not very hard to see that there are equally manychoices in both constructions.

The next proposition will guide us how to turn contact r-surgery with r > 0into a sequence of contact (±1)-surgeries.

Proposition 11.2.12 (Ding–Geiges, [17]). Fixt r = pq > 0 and an integer

k > 0. Then contact r-surgery on the Legendrian knot K is the same ascontact 1

k -surgery on K followed by contact pq−kp -surgery on the Legendrian

push-off K ′ of K.

By choosing k > 0 large enough, the above proposition provides a way toreduce 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 thealgorithm is complete once we know how to turn contact 1

k -surgery into(±1)-surgeries.

Lemma 11.2.13 (Ding–Geiges, [17]). Let K1, . . . ,Kk denote k Legendrianpush-offs of the Legendrian knot K. Contact 1

k -surgery on K is thenisotopic to performing contact (+1)-surgeries on the k Legendrian knotsK1, . . . ,Kk.

Exercises 11.2.14. (a) Verify that the above algorithm is correct on thetopological level, that is, the algorithm provides a surgery presentation of a3-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 notunique. Show that after applying the Cancellation Lemma 11.2.6 sufficientlymany 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 Steincobordism W between Y and Y ′ such that H1(Y

′; Z) = 0. (Hint: Start witha contact surgery diagram L of (Y, ξ) and for every knot Li in L consider aLegendrian knot Ki linking Li once, not linking the other knots in L, andhaving tb(Ki) = 1. Adding Weinstein handles along Ki we get W ; checkthat the resulting 3-manifold Y ′ after the handle attachment is an integralhomology sphere. For more details see [159].)

(d) Find an open book decomposition of #k(S1 × S2) 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 contactstructure 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− ⊂ (S3, ξst).Recall that integral surgery can also be regarded as (4-dimensional) 2-handleattachment to D4, hence the diagram represents a compact 4-manifold Xwith ∂X = Y . There is, however, an additional structure on X. It isfairly 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 Akbulutand the first author outlined in Section 10.2. (Also take the refinement ofPlamenevskaya [146] given in Proposition 10.2.4 into account.) Recall thatan achiral Lefschetz fibration on X naturally provides an open book de-composition obL on ∂X = Y . Next we would like to show that ξ (as theresult of contact (±1)-surgeries) on Y is compatible with this open bookdecomposition. Notice that this step will complete a portion of the proofof Giroux’s Theorem 9.2.11 about relating open book decompositions andcontact structures. Let ξL denote the contact structure (unique up to iso-topy by Part(a) of Theorem 9.2.11, see also Proposition 9.2.7) compatiblewith the open book decomposition obL. Our main result is now

Theorem 11.3.1. The contact structures ξ and ξL on Y are isotopic, henceξ is compatible with the open book decomposition obL defined above.


In the light of Theorem 5.2.1 of Torisu we would like to show that both ξand ξL admit the properties listed under (i) and (ii) of that theorem. Thisis obviously satisfied (and explicitly stated in [168]) for ξL, hence we onlyneed to verify them for ξ.

Lemma 11.3.2 ([160]). The restrictions ξ|Uito the handlebodies Ui

(i = 1, 2) of the Heegaard decomposition induced by the open book de-composition obL are tight. The dividing set of the convex surface Σ ⊂ Ywith respect to ξ is isotopic to the binding of the open book decomposition.

Proof. Consider the open book decomposition found on S3 induced bythe Lefschetz fibration D4 → D2 in the course of the algorithm presented inSection 10.2. Recall that this Lefschetz fibration is given by the factorizationof the monodromy of the (p, q) torus knot defined by the knot in squarebridge position. Since the monodromy of this open book decomposition isthe product of right-handed Dehn twists only, the corresponding contactstructure is isotopic to ξst. In addition, this open book decompositioninduces a Heegaard decomposition of S3, and the contact handlebodies ofthis Heegaard decomposition — since they are contained by the tight S3

— are tight. The Heegaard decomposition S3 = V1 ∪ V2 can be chosen insuch a way that L is contained in V1. Therefore (ii) of the assumptions ofTheorem 5.2.1 obviously holds, since surgery along L will not change theconvex surface ∂V1 = ∂V2, and the binding of the open book decompositionremains unchanged. We only need to check (i), that is, that the contactstructures ξ|Ui

are tight for i = 1, 2. By our choice U2 = V2 and ξ|U2= ξst|V2

,hence we only need to deal with ξ|U1

. Consider Legendrian push-offs for allLegendrian knots in L+ in such a way that these push-offs are in V2. Thiscan be done, since the contact framings of the knots in L coincide with thepage framing they inherit from the open book decomposition. Thereforea contact push-off can be assumed to lie on a page, and this page can bechosen to be in V2. 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|U1

. It is easy to see that (Y ′, ξ′) is tight: by theCancellation Lemma 11.2.6 it can be given by doing (−1)-surgery alongL− ⊂ (S3, ξst), therefore (Y ′, ξ′) is Stein fillable, hence tight. Since ξ|U1 iscontained 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 ξL and ξsatisfy conditions (i) and (ii) of Theorem 5.2.1, hence the theorem impliesthat ξ and ξL are isotopic.


Remark 11.3.3. A similar theorem was proved by Gay [53] in the casewhen no (+1)-surgeries are present in the picture.

Now Theorem 11.3.1 allows us to find open book decompositions for allcontact structures given by contact (±1)-surgery diagrams. Notice also thatwe just proved that every 3-manifold admits an open book decomposition:presenting Y as the boundary of a 4-dimensional handlebody with a unique0-handle and some 2-handles, we get a contact surgery diagram of Y withsome contact structure. Turn this diagram into (±1)-surgeries and apply theabove theorem to find an open book decomposition on Y . (This operationwill change the 4-dimensional handlebody, though.) As an easy applicationwe show that

Proposition 11.3.4. Contact (+1)-surgery on the Legendrian unknot pro-vides a tight structure on S1 × S2.

Proof. After performing the algorithm given in Section 10.2 we get anachiral Lefschetz fibration X → D2 with fiber diffeomorphic to the Seifertsurface of the (2, 2) torus knot, i.e., the annulus A. The 4-manifold Xis built from the Lefschetz fibration D4 → D2 by attaching a 2-handlealong 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 twisttC along C, the total monodromy of the induced open book decompositionon ∂X = S1 × S2 is equal to tC · t−1

C = 1. The reason for the negativeexponent 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


on the Legendrian unknot is compatible with the open book decompositiondefined by the identity element 1 ∈ ΓA. Corollary 9.2.15 now implies that itis Stein fillable, hence the proof is complete. In addition, by the classificationof tight contact structures on S1 × S2 the above argument also shows thatthe 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 openbook decompositions. (Hint: Notice that the curve L′ can be given as thepush-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 Ozsvath–Szabo invariants.(Hint: Use Lemma 14.4.10.) Prove tightness for the contact structure givenby (+1)-surgery along the k-component Legendrian unlink.

(c) Show that any solid genus-g handlebody admits a contact structurewhich can be embedded into a Stein fillable structure on a closed 3-manifold.

Our next application concerns computability of homotopic invariants ofcontact structures on a 3-manifold Y . Recall form Chapter 6 (cf. also [65])that two oriented 2-plane fields ξ1 and ξ2 on Y are homotopic if and onlyif their induced spinc structures tξi and 3-dimensional invariants d3(ξi) areequal. If c1(tξ) is nontorsion then d3(ξ) does not admit a Q-lift, but forc1(tξ) torsion, this latter invariant can be lifted to Q and can be computedas

1

4

(c21(Xi, Ji) − 3σ(Xi) − 2χ(Xi)

)

where (Xi, Ji) are almost-complex 4-manifolds with ∂Xi = Y such that theoriented 2-plane fields of complex tangencies of Ji along ∂Xi are homotopicto ξi. The surgery picture together with Theorem 11.3.1 easily provides sucha 4-manifold X: Suppose that (Y, ξ) is given by (±1)-surgery on L = L+ ∪L− ⊂ (S3, ξst), and let X1 denote the 4-manifold defined by the diagram.As explained in Section 10.2, X1 admits an achiral Lefschetz fibrationstructure. Consider the oriented 2-plane field of tangents of fibers awayfrom the set C of critical points of the fibration. By taking the orthogonalcomplement for some metric, this oriented 2-plane field provides an almost-complex structure on X1 − C: define J as counterclockwise 90 rotationon these planes. This almost-complex structure obviously extends throughthose points of C which admit orientation preserving complex charts —just use the local model. At points of C with oppositely oriented coordinatecharts (corresponding to contact (+1)-surgeries) the two branches of the


oriented singular fiber provide an orientation for X incompatible with theone originally fixed. The obstruction for extending J through such pointscan be computed using a local model, as explained in [66, Lemma 8.4.12] orin [160]. In conclusion, for these points of C we need to take the connectedsum of X1 with CP2 with its standard complex structure for extending thealmost-complex structure defined on X1 −C. Consequently X = X1#qCP2

with the extended almost-complex structure is a good choice of (X,J) forthe given contact structure (Y, ξ). Here q denotes the number of componentsin L+. By repeating the proof of [65, Proposition 2.3] verbatim (see alsoProposition 8.2.4) we get

Theorem 11.3.6 (Gompf, [65]). The first Chern class c1(X,J) ∈ H2(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 alongY = ∂X, the cohomology class c1(ξ) is equal to the restriction of the abovec1(X,J) to ∂X. The class c1(X,J) is specified by Theorem 11.3.6, and thedescription of H1(Y ; Z) in terms of a surgery diagram then provides c1(ξ).Note that here X is simply connected, hence the spinc structure sJ inducedby J is specified by c1(X,J). In this way the induced spinc structure tξ isgiven as sJ |∂X . If c1(ξ) ∈ H2(Y ; Z) is torsion, then for appropriate n ∈ N theclass PD

(nc1(X,J)

)∈ H2(X,∂X; Z) is the image of a class α ∈ H2(X; Z),

hence c21(X,J) can be computed as 1n2α

2 ∈ Q as discussed in Section 6.3.Notice also that both c1(X1 − C, J) and the induced spinc structure sJ

extend uniquely through the points of C, hence for practical purposes wecan work with this extended cohomology class c ∈ H2(X1; Z), although it isnot the first Chern class of any almost-complex structure. When computingd3(ξ), this fact results a correction term in the formula. In conclusion, forξ with torsion induced spinc structure tξ all terms in the formula for d3(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) isgiven by contact (±1)-surgery along the link L = L+ ∪ L− ⊂ (S3, ξst).Let X1 denote the 4-manifold defined by the diagram and suppose thatc ∈ H2(X1; Z) is given by c

([ΣK ]

)= rot(K) on [ΣK ] ∈ H2(X1; Z), where

ΣK is the surface corresponding to the surgery curve K ⊂ L. If therestriction c|∂X1

to the boundary is torsion and L+ has q components then

d3(ξ) =1

4

(c2 − 3σ(X1) − 2χ(X1)

)+ q.


Proof. Since σ(X1) = σ(X1 − x1, . . . , xk

)and χ(X1) = q + χ

(X1 −

x1, . . . , xq)

for the critical points x1, . . . , xq of the achiral Lefschetzfibration X1 → D2 which lie on incorrectly oriented coordinate charts, theformula 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 ofthe achiral Lefschetz fibration or the open book decomposition providedby the surgery diagram. As shown in Chapter 8, the complex structure ofD4 (inherited from C2) extends to all the 2-handles attached with contactframing −1. We do not have such an extension for the contact (+1)-framed2-handles, but there is no obstruction to finding an appropriate almost-complex structure on these handles away from a point. In conclusion, wehave an almost-complex structure on X1 − x1, . . . , xq where q is the car-dinality of L+ — the knots on which we do contact (+1)-surgery. Since aspinc structure (like a 2-cohomology element) extends through a point in a4-manifold, we have a spinc structure s on X1 extending the spinc structuretξ ∈ Spinc(Y ) induced by ξ. We want to determine c1(s) on the homo-logy classes given by the Legendrian knots in L. For this computation, fixL ⊂ (S3, ξst), perform contact (+1)-surgery on it and consider the resulting4-manifold XL with spinc structure sL ∈ Spinc(X). Let k denote the valueof c1(sL) on a generator for H2(XL; Z). (To be precise, we need to fix anorientation for L, which provides a canonical generator for H2(XL; Z) ∼= Z.)Define u as the obstruction to extending the almost-complex structure fromXL − pt. to XL, i.e., the 3-dimensional invariant of the oriented 2-planefield induced on the boundary S3 of the neighborhood of the point is u.

Proposition 11.3.8 ([18, 99]). If tb(L) 6= 0 then k = rot(L) and u = 12 .

Proof. Consider 2n Legendrian push-offs of L and call them L1, . . . , Lnand L′

1, . . . , L′n. Do contact (−1)-surgeries along Li and (+1)-surgeries

along L′i. According to the Cancellation Lemma 11.2.6 the result is (S3, ξst)

again. On the other hand, simple homological computation shows that the3-dimensional invariant of the result of the surgery is

1

4(n(k2 − rot(L)

)− n2 tb(L)

(k − rot(L)

)2 − 2) + n

(u− 1

2

)− 1

2.

Since d3(S3, ξst) = −1

2 , the above expression implies u = 12 and k = rot(L)

provided tb(L) 6= 0.


Remark 11.3.9. In fact, we need to use the above expression for n = 1and n = 2 only to draw that conclusion. Note that since u can be easilyshown to be independent of L, for tb(L) = 0 the above argument givesk = ± rot(L); a more detailed study of the almost-complex structure onXL−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) 6= 0 case is sufficient.

The formula of Theorem 11.3.7 above gives us a way to distinguishcontact structures given by surgery diagrams on a fixed 3-manifold. Forexample, let L be n unlinked copies of the knot given by Figure 11.3, andtake (Y, ξn) to be (+1)-surgery on L. Simple computation verifies

Lemma 11.3.10. Y = S3 and d3(ξn) = n− 12 .

Proof. By turning the contact framing coefficients to Seifert framings, wesee that Y is given by (−1)-surgery on the n-component unlink, hence we canblow all surgery curves down, showing that Y = S3. The corresponding 4-manifold X1 therefore has σ(X1) = −n, χ(X1) = n+1 and since L = L+, wehave q = n. Easy computation shows that c2 = −n; by plugging these valuesinto 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 11.6provide a sequence of contact structures ξ−n on S3 with d3(ξ−n) = −n− 1

2 .

Proof. Figure 11.6 shows that the manifold we get after the surgery isdiffeomorphic to S3. Application of the formula for the 3-dimensionalinvariant d3 now implies the result.

Since S3 admits a unique tight contact structure ξst and d3(ξst) = −12 , the

contact structures ξn for n ∈ Z−0 encountered above are all overtwisted.

Exercises 11.3.12. (a) By finding the overtwisted disks show directly thatthe contact structures (S3, ξn)

(n ∈ Z−0

)of the above two Lemmas are

overtwisted.

(b) Find an overtwisted contact structure ξ0 on S3 homotopic to ξst. (Hint:Take the connected sum of ξ1 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 spinc

structures induced by the contact structures. Verify that reflection inducesa contactomorphism.)

(d) Find open books compatible with the contact structures given by Fig-ures 11.7(a) and (b).


+1

−1

smoothly

−5

−1

−1

−1

Figure 11.6. The contact 3-manifold (S3, ξ−1)

−1 −1

(a) (b)

Figure 11.7. Contactomorphic, nonisotopic contact structures


The map (S3, ξ) 7→ d3(ξ) + 12 gives a bijection between the space of over-

twisted contact structures and Z. To see this we only need to verify thatfor any oriented 2-plane field ξ on S3 the quantity d3(ξ) + 1

2 is an integer.Recall that

d3(ξ) =1

4

(c21(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

(c21(X,J) − σ(X)

)is an even integer since c1(X,J) is

a characteristic vector, and 12

(σ(X) + χ(X)

)= b+2 (X) − 1

2 . Therefore wehave

Corollary 11.3.13. The above lemmas together with Exercise 11.3.12(b)show surgery diagrams for all overtwisted contact structures ξn (n ∈ Z) onthe 3-sphere.

Notice that this corollary concludes the proof of Theorem 11.2.5.

Exercise 11.3.14. Consider a Legendrian knot L ⊂ (S3, ξst) and its Leg-endrian push-off L′. Stabilize L′ twice to get L1 and perform contact (+1)-surgery on L and L1. Determine the resulting 3-manifold Y and computed3(ξ) for the resulting contact structure ξ. (Hint: See [19].)

Following similar lines, in fact, we can produce surgery diagrams for allovertwisted contact structures on any 3-manifold presented by a surgerydiagram. This presentation (given in [18]) provides a new proof of a classicalresult 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 structurehomotopic to ξ. The contact structure can be chosen to be overtwisted.

Exercises 11.3.16. (a) Let L0 ⊂ (S3, ξst) be the Legendrian unknot and L1

another Legendrian unknot linking it k times (k ∈ Z). Add two zig-zags tothe Legendrian push-off L′

2 of L1 and get L2. Perform contact (+1)-surgeryon L0, L1 and L2. Prove that the resulting manifold is diffeomorphic toS1 ×S2. Determine the spinc structure of the resulting contact structure ξ.

(b) Using Exercise 11.3.14 and the above result verify Proposition 11.3.15for S1 × S2.

(c) Prove Proposition 11.3.15 in general. (Hint: See [18].)


Recall that according to Eliashberg’s result, isotopy classes of overtwistedcontact structures and homotopy classes of oriented 2-plane fields are inone-to-one correspondence. Therefore the solution of Exercise 11.3.16(c)provides a surgery diagram for any overtwisted contact structure on a closed3-manifold.

−1

.....

.....+1

−1 −1 −1

m

−1

Figure 11.8. Contact structure ξ−(2m+1) on S3

Exercise 11.3.17. Show that the contact surgery diagram depicted inFigure 11.8 gives a contact structure on S3 with d3 = 1

2 − 2(m + 1),where m ≥ 0 is the number of unknots in the figure with vanishing rotationnumber. (Hint : Compute d3 and use the classification of overtwisted contactstructures.) Notice that this surgery diagram represents some overtwistedcontact structures on S3 using unknotted surgery curves and only one (+1)surgery curve. This example also illustrates (Stein) cobordisms betweenvarious 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 typesof fillings, and give a family of tight, nonfillable contact structures. Theconstruction of these latter examples utilizes contact surgery, while tightnessis proved by computing contact Ozsvath–Szabo invariants (see Chapter 14).In the last section we will concentrate on topological restrictions a contact3-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|ξdoes 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 isthe one compatible with ξ.) (Y, ξ) is strongly symplectically fillable if it isthe ω-convex boundary of a compact symplectic manifold (W,ω). In otherwords, ω is exact near the boundary and its primitive α (i.e., a 1-form withdα = 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 dilatingvector field for the boundary (defined near ∂X) pointing outwards. (Y, ξ)is holomorphically fillable if there is a compact complex surface (X,J) suchthat 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-convexboundary of a Stein surface.

202 12. Fillings of contact 3-manifolds

Remarks 12.1.2. (a) Without imposing the compactness condition onW , the above definition of (weak or strong) symplectic fillability would besatisfied by all closed contact 3-manifold (Y, ξ): Consider simply Y × (0, 1]equipped with the symplectic structure it inherits from the symplectizationof (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 aStein filling. Therefore the two last notions of fillability in Definition 12.1.1are the same.

(c) Notice that holomorphic/Stein fillability implies strong fillability, whichin turn implies weak fillability. For a related discussion on various fillabilitynotions 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 ofChapter 7 we can attach Weinstein handles to a strong symplectic fillingalong Legendrian knots in a way that the symplectic structure extends tothe handle and the new symplectic 4-manifold strongly fills its boundary.In this gluing process, however, the symplectically dilating vector field isused only in a neighborhood of the attaching circle. It turns out that ifL ⊂ (Y, ξ) is Legendrian and (W,ω) is a weak filling of (Y, ξ) then there isalways 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 notstrongly fillable: for example, the contact tori (T 3, ξn) with n ≥ 2 all havethis property [29]. (For an even larger collection of such contact 3-manifoldssee [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 admita Stein structure anymore, but such an operation does not affect fillabilityproperties of the boundary contact 3-manifold.

Example 12.1.4. The Legendrian surgery diagram of Figure 12.1 gives astrong (in fact, Stein filling) of the boundary of the nucleus Nn with theinherited contact structure.

Suppose that (W,ω) is a weak filling of (Y, ξ). It is obvious that if ω is notexact 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−zagsn−k

zig−zagsk

−1

−1

Figure 12.1. Stein structure on the nucleus Nn

becomes a strong filling, see [30]. In the special case of rational homologyspheres therefore we have

Theorem 12.1.5 (Ohta–Ono, [127]). Suppose that b1(Y ) = 0. The sym-plectic structure ω on a weak symplectic filling W of any contact structureξ on Y can be extended to W ∪ Y × [0, 1] to a strong filling of (Y, ξ). Inconclusion, a contact structure on a rational homology sphere Y is weaklyfillable if and only if it is strongly fillable.

According to a recent result of Eliashberg [30] a weak filling can besymplectically embedded into a closed symplectic 4-manifold. This theoremturned 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 beembedded into a closed symplectic 4-manifold.


Proof. Consider a surgery presentation L = L+ ∪ L− ⊂ (S3, ξst) 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 theknots 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 manifoldcan 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,ωZ) 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 theclosed 4-manifold U = (Z − intX) ∪Y ′ W ′. Since (W,ω) is a symplecticsubmanifold of (W ′, ω′), this provides a symplectic embedding of (W,ω) intothe closed symplectic 4-manifold U . Notice that by adding more Weinsteinhandles 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 symplecticfilling follows by a trivial argument.

Theorem 12.1.7 (Eliashberg, [30, 42]). If (W,ω) is a weak symplecticfilling of (Y, ξ) then (W,ω) embeds symplectically into a closed symplectic4-manifold (U,ωU ).

Proof. According to Exercise 11.2.14(c) the weak symplectic filling embedsfirst 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 tomodify ω′ near ∂W ′ to achieve that the new symplectic form ω1 providesa strong symplectic filling of (Y ′, ξ′). The application of Theorem 12.1.6now provides a symplectic embedding of (W ′, ω1) into a closed symplectic4-manifold, and since (W,ω) is a symplectic submanifold of (W ′, ω1), theproof is complete.

It is still a question of central importance in contact topology whether agiven contact structure is fillable or not (in any of the above sense) andwhich 3-manifolds support fillable contact structures.

The previous chapters provided a very powerful topological tool for con-structing Stein manifolds: attach 2-handles to nS

1 × D3 along a Leg-endrian link with framing −1 relative to the contact framing. (Here∂(nS

1 ×D3) = #nS1 × S2 is equipped with its unique tight contact struc-

ture.) In fact, every Stein domain can be given in this way. This approachhas 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; r1, . . . , rk) with one of its orientations admits a Stein fillable contactstructure. If g ≥ 1 then M admits Stein fillable contact structures witheither of its orientations.

According to a result of Eliashberg, Stein fillability needs to be determinedonly for prime 3-manifolds:

Proposition 12.1.9. The connected sum (Y1, ξ1)#(Y2, ξ2) is Stein fillableif and only if both (Yi, ξi) are Stein fillable.

According to a theorem of Eliashberg and Gromov, a fillable contactstructure (in any of the above sense) is tight.

Theorem 12.1.10 (Eliashberg–Gromov). A weakly symplectically fillablecontact 3-manifold (Y, ξ) is tight.

Proof (sketch). Let (W,ω) be a symplectic filling of (Y, ξ) and supposethat (Y, ξ) contains an overtwisted disk. Choose a disk D with Legendrianboundary and the property that tb(∂D) = 2, and attach a Weinstein handlealong ∂D to the weak filling (W,ω). The resulting 4-manifold W ′ will be aweak symplectic filling of the surgered contact 3-manifold (Y ′, ξ′) containinga sphere with self-intersection (+1). Now embed (W ′, ω′) into a closedsymplectic 4-manifold U with b+2 (U) > 1. The adjunction inequality ofTheorem 13.3.3 for the sphere of positive self-intersection now provides thedesired contradiction.

Remark 12.1.11. The first proof of the above theorem is due to Eliashbergand 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 hardto find counterexample to this expectation, since the only tool for provingtightness of a given (Y, ξ) was to show that it is fillable. The state traversalmethod tightness, and this method led to the discovery of the first tightbut not fillable contact structures [44]. The introduction of Ozsvath–Szaboinvariants then gave a very effective way for examining tightness propertiesof contact structures on closed manifolds, leading to a plethora of examplesof tight nonfillable contact structures.


12.2. Nonfillable contact 3-manifolds

Not all contact structures are fillable and there are examples of 3-manifoldswhich do not admit any symplectic fillings.

Theorem 12.2.1 (Lisca, [94]). The Poincare homology 3-sphere with itsnatural orientation reversed admits no fillable contact structure.

Proof. The Poincare homology sphere is diffeomorphic to the Brieskornsphere 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 aclosed symplectic manifold (as it is given in Theorem 10.3.1). The fact thatΣ(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 E8-plumbing given by the plumbing graph of Figure 1.5 then W ∪ (−E) isa negative definite closed 4-manifold with nonstandard intersection form,contradicting Donaldson’s famous diagonalizability result [20]. ThereforeW cannot exist.

Exercise 12.2.2. Show that if(Zn, (−En)

)(n = 6, 7, 8) is a sublattice of a

negative definite lattice (Zk, Q) then Q cannot be diagonal. (Hint: Noticethat −E6 is contained in all these lattices. For a solution see [95].)

A similar argument shows that the boundary of the positive definite E6-and E7-plumbing cannot be the boundary of a Stein domain. Notice thatin the light of Proposition 12.1.9 we have many 3-manifolds which arenot boundaries of Stein domains — just take connected sum with one ofthe above mentioned nonfillable manifolds. For example, the 3-manifoldΣ(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] theoriented 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 givenby 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 thatY supports no fillable contact structures, hence Figure 1.6 must define anonfillable structure. In the proof of tightness we will make use of thecontact Ozsvath–Szabo invariants. For an overview of these invariants and

12.2. Nonfillable contact 3-manifolds 207

Ozsvath–Szabo homology see Appendix 14; here we will freely use the resultdiscussed there. Recall that HF (Y ) denotes the Ozsvath–Szabo homology

group of the closed, oriented 3-manifold Y , while c(Y, ξ) ∈ HF (−Y ) is thecontact invariant of (Y, ξ).

Proposition 12.2.3. The contact 3-manifold (Y, ξ) given by Figure 1.6 hasnonvanishing contact Ozsvath–Szabo 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 asFW(c(S3, ξst)

), where W is the cobordism of the handle attachment with

reversed orientation. Therefore injectivity of FW gives the nonvanishingof the invariant. The cobordism W can be given by a single 2-handleattachment along the left-handed trefoil knot with framing −2. Denotethe left-handed trefoil by T . Then the surgery exact triangle reads as

HF (S3) HF(S3−2(T )

)

HF(S3−1(T )

)

FW

Since HF(S3−n(T )

)= HF

(S3n(T )

), the genus of T is 1 and S3

5(T ) is a

lens space, Propostion 14.3.5 implies that dim HF (S3−n(T )) = n, hence the

above triangle translates to

Z2 Z2 ⊕ Z2

Z2

FW

therefore exactness implies the injectivity of FW , concluding the proof.

Exercises 12.2.4. (a) Show that S35(T ) is a lens space. (Hint: Use the

presentation of S31(T ) as plumbing on the positive definite E8-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 structureon the boundary of the positive definite E6-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 theboundary of the positive definite E6-plumbing using Kirby calculus.)


(c) Show that if (YK , ξK) is given by contact (+1)-surgery on (Y, ξ) alonga Legendrian knot K and (Y, ξ) is not fillable then (YK , ξK) is not fillableeither. (Hint: Remember that contact (+1)-surgery along a Legendrianknot 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(YK , ξK), 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(Yk, ξk). According to the above exercise these structures are all nonfillable.

Exercise 12.2.5. Show that Yk can be given by the surgery diagram ofFigure 12.2. Conclude that

∣∣H1(Yk; Z)∣∣ = k + 1. (Hint: Convert the

copiesk

......

k1

+1

+1

...

Figure 12.2. Tight, nonfillable contact 3-manifold (Yk, ξk)


Legendrian surgery diagram into a smooth diagram and slide the trefoilsover each other.)

Proposition 12.2.6 ([101]). The contact Ozsvath–Szabo invariant of(Yk, ξk) 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 isjust (S3, ξst) and for k = 1 we can apply Proposition 12.2.3. Notice that(Yk+1, ξk+1) is given as contact (+1)-surgery on (Yk, ξk), giving rise to a map

FW : HF (−Yk) → HF (−Yk+1) with the property that FW(c(Yk, ξk)

)=

c(Yk+1, ξk+1). As the surgery diagram of Figure 12.3 shows, the thirdmanifold in the corresponding surgery triangle is S3

−1(T ) again. Since

dim HF (−Yk) ≥∣∣H1(Yk; Z)

∣∣ = k + 1 and HF (−Y0) = Z2, the triangle

HF (−Yk) HF (−Yk+1)

Z2

FW

shows that HF (−Yk) = Zk+12 for all k ∈ N and FW is injective. Therefore

by induction c(Yk+1, ξk+1) 6= 0, finishing the proof. For related results see[101].

The above results might give the impression that nonfillability can followonly from some strong topological properties of the underlying 3-manifold,and nonfillability must hold for all contact structures on a given manifoldat the same time. Below we discuss a family of examples of 3-manifoldsadmitting both fillable and tight nonfillable structures. Let Yn,g → Σg

denote the circle bundle with Euler number n over the genus−g surfaceΣg. Honda [77] gave a complete classification of tight contact structures onthese 3-manifolds. He showed that all tight structures are fillable, with theexception of one for n = 2g > 0 and two for n > 2g > 0. Using Seiberg–Witten gauge theory it has been verified that these exceptional structuresare, in fact, nonfillable:

Theorem 12.2.7 ([99]). Suppose that g > 0 and n ≥ 2g. Then thevirtually overtwisted contact circle bundles Yn,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


−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

1k+

Figure 12.3. Kirby calculus in the surgery triangle

sequence of exercises we show a proof of Theorem 12.2.7 in the simplestpossible case: when g = 1 and n = 2. Then we show some examples of tightnonfillable structures on the type of Seifert fibered 3-manifolds for whichthe classification result of Ghiggini holds.

Exercises 12.2.8. (a) Show that the surgery diagram of Figure 12.4 givesa contact structure on Y2,1. (Hint: Turn contact surgery coefficients intoSeifert framings, put dots on the 0-framed unknots and compare the resultwith the diagram of Figure 2.11. In doing so one might need to apply thetransformation of Figure 10.8.)


+1

+1

+1

Figure 12.4. Tight nonfillable contact circle bundle

(b) Verify that ξ is nonfillable. (Hint: Consider the diagram withoutthe two Legendrian unknots. Verify that it gives a contact structure on−Σ(2, 3, 4). Finally show that contact (+1)-surgery on a nonfillable contactstructure produces a nonfillable structure, cf. Exercise 12.2.4(c).)

(c) By computing the contact Ozsvath–Szabo invariants of the contactstructure defined by the surgery diagram of Figure 12.4, show that it istight. (Hint: Use Lemma 14.4.10 and the result of Exercise 14.3.11(c).)


(d) Verify that Figure 12.5 gives the same contact structure as definedby Figure 12.4. (Hint: Show that the neighborhood of K in Figure 12.5containing the linking Legendrian knots K1 and K2 remains tight after thesurgeries on K1 and K2. Since it is glued with the same framing as (+1)-surgery on K, the solution follows from uniqueness of contact surgery withcoefficient of the form 1

k .)

K 2

1K

K

+1

+1

+1

+1

−1

Figure 12.5. Another surgery diagram for the same structure as in Figure 12.4


(e) Show that the contact structures ξα (2 ≤ α integer) defined by Fig-ure 12.6 on the 3-manifolds Yα are all tight. Notice that there are α manychoices to turn the (−α)-surgery into a (−1)-surgery by adding zig-zags tothe 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 withinduced spinc structure tξ ∈ Spinc(Y ) such that the Seiberg–Witten modulispace MY (tξ) is a smooth manifold consisting of reducible solutions only.Then any weak symplectic filling W of (Y, ξ) satisfies b+2 (W ) = 0 and themap H2(W ; R) → H2(∂W ; R) induced by the inclusion ∂W ⊂W is zero.

Remark 12.2.10. The crux of the argument is that with such moduli spacethe Seiberg–Witten equations over the 3-manifold admit a perturbationwith no solutions, and such perturbation can be extended to the symplecticfilling unless the topological properties listed in the theorem hold for thefilling W . But an extension would imply vanishing SW(W,ξ)-invariants for aweak symplectic filling, contradicting Theomem 13.2.2 of Kronheimer andMrowka.

Exercise 12.2.11. Determine the spinc structure induced by ξα.

By applying results of Mrowka, Ozsvath and Yu [123] the solution of theabove exercise can be used to verify that the assumptions of Theorem 12.2.9do hold for the contact structures ξα. Notice that the surgery descriptioninvolves several contact (±1)-surgeries and one contact (−α)-surgery. Thislatter surgery, however, is not unique. By introducing zig-zags on thecorresponding 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: Recalldefinitions from Section 2.3 and perform handleslides. Notice that all thesurgery 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 contactstructures of Figure 12.6 with that fixed α there are at least n noncontac-tomorphic. (Hint: Determine c1 of the resulting contact structures with


+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 −Yα is the boundary of a negative definitemanifold with intersection form containing −E8. (Hint: Embed the 4-manifold given by the surgery diagram into a blown-up CP2 and computethe intersection form of the complement.)

Theorem 12.2.13. The contact structures ξα defined by Figure 12.6 aretight and nonfillable.

Proof. The same idea as in the proof of Theorem 12.2.1 now shows thatthe contact structures ξi (i = 1, . . . , α − 1) given by Figure 12.6 on Yα arenonfillable. Since these structures can be given as contact (−1)-surgeryon the contact structure given by Figure 12.5 and this latter structure hasnonvanishing contact Ozsvath–Szabo invariants, tightness of (Yα, ξi) followsfrom 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 Yn with atleast n pairwise noncontactomorphic tight contact structures, none of themweakly symplectically fillable.

Notice that by the work of Gompf all these manifolds admit Stein fillablecontact structures. For related results see [101].

12.3. Topology of Stein fillings

We switch perspective now, and instead of examining fillability properties of3-manifolds, we study topological properties of the fillings. The motivatingproblem of this section can be summarized as:

Problem 12.3.1. Fix a contact 3-manifold (Y, ξ) and describe all Steinfillings of (Y, ξ).

Remark 12.3.2. Similar questions for weak (or strong) fillings are notexpected 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 withself-intersection 0) then by taking symplectic normal sums we can change thetopology of the filling drastically. In some cases (when no such submanifoldsare 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 surfacesingularities, see Remark 12.3.8 below.


Let us begin the study of Stein fillings by a simple observation. If Wis a Stein filling of Y then π1(Y ) → π1(W ) is surjective since W can bebuilt on Y × [0, 1] by attaching 2-, 3- and 4-handles only; in particularb1(W ) ≤ b1(Y ). In the following we will list contact 3-manifolds for whichStein fillings have been determined (up to diffeomorphism). We start witha famous result of Eliashberg.

Theorem 12.3.3 (Eliashberg). A Stein filling of S3 with its standardcontact structure ξst is diffeomorphic to D4.

Proof. Let us fix a Stein filling W of S3. By considering a neighborhoodof a point p ∈ CP2 together with a Liouville vector field and using thesymplectic cut-and-paste operation we get that Z = W ∪S3 (CP2 −D4) is asymplectic 4-manifold. Notice that CP2−D4 and so Z contains a symplecticsphere 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 wealso get that π1(W ) = 1, therefore Z is homotopy equivalent, hence (bya theorem of McDuff) symplectomorphic to CP2. In CP2, however, twosymplectic spheres representing the generator of H2(CP2; Z) are isotopic,showing that W is diffeomorphic to CP2 − CP1 = D4.

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 ξst itinherits from (S3, ξst) admits a unique (up to diffeomorphism) Stein fillingfor p 6= 4, which can be given as (−1)-surgery on a Legendrian unknot withtb = p−1 and rot = 2−p. For other L(p, q)’s (still with the quotient of thestandard contact structure (S3, ξst)) Lisca [96] gave a complete descriptionof Stein fillings — in general, however, uniqueness fails to hold.

Exercises 12.3.4. (a) Verify that the boundary of the Kirby diagram ofFigure 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 theresulting 4-manifold is the complement of a quadric in CP2. Equip this4-manifold with a Stein structure.

(c) Show that both the disk bundle over S2 with Euler class −4 and thecomplement of the quadric in CP2 provide Stein fillings of L(4, 1) for somecontact structures. By determining their homotopy types, show that thetwo contact structures coincide. In fact, the above two distinct examplescomprise a complete list of Stein fillings for (L(4, 1), ξst) up to diffeomor-phism.


−10

Figure 12.7. 4-manifold with lens space boundary

Remark 12.3.5. For the fixed lens space L(p, q) consider the continuedfraction expansion [b1, . . . , bk] of p

p−q . Elements of the set

Zp,q =

(n1, . . . , nk) ∈ Zk | [n1, . . . , nk] = 0 and 0 ≤ ni ≤ bi

give rise to Stein fillings of (L(p, q), ξst) as follows: First consider a linearchain of k unknots framed by ni (providing W with ∂W = S1 ×S2) and onthat do Legendrian surgery on unknots linking the circles of the chain —there are bi − ni such circles linking the circle with framing ni. Surgeringout the 4-manifold W given by the linear chain, i.e., replacing W with a0-handle and a 1-handle we get a Stein filling of L(p, q). The determinationof the homotopy type of the contact structure on the boundary shows thatwhat we constructed are fillings of ξst (cf. the classification result of Hondain Section 11.1). Now Lisca proves that any Stein filling of this contact lensspace is diffeomorphic to one of the manifolds constructed above. This laststep is carried out by embedding a filling into a rational surface and showingthat the complement is standard, similar to the argument presented in theproof of Theorem 12.3.3. For further details see [96].

Using the same main ideas as above, Ohta and Ono described Stein fillingsof links of simple and simple elliptic singularities (again with specific contactstructures). All these fillings happened to have b+2 = 0 and could beembedded into rational or ruled surfaces. In particular:

Theorem 12.3.6 (Ohta–Ono, [127]). The Poincare homology 3-sphereΣ(2, 3, 5) (with its contact structure inherited from S3) admits a unique (upto diffeomorphism) Stein filling which is the negative definite E8-plumbing.The same uniqueness holds for the boundary of the negative definite E6-and E7-plumbings.


Remark 12.3.7. In [127] it was shown that simple (or ADE) singularitieswith the contact structure given by the link of the singularity admit a uniqueStein filling (up to diffeomorphism). For a simple elliptic singularity Lk —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 singularityit has been proved [128] that (i) Lk admits a strong symplectic filling Xwith c1(X) = 0 if and only if 0 < k ≤ 9 and such X (which can be regardedas the generalization of smoothing) is unique up to diffeomorphism unlessk = 8, when there are two possibilities, and (ii) for k ≥ 10 a minimalstrong symplectic filling is unique up to diffeomorphism, and such a fillingis diffeomorphic to the minimal resolution. Finally for k ≤ 9 a minimalfilling is diffeomorphic either to the minimal resolution or to a smoothing(i.e., a filling with c1 = 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 ofthe 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 strongfillings the same cut-and-paste argument works.

The key common feature of the above results is that in each case anyfilling can be embedded into a closed symplectic 4-manifold with κ = −∞.In particular, all the above fillings have b+2 = 0. Next we will describesome particular cases when the above approach fails (for example, becauseof the existence of fillings with b+2 > 1). Using ad hoc arguments ofembedding Stein fillings of T 3 and ±Σ(2, 3, 11) into homotopy K3-surfacesone gets strong constraints on the topology of such Stein manifolds [158].These methods, however, seem to be insufficient in greater generality. Forexample, the contact structures on the 3-torus T 3 have been classified byKanda and Giroux [62, 79] by showing that any (T 3, ξ) is contactomorphicto one of (T 3, ξn) where ξn = ker

(cos(2πnt) dx + sin(2πnt) dy

)(n ≥ 1).

Using delicate results of Gromov, Eliashberg showed [29] that (T 3, ξn) is notstrongly fillable once n ≥ 2. For n = 1, Figure 12.8 gives a Stein filling of(T 3, ξ1). Using a version of the cut-and-paste argument outlined above, onecan show

Proposition 12.3.9 ([158]). If W is a Stein filling of (T 3, ξ1) then W ishomeomorphic to T 2 ×D2.


0

Figure 12.8. Stein structure on D2× T 2

For the understanding of Stein fillings of ±Σ(2, 3, 11) the Ozsvath–Szabohomology groups of these manifolds seem to play a crucial role. First wediscuss a slightly more general result, since by embedding Stein fillings intosymplectic 4-manifolds and applying product formulae for Ozsvath–Szaboinvariants, one can show

Theorem 12.3.10. Suppose that Y is a rational homology sphere withHF (Y, t) = Z2 for t ∈ Spinc(Y ). If W is a Stein filling of (Y, ξ) such that

tξ = t then b+2 (W ) = 0. If HF (Y, t) = Z32 then for any Stein filling W with

b+2 (W ) > 0 we have c1(W ) = 0.

Remark 12.3.11. The proof of this statement falls aside from the maintopic of these notes, and we do not present it here. We just note that theproof rests on the embeddability of Stein fillings into Lefschetz fibrations.The fact HF (Y, t) = Z2 is equivalent to HFred(Y, t) = 0, while HF (Y, t) =Z3

2 is the same as HFred(Y, t) = Z2.

Computation shows that HF(± Σ(2, 3, 11)

) ∼= Z32, hence a Stein filling of

it with b+2 > 0 has c1 = 0. Surgery on the K3-surface together with thehomeomorphism characterization of the K3-surface using Seiberg–Witteninvariants due to Morgan and Szabo given in Theorem 3.3.11 provides:


Proposition 12.3.12. If W is a Stein filling of −Σ(2, 3, 11) thenb2(W ) = 2. If W is a Stein filling of Σ(2, 3, 11) then either b+2 (W ) = 0or b2(W ) = 20.

Proof (sketch). It is not very hard to prove that −Σ(2, 3, 11) does notbound negative definite 4-manifold: there exists a compact 4-manifold N2

with ∂N2 = −Σ(2, 3, 11) such that three disjoint copies of N2 are embeddedin K3 in a way that the intersection form of K3−3 N2 is 2E8. (For a Kirbydiagram of N2 see Figure 2.14.) If ∂X = −Σ(2, 3, 11) and X is negativedefinite then the closed negative definite 4-manifold (K3 − 3 N2) ∪ 3 Xwould contradict Donaldson’s diagonalizability theorem [20]. If W is aStein filling of −Σ(2, 3, 11) then for Z = (K3−N2)∪W a suitable productformula of the Seiberg–Witten invariants and Theorem 3.3.11 implies thatZ 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) andb+2 (W ) > 0 then the intersection form QW is equal to 2E8 ⊕ 2H. (For moredetails see [158].)

In fact, it is reasonable to conjecture that ifW is a Stein filling of −Σ(2, 3, 11)then it is diffeomorphic to N2, and a Stein filling of Σ(2, 3, 11) is diffeomor-phic either to the smoothing or to the resolution of the isolated singularityx2 + y3 + z11 = 0 ⊂ C3 — similar to the case of simple and simple ellipticsingularities. Notice that in the above arguments we did not make use ofthe particular choice of the contact structures on ±Σ(2, 3, 11). A recent re-sult of Ghiggini and Schonenberger [59] classifies tight contact structures oncertain Seifert fibered spaces — including ±Σ(2, 3, 11). According to theseresults, −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. TheSeifert 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 Steinfillings in general. According to the next result we cannot expect a finitelist as a solution of Problem 12.3.1, since

Proposition 12.3.13 ([132]). For g ≥ 2 the element ∆2g ∈ Γg,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 correspondingopen book decomposition admits infinitely many distinct Stein fillings.


−1

−1

Figure 12.9. Stein structure on the nucleus N2

In the theorem ∆g ∈ Γg,1 denotes the right-handed Dehn twist along asimple closed curve parallel to the unique boundary component of Σg,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 Σh × S2 along two copies ofΣh×∗ ⊂ Σh×S2 and two (four for even g) copies of ∗×S2 ⊂ Σh×S2.The fibration map can be given by perturbing the composition of thebranched cover map with the projection to the second factor. It is easyto see that the resulting fibration has a section of square −1, hence gives afactorization of ∆g ∈ Γg,1, cf. Section 15.2. Taking a twisted fiber sum oftwo copies of this fibration we get factorizations of ∆2

g ∈ Γg,1. The twistingcan be chosen in such a way that the resulting 4-manifolds have differenttorsion in their first homologies, cf. the proof of Theorem 10.3.5. Nowtaking the complement of a section and a regular fiber we get Lefschetz fi-brations over D2 with nonclosed fibers, hence infinitely many Stein fillingsof the contact 3-manifold given by ∆2

g. The fillings are distinguished by thetorsion of their first homologies.

We close this section with a general result concerning the topology ofStein fillings:


Theorem 12.3.14 ([159]). For a given contact 3-manifold (Y, ξ) thereexists 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 Wof (Y, ξ) — which resembles the c21-invariant of a closed complex surface —is bounded from below.

Remark 12.3.15. The idea of the proof is roughly as follows: Supposethat W1, . . . ,Wn, . . . is the possibly infinite list of Stein fillings of (Y, ξ).Consider the Kahler embeddings Wi → Xi where Xi are minimal surfacesof general type. Our aim is to control the topology of Wi. So fix a fillingW1 and consider T1 = X1 − intW1. Now for any other Stein filling W of(Y, ξ) we can form Z = T1 ∪W , and according to Theorem 7.1.9 this is asymplectic 4-manifold (with b+2 (Z) > 1). Therefore minimality of Z wouldimply c21(Z) ≥ 0, giving the desired lower bound for 3σ(W ) + 2χ(W ) interms of invariants of the fixed 4-manifold T1. Minimality of Z is, however,hard to prove — although it seems to be true —, so rather we have to usea larger (still finite) set of test manifolds Xi − intWi to compare the Steinfilling W with. Also we may relax the minimality requirement by trying toprove that the number of blow-ups contained in the symplectic 4-manifoldZ is bounded by some number depending only on (Y, ξ). In the computationthe 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 givencontact 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. Such3-manifolds are, for example, the ones carrying positive scalar curvature, orhaving vanishing reduced Floer homologies, e.g. lens spaces or boundaries ofcertain plumbings along negative definite plumbing diagrams [124, 139, 143].This observation leads us to the following conjecture.

Conjecture 12.3.16. The set

C(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 alsoreview 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 topologicalproblems. The last section is devoted to a discussion centering aroundthe adjunction inequality. For a more complete discussion of the topicsappearing 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 andb+2 (X) − b1(X) is odd. Below we outline the construction of a mapSWX : Spinc(X) → Z, the Seiberg–Witten invariant of X, which turns outto be a diffeomorphism invariant, that is, for a diffeomorphism f : X1 → X2

and spinc structure s ∈ Spinc(X2) we have SWX2(s) = ±SWX1(f∗s). The

value SWX(s) counts solutions of a pair of equations for pairs of connec-tions and sections of bundles naturally associated to the spinc structure s.In the following we will assume that the reader is familiar with basic notionsof differential geometry, such as connections, covariant differentiation andLevi–Civita connections.

Fix a metric g on X and suppose that the spinc structure s ∈ Spinc(X)is given by the hermitian spinor bundles W± → X with Clifford multi-plication c : T ∗X → HomC(W+,W−) satisfying c(v)∗c(v) = −|v|2 idW+.The fixed metric induces a connection, the Levi–Civita connection onTX and on all bundles associated to it. By fixing the connection A on

224 13. Appendix: Seiberg–Witten invariants

L = detW+(∼= detW−) we get a coupled connection on W± and hence acovariant 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±) andζ ∈ Γ(T ∗X), and then /∂Aψ = c(ζ)w ∈ Γ(W∓).

We recall that the metric g on X gives rise to the Hodge star operator∗g : Λi(X) → Λ4−i(X). On two forms ∗2

g = idΛ+(X), and a 2-form ω ∈ Λ2(X)is self-dual (anti-self-dual, or ASD) if ∗gω = ω (resp. ∗gω = −ω). Theself-dual part of ω, which is equal to 1

2 (ω + ∗gω), is denoted by ω+. TheCliffor multiplication naturally extends to 2-forms and provides a bundleisomorphism 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 thisendomorphism will be denoted by q(ψ). Now we are in the position to writedown the Seiberg–Witten equations. Let η ∈ Λ+(X) be a fixed self-dual2-form. For a spinor ψ ∈ Γ(W+) and connection A on L = detW+ theη-perturbed Seiberg–Witten equations read as follows:

/∂Aψ = 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,S1) acts on the space A(L) × Γ(W+) =U(1)-

connections on L×Γ(W+) by g(ψ,A) = (gψ,A− 2dgg ), and it is not hard

to see that this action maps Seiberg–Witten solutions to Seiberg–Wittensolutions. 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 theη-perturbed Seiberg–Witten equations is called the Seiberg–Witten modulispace Mη(s). The union ∪η∈Λ+(X)Mη(s) is the parameterized moduli spaceM(s).

13.1. Seiberg–Witten invariants of closed 4-manifolds 225

These spaces admit suitable topologies; for technical reasons we actuallyconsider solutions in some Sobolev completions, but this subtlety will beignored in the following.

Definition 13.1.3. The pair (A,ψ) ∈ B is reducible if ψ ≡ 0 and irreducibleotherwise. The set of irreducible elements in B is denoted by B∗. We defineM∗(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. Thestructure 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

(c21(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 andb+2 (X) > 0 the moduli space Mη(s) consists of irreducible solutions only.The Sard–Smale theorem implies that for generic η the moduli space isa compact smooth manifold of dimension d(s). After fixing a homologyorientation on X (that is, an orientation for H2

+(X; R) ⊗ H1(X; R)) themoduli space admits a canonical orientation. Therefore for generic η theoriented, compact submanifold Mη(s) gives rise to a homology class inH∗(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 ofsolutions modulo gauge equivalence of the Seiberg–Witten equations. Ford(s) > 0 we evaluate suitable cohomology classes of the cohomology ringH∗(B∗

X ; Z) on Mη(s). In this way we produce a number SWX(s) ∈ Z forwhich the following result holds:

Theorem 13.1.5 (Seiberg–Witten, [174]). If b+2 (X) > 1 then the valueSWX(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 chosenmetric and perturbation does not apply, since in that case a 1-parameterfamily of moduli spaces might contain reducible solutions. Such solutionsare fixed points of a nontrivial subgroup of the gauge group and thereforerequire special attention. For a thorough discussion the reader is advised toturn to [119, 149]. Recall that an element K ∈ H2(X; Z) is characteristicif for all x ∈ H2(X; Z) we have that K(x) ≡ QX(x, x) (mod 2). The set ofcharacteristic elements in H2(X; Z) will be denoted by CX .


Definition 13.1.6. A class K ∈ CX is a basic class if SWX(s) 6= 0 fors ∈ Spinc(X) with c1(s) = K. The set of basic classes will be denotedby BX . The manifold X is of simple type if K ∈ BX implies that K2 =3σ(X) + 2χ(X).

The following proposition summarizes some of the basic properties ofBX and SWX . Here (for simplicity) we assume that X is of simple type.Recall that in the definition we assumed that b+2 (X) − b1(X) is odd andb+2 (X) is greater than 1.

Proposition 13.1.7.

1. The set BX of basic classes is finite and K ∈ BX if and only if−K ∈ BX . In fact,

SWX(−K) = (−1)14(σ(X)+χ(X))SWX(K).

2. If BX 6= ∅ and Σ ⊂ X is an embedded surface representing thehomology class [Σ] with [Σ]2 ≥ 0 and Σ 6= S2, then

2g(Σ) − 2 ≥ [Σ]2 + |K( [Σ])|

for all K ∈ BX . If BX 6= ∅ and Σ ⊂ X is an embedded spherethen [Σ]2 < 0. The above inequality is usually called the adjunctioninequality, since it generalizes the formula of Theorem 3.1.9.

3. If X is a symplectic manifold then ±c1(X,ω) ∈ BX . For a minimalsurface of general type BX =

± c1(X)

. Moreover, in both cases

SWX

(± c1(X)

)= ±1.

4. If X admits a positive scalar curvature metric, or decomposes asX = X1#X2 with b+2 (X1), b

+2 (X2) > 0 then BX = ∅.

5. More generally, if X = X1 ∪N X2 with b+2 (X1), b+2 (X2) > 0 and N

admits a metric of positive scalar curvature then BX = ∅. Saying thisproperty in a different way, if X = X1∪NX2, N admits positive scalarcurvature metric and BX 6= ∅ then either b+2 (X1) = 0 or b+2 (X2) = 0.

6. If X = Y#CP2 then BX = L ± E | L ∈ BY , where H2(X; Z)is identified with H2(Y ; Z) ⊕ H2(CP2; Z) and E is the generator ofH2(CP2; Z).

According to the following theorem, the assumption on the simple typeproperty 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 ofsimple 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 Σ1,Σ2 ⊂ X are two embedded surfaces withgenera g(Σ1) and g(Σ2) in the symplectic 4-manifold X with [Σ1] = [Σ2],[Σ]2 ≥ 0 and Σ1 is a symplectic submanifold then g(Σ2) ≥ g(Σ1). (Thisinequality is usually referred to as the “Symplectic Thom Conjecture”. Forthe 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 tostate the result, let us assume that (X,ω) is a given symplectic 4-manifoldand J is a compatible almost-complex structure. Suppose furthermore thatb+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 SWX(K) 6= 0 for a givencohomology class K ∈ H2(X; Z). Assume furthermore that the class c =12

(K − c1(X,ω)

)is nonzero in H2(X; Z). Then for a generic compatible

almost-complex structure J on X the class PD(c) ∈ H2(X; Z) can berepresented by a pseudo-holomorphic submanifold.

In fact, Taubes proved much more. By defining a rather delicate wayof counting pseudo-holomorphic submanifolds representing a fixed homo-logy class PD(c) ∈ H2(X; Z), he proved that this number and the valueSWX(c1(X,ω)+2c) are equal. In many applications only the direction thata nonvanishing Seiberg–Witten invariant implies the existence of pseudo-holomorphic curves is used. Note that the curve Σ representing PD(c) isnot given to be connected. This observation becomes important if one wantsto apply the adjunction formula to compute the genus of Σ. By Proposi-tion 13.1.7 we have that −c1(X,ω) ∈ BX , consequently Theorem 13.1.10implies, in particular, that the Poincare dual of −c1(X,ω) can be represen-ted by a pseudo-holomorphic submanifold C (assuming it is nonzero). Sincea pseudo-holomorphic submanifold is always symplectic, the above reason-ing shows that −c1(X,ω) · [ω] =

∫C ω > 0 for manifolds with b+2 (X) > 1

and c1(X,ω) nonzero. Furthermore, it can be shown that if b+2 (X) > 1,then a class e ∈ H2(X; Z) with e2 = −1, c1(X,ω) · PD(e) = 1 andSWX

(c1(X,ω) + 2PD(e)

)6= 0 can be represented by a symplectic sphere;

consequently X is nonminimal. (The fact c1(X,ω) + 2PD(e) ∈ BX guar-antees the existence of a pseudo-holomorphic representative for e. The two


other assumptions — together with the adjunction formula — ensure thatthis representative is a sphere, see the proof of Corollary 13.1.13.) As afurther 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 13.1.8and [84]. Theorem 13.1.10 also proves the inequality in Theorem 3.1.12: IfK is a basic class, then c = 1

2

(K−c1(K,ω)

)can be represented by a pseudo-

holomorphic (in particular symplectic) submanifold (unless c = 0), hencec · [ω] ≥ 0. Reversing the sign of K if necessary, we can assume K · [ω] ≤ 0,so c1(X,ω) · [ω] ≤ K · [ω] ≤ 0, which proves the inequality. Note that equal-ity implies c · [ω] = 0, hence c = 0, and consequently, K = c1(X,ω) (orK = −c1(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 invariantsdepend on the chosen metric and perturbation. After the appropriatemodifications, the theorems and properties discussed above extend to thecase of b+2 (X) = 1. For the sake of brevity, however, here we will omit thediscussion of these extensions; see [145] for a nice review of the b+2 (X) = 1case.

Corollary 13.1.12. Suppose that the symplectic 4-manifold X satisfyingb+2 (X) > 1 is minimal. Then c21(X) ≥ 0.

Proof. According to Theorem 13.1.10 the Poincare dual of the class−c1(X,ω) can be represented by an embedded J-holomorphic submanifoldC = ∪ni=1Ci; here Ci are the connected components of C. Now the adjunc-tion formula for Ci reads as 2g(Ci)− 2 = [Ci]

2 − c1(X,ω)[Ci] = 2[Ci]2. Now

[Ci]2 ≥ 0 holds, since [Ci]

2 < 0 implies g(Ci) = 0 and [Ci]2 = −1 contradict-

ing minimality. Therefore c21(X,ω) =∑

[Ci]2 ≥ 0, concluding the proof.

Corollary 13.1.13. If the symplectic 4-manifold X smoothly decomposesas Y#CP2 then it contains a symplectic (−1)-sphere, i.e., it is not minimalas a symplectic 4-manifold.

Proof. According to Proposition 13.1.7(6.) we know that BX = L ± E |L ∈ BY . Therefore ±c1(X) = ±(L − E) for some L ∈ BY ; now applyTheorem 13.1.10 for K = L + E. We get that 1

2

(K − c1(X)

)= E

can be represented by a J-holomorphic (hence symplectic) submanifold,furthermore by the adjunction formula E2 = −1 and c1(X) · E = 1 give

2g(E) − 2 = E2 − c1(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 classK ∈ BX with the property that

(K − c1(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 formY 3 × R, Seiberg–Witten Floer homologies can be defined for closed oriented3-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 originalSeiberg–Witten equations — developed by Kronheimer and Mrowka [86]— turns out to be extremely useful. Here we restrict ourselves to a quickreview of the invariants, for a more complete discussion see [86, 89]. Let Xbe a given compact 4-manifold with nonempty boundary and fix a contactstructure ξ on ∂X. Define Spinc(X, ξ) to be all spinc structures on X whichrestrict to the spinc structure tξ induced by ξ.

Remark 13.2.1. Recall that the set of spinc structures on X forms aprincipal H2(X; Z)-space and for a 4-manifold it is never empty. As itfollows from the long exact sequence of cohomologies of the pair (X,∂X),the above defined set Spinc(X, ξ) is a principal H2(X,∂X; Z)-space.

The invariant SW(X,ξ) will map from Spinc(X, ξ) to Z and is roughly de-fined as follows. Consider the symplectization of (∂X, ξ) and glue it to Xalong ∂X × 1 to get X+. By choosing an almost-complex structure forξ, 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 spinc

structure defines a spinor Ψ0 and a spin connection A0, for a spinc struc-ture s ∈ Spinc(X, ξ) extend these over X+. Take the space of pairs (A,Ψ)— spin connections and spinors for the fixed spinc structure s — whichsolve the usual (perturbed) Seiberg–Witten equations on the noncompactRiemannian manifold (X+, g) and are close to (A0,Ψ0) in an appropriateL2-sense. After dividing with the appropriate gauge group G we get themoduli space MX+,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) isdefined by counting the number of elements (with sign) in MX+,g(s). To get


a well-defined invariant, we need to show independence of the choices (met-ric, almost-complex structure, extensions, perturbation) made throughoutthe definition. This argument follows the usual cobordism method appliedin the closed 4-manifold case. The two notable differences from the closedcase are:

• There are no reducible solutions (i.e., points in the moduli space withvanishing spinor component) since Ψ0 is nonzero on X+ −X and forany element (A,Ψ) in the moduli space Ψ is close to Ψ0. Thereforethe gauge group acts freely, the index formula provides the actualdimension of a (smoothly cut out) moduli space and there is no needto assume anything about b+2 (X).

• In the case dim MX+,g(s) > 0 the invariant SW(X,ξ)(s) is defined tobe zero, since there is no reasonable constraint with which one couldcut down the dimension. (The cohomology class used in the closedcase vanishes for (X, ξ).)

The main result of [86] concerning SW(X,ξ) is the the generalization ofTheorem 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 sω is the spinc structure induced by analmost-complex structure compatible with the symplectic form ω thenSW(X,ξ)(sω) = 1. Moreover, if (X,ω) is as above and SW(X,ξ)(s) 6= 0then [ω] ∪ (s − sω) ≥ 0 with equality only if s = sω.

Notice that the last assertion implies that if ω is exact (for example, if(X,ω) is a Stein filling of (∂X, ξ)) then sω is the only spinc structure withnonzero SW(X,ξ)-invariant. In addition, these invariants can be used toprove the adjunction inequalities of the type of Proposition 13.1.7(2.) forweak symplectic fillings.

Let us take a contact 3-manifold (Y, ξ) and consider the noncompact4-manifold X = Y × (−∞, 0] with contact type boundary. The ideasoutlined above now produce a contact invariant cSW (Y, ξ) of the contactmanifold in the appropriate Seiberg–Witten Floer cohomology of Y . Thisinvariant can be shown to share many properties with the contact invariantc(Y, ξ) ∈ HF (−Y ) to be discussed in Section 14.4. In this volume we willrestrict our attention to the Heegaard Floer theoretic contact invariants, forthe exact definition and some basic properties of cSW 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 − c1(X)[C].

Remark 13.3.2. In complex algebraic geometry it is customary to usethe canonical bundle KX instead of c1(X). Since in H2(X; Z) we havec1(X) = −KX (one originates from the tangent, while the other from thecotangent bundle), the formula reads as −χ(C) = C2 +KX · C.

It is not hard to see that the above formula holds for a J-holomorphicsubmanifold of an almost complex 4-manifold (X,J). In particular,

−χ(Σ) = [Σ]2 − c1(X,ω)[Σ]

holds for a symplectic submanifold of a symplectic 4-manifold (X,ω). Thisequality 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) withb+2 (X) > 1. If g(Σ) > 0 then

[Σ]2 +∣∣c1(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 contact3-manifold (Y, ξ) and Σ ⊂ W is a homologically nontrivial surface withg(Σ) > 0 then

[Σ]2 +∣∣c1(W,ω)[Σ]

∣∣ ≤ −χ(Σ).

Proof. Embed the symplectic filling into a closed symplectic 4-manifold Xwith b+2 (X) > 1 and apply Theorem 13.3.3.

Theorem 13.3.3 follows from the fact that c1(X,ω) of a symplectic 4-manifold is a basic class and


Theorem 13.3.5 (The adjunction inequality). Suppose that X is a smooth,closed 4-manifold. If K ∈ H2(X; Z) is a basic class of the 4-manifold Xwith 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–Szabo–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 (Ozsvath–Szabo, [133]). Suppose that Σ is a smooth,embedded, closed 2-dimensional submanifold in the smooth 4-manifold Xand for a basic class K we have χ(Σ) − [Σ]2 − K

([Σ])

= 2n < 0. Let εdenote the sign of K

([Σ]). Then the cohomology class K + 2εPD

([Σ])

isalso a basic class.

The most spectacular application of these results is the proof of the Sym-plectic Thom Conjecture due to Ozsvath and Szabo, which improves theresult of Exercise 13.1.9 by dropping the assumption on the self-intesectionof the surface.

Theorem 13.3.7 (Ozsvath–Szabo, [133]). If Σ1,Σ2 ⊂ X are two 2-dimensional connected submanifolds of the symplectic 4-manifold (X,ω),the homology classes [Σi] are equal and Σ1 is a symplectic submanifold,then the genus of Σ2 is not smaller than the genus of Σ1.

In addition, the form of the adjunction inequality given in Theorem 13.3.5implies an improved version of Corollary 13.3.4, already encountered in theintroduction:

Theorem 13.3.8. If Σ is a smoothly embedded closed, oriented 2-dimen-sional submanifold in the Stein surface S then

[Σ]2 − c1(S)[Σ] ≤ −χ(Σ)

unless Σ is a nullhomologous sphere.

Proof. Recall that a Stein surface can always be symplectically embeddedinto a symplectic 4-manifold X, therefore for [Σ]2 ≥ 0 the statement followsfrom the usual adjunction inequality (together with the fact that c1(X) of

13.3. The adjunction inequality 233

a symplectic 4-manifold is a basic class). In the case of negative [Σ]2 we usethe embedding of S into a minimal surface X of general type. Assumingg(Σ) > 0 the relation of Theorem 13.3.6 implies that either the inequality issatisfied or c1(X)± 2PD([Σ]) is a basic class. (The sign here is determinedby the sign of c1(X)

([Σ]).) But for a minimal surface of general type there

are only two basic classes, which are ±c1(X). Therefore we have either[Σ] = 0 or the difference of the two basic classes c1(X) and −c1(X) (whichis 2c1(X)) is equal to 2PD

([Σ]). This latter case, however, provides a

contradiction since it implies that c21(X) = [Σ]2 is negative, which cannothold for a minimal surface of general type. Finally if g(Σ) = 0 then theabove principle provides [Σ]2 ≤ −2 since a sphere with self-intersection −1would 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 (atleast partially) can be computed from surgery diagrams. In this appendixwe outline the construction of such invariants — for a complete discus-sion the reader is referred to the original papers of Ozsvath and Szabo[135, 136, 137, 138]. To set up the stage, first we discuss Ozsvath–Szabohomology groups of oriented, closed 3-manifolds (together with maps in-

duced by oriented cobordisms). The definition of the group HF (Y ) for a3-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 Ozsvath–Szabo invariants and verify some of the basicproperties of this very sensitive invariant. A few model computations arealso given.

14.1. Topological preliminaries

Recall that a closed, oriented 3-manifold Y can be decomposed as a union oftwo solid genus-g handlebodies Y = U0∪ΣgU1: consider a Morse function onY and define U0 as the union of the 0- and 1-handles while U1 = 2-handles∪ 3-handle. In fact, the 1-handles can be recorded on the genus-g surfaceΣg by their cocores, while the 2-handles by their attaching circles. Hencethe handlebody decomposition can be presented on Σg by two g-tuples ofembedded simple closed curves α1, . . . , αg and β1, . . . , βg which satisfythat the α’s (and the β’s) are disjoint among themselves and form a linearlyindependent system in H1(Σg; Z). Of course, the α-curves might intersectthe β-curves. In conclusion, a 3-manifold can be described by a Heegaard

236 14. Appendix: Heegaard Floer theory

diagram(Σg, αigi=1, βi

gi=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 asurgery diagram. As we saw, any rational surgery can be transformed into asequence of integral surgeries; in the following we will describe an algorithm(given in [135]) for finding a Heegaard diagram of a 3-manifold given byintegral surgery on a knot. (The general case of surgery on a link followssimilar ideas.) Suppose that Y is given by an integral surgery on K ⊂ S3

and consider the following Heegaard diagram of S3 − νK: Fix a projectionof K to some plane. Consider a tubular neighborhood of K in R3 andadd vertical tubes for every crossing of the given projection, as it is shownby the upper diagrams of Figure 14.1. By an isotopy, the resulting subsetUK ⊂ R3 can be regarded as an ε-neighborhood of the knot projection, cf.the lower diagrams in Figure 14.1. In fact, UK 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 S3 alsoa genus-g handlebody. This last statement can be easily verified by addingg 3-dimensional 2-handles along the curves αi encircling the “holes” of UK

14.1. Topological preliminaries 237

(see Figure 14.2 again) turning UK into a solid ball. Notice that the α-curves are the cocores of the 1-handles of the complementary handlebodyS3 − UK . This specifies the α-curves of the diagram for S3 − νK. On

1α 2α

α 3

α 4

Figure 14.2. The α-curves of a Heegaard decomposition of the complement for the trefoilknot

the other hand the meridians of the vertical tubes that we attach (as it isshown by Figure 14.1) give rise to the β-curves since we can think of themas the attaching circles of the 2-handles in S3 − νK. By attaching handlesalong the α-curves we fill the complement of UK (minus a point), whileby attaching 2-handles along the β-curves we fill the vertical tubes insideUK . Therefore the Heegaard diagram

(Σg, αigi=1, βjg−1

j=1

)provides the

knot complement S3 − νK. Now taking a simple closed curve defining any(integral) surgery along K as βg we get a Heegaard diagram for the surgeredmanifold. For example, if we choose the meridian of K as βg then this choicecorresponds to a trivial surgery along K so that we get a Heegaard diagramof S3. For another example see Figure 14.3.

Exercise 14.1.1. Determine the knot and compute the surgery coefficientof the surgery corresponding to the Heegaard diagram of Figure 14.3.

It is natural to wonder when do two Heegaard diagrams represent thesame 3-manifold. It is fairly easy to list moves which do not change theresulting 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


β1

β2

β3

β4

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 andα, 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 diagramsrepresent diffeomorphic 3-manifolds then one diagram can be transformedinto the other by a finite sequence of isotopies, handle slides and stabiliza-tions/destabilizations [135]. This observation can be used to show that aquantity 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 underthe 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 Y1 to Y2 can bedecomposed as a sequence of attaching 1-, 2- and 3-handles. By assumingorientability ofW , the gluing of 1-handles (and so of 3-handles) is essentiallyunique, 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 ⊂ Y1 is a framed knot,W is the cobordism given by the 2-handle attachment along K with thegiven framing and consider a Heegaard diagram

(Σg, αigi=1, βjg−1

j=1

)for

Y1 − νK. (This can be given by implementing the algorithm describedabove.) Let γj = βj for j = 1, . . . , g − 1, βg = meridian of K and γg =thecurve representing the framing of K fixed before. Then the Heegaarddiagrams (

Σg, αigi=1, βjgj=1

),(Σg, αigi=1, γjgj=1

)

represent Y1 and Y2.

Exercise 14.1.2. Verify that the Heegaard diagram(Σg, βigi=1, γjgj=1

)

defines #g−1(S1 ×S2). (Hint: Displace γj by an isotopy to make it disjoint

from βj (j = 1, . . . , g − 1). Destabilize (βg, γg) and use induction.)

Therefore, the 4-manifold X we get by attaching [0, 1]×Uα, [0, 1]×Uβ and[0, 1] × Uγ to Σg × solid triangle ⊂ C along the sides I × Σg, has threeboundary components: Y1, Y2 and #g−1(S

1 × S2). (Here Uα, Uβ and Uγstand for the handlebodies defined by the corresponding sets of curves.)Now filling the boundary component #g−1(S

1 ×S2) with g−1(S1 ×D3) we

get a cobordism from Y1 to Y2, which can be proved to be diffeomorphicto the given cobordism W we started with. Therefore the cobordism Wbuilt on Y by attaching a 2-handle along K ⊂ Y can be represented by theHeegaard triple (

Σg, αigi=1, βjgj=1, γkgk=1

),

where(Σg, αigi=1, βjg−1

j=1

)is a Heegaard diagram for Y −νK and γkgk=1

is given from βkgk=1, the surgery curve K and the framing as describedabove.

14.2. Heegaard Floer theory for 3- and 4-manifolds

Let Y be a given closed, oriented 3-manifold and fix a Heegaard diagram(Σg, αigi=1, βjgj=1

)for Y . Without loss of generality we can assume that

each αi intersects each βj transversely. Let us consider the tori

Tα = α1 × . . . × αg, Tβ = β1 × . . . × βg

in the g-fold symmetric power Symg(Σg). This symmetric power (whichis a smooth manifold of dimension 2g) can be equipped with a symplectic


structure ω and the Floer homology group HF (Y ) is supposed to measurehow the above two (totally real) tori intersect each other “in the symplectic

sense”. More precisely, define CF (Y ) as the free Abelian group generatedby the intersection points Tα∩Tβ. (It is easy to see that since the curves αiand βj intersect transversely, so do the tori Tα and Tβ.) We consider twointersection points to be “removable” if there is a “holomorphic Whitneydisk” showing how to get rid of them. More formally, fix an ω-tame almost-complex structure J on Symg(Σg) and define a differential ∂ : CF (Y ) →CF (Y ) as follows: for x, y ∈ CF (Y ) the matrix element 〈∂x, y〉 counts theJ-holomorphic maps u : ∆2 → Symg(Σg) from the unit disk ∆2 ⊂ C (up toreparametrization) with

u(i) = x, u(−i) = y,

u(z) ∈ Tα if z ∈ ∂∆2 and ℜe z < 0,

u(z) ∈ Tβ if z ∈ ∂∆2 and ℜe z > 0.

In order to get a sensitive invariant, we need to choose a base point z0 ∈Σg − ( ∪i αi

⋃∪jβj) and require u(∆2) ∩ z0 × Symg−1(Σg) = ∅, that is,the holomorphic disk should avoid the divisor z0 × Symg−1(Σg) definedby the base point. If the space of these maps (up to reparametrization) isnot 0–dimensional, we define 〈∂x, y〉 to be zero, otherwise

〈∂x, y〉 =

#

holomorphic disks from x to y disjoint from z0 × Symg−1(Σg).

Remark 14.2.1. Using a delicate construction (and fixing some auxiliarydata) a sign can be attached to any map of the above type in a 0–dimensionalspace, and in the definition of 〈∂x, y〉 we count the holomorphic maps withthose signs. Alternatively, we can use Z2-coefficients, which turns out tobe sufficient for our present purposes, therefore we will always restrict ourattention to this special case.

The complex(CF (Y ), ∂

)splits as a sum ⊕t∈Spinc(Y )

(CF (Y, t), ∂

)of sub-

complexes: an intersection point x ∈ Tα ∩ Tβ and the fixed base pointz0 ∈ Σg naturally determines a spinc structure sz0(x) ∈ Spinc(Y ) in thefollowing way: Suppose that the Heegaard diagram is induced by a Morsefunction f : Y → R and fix a Riemannian metric g0 on Y . Then an in-tersection point x ∈ Tα ∩ Tβ can be regarded as a choice of gradient lines


for f (with respect to g0) connecting index-2 and index-1 critical points off : choose those gradient flow lines which pass through the coordinates ofx = (x1, . . . , xg) ∈ Tα ∩ Tβ in Σg. The base point z0 specifies a gradientline connecting the minimum and maximum of f , therefore on the comple-ment of the neighborhood of these paths the gradient ∇f defines a nowherevanishing vector field. Since along any of these paths the indices of thecritical points have opposite parity, the resulting vector field extends to Y ,giving rise to a well-defined spinc structure on the 3-manifold. It is not veryhard to verify that there is a topological Whitney disk connecting x andy if and only if sz0(x) = sz0(y). Using Gromov’s compactness theorem itcan be shown that for generic choices ∂2 = 0, hence the Floer homologyHF (Y, t) = H∗

(CF (Y, t), ∂

)can be defined for all t ∈ Spinc(Y ).

Theorem 14.2.2 (Ozsvath–Szabo, [136]). Let Y be a given closed oriented3-manifold equipped with a spinc structure t ∈ Spinc(Y ). The Ozsvath–

Szabo homology group HF (Y, t) is a topological invariant of the spinc 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 Symg(Σg) and base point z0 ∈ Σg. The in-dependence from the chosen almost-complex structure is essentially built inthe definition: it is a general feature of Floer homology groups associated tointersecting Lagrangian submanifolds in symplectic manifolds. (AlthoughTα and Tβ are not Lagrangian in Symg(Σ), the general theory still ap-plies because of special features of this particular case.) The independencefrom the Heegaard decomposition requires to show that the groups do notchange under isotopies, handle slides and stabilization. The independenceof isotopies is again a consequence of some general facts regarding Floer ho-mologies: any isotopy can be decomposed into a Hamiltonian isotopy andanother one which can be represented by the change of the almost-complexstructure on Σg. By a good choice of the base point, independence fromstabilization is a fairly easy exercise, while handle slide invariance requiresto work out a special case and a way to implement this special case undergeneral circumstances. Finally, the change of base point can be reduced to asequence 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 b1(Y ) > 0 one also has to assume a certain admissibilityof the Heegaard diagram, which can always be achieved by appropriateisotopies of the α- and the β-curves. This condition is needed for having


finite sums in the definition of the boundary operator ∂ and in the proof ofindependence of choices. For details see [135].

Proposition 14.2.4. The set

t ∈ Spinc(Y ) | HF (Y, t) 6= 0

is finite forany 3-manifold Y . In particular, the vector space

HF (Y ) = ⊕t∈Spinc(Y )HF (Y, t)

is finite dimensional.

Proof. After fixing an admissible Heegaard diagram, there are only finitelymany intersection points in Tα ∩ Tβ, hence the chain complex CF (Y ) isfinite dimensional, implying the result.

Examples 14.2.5. (a) Consider the lens space L(p, q). It admits a genus-1 Heegaard decomposition with α1 and β1 intersecting each other in ppoints. These points all correspond to different spinc structures, thereforethe boundary map ∂ vanishes for any t ∈ Spinc

(L(p, q)

), and so we have

that HF (L(p, q), t) = Z2. In particular, HF (S3) = Z2.

(b) It is not hard to see that HF (Y, t) ∼= HF (−Y, t). If (Y, t) decomposesas a connected sum (Y1, t1)#(Y2, t2) then

HF (Y, t) = HF (Y1, t2) ⊗Z2 HF (Y2, t2).

(c) It can be shown that if t ∈ Spinc(Y ) is torsion, that is, c1(t) ∈ H2(Y ; Z)

is a torsion element, then HF (Y, t) is nontrivial. In particular, if Y is a

rational homology sphere (i.e., b1(Y ) = 0) then HF (Y, t) is nonzero forall t ∈ Spinc(Y ). Since any 3-manifold admits torsion spinc structure, the

above nontriviality statement implies that HF (Y ) 6= 0 for any 3-manifold Y .

(d) The 3-manifold S1 × S2 admits a genus-1 Heegaard decompositionwith two parallel circles as α- and β-curves. This Heegaard decomposition,however, is not admissible. The diagram of Figure 14.5 gives an admissibleHeegaard diagram for S1 × S2. By analyzing the possible holomorphicdisks we get that for the spinc structure t0 with vanishing first Chernclass HF (S1 × S2, t0) ∼= Z2 ⊕ Z2 holds, while for all other spinc structures

the Ozsvath–Szabo homology group vanishes. Consequently HF(#k(S

1 ×S2), t

)is zero unless c1(t) = 0, and for c1(t0) = 0 we have HF

(#k(S

1 ×S2), t0

) ∼= Z2k

2

( ∼= H∗(Tk; Z2)

).


α

β

Figure 14.5. Admissible Heegaard diagram for S1× S2

As we saw in Proposition 14.2.4, the groups are nontrivial only for finitelymany spinc structures in Spinc(Y ). In fact, the particular geometry of Yprovides a constraint for the nontriviality of the Ozsvath–Szabo homologygroups:

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) isnontrivial for a spinc structure t then |⟨c1(t), [Σ]

⟩ | ≤ −χ(Σ). If Σ ∼= S2

then HF (Y, t) 6= 0 implies that⟨c1(t), [Σ]

⟩= 0.

Similar ideas provide invariants for 4-dimensional manifolds. Supposethat (W, s) is a spinc cobordism between (Y1, t1) and (Y2, t2). Standardmanifold topology implies thatW can be decomposed asW = W1∪W2∪W3,where Wi can be built using 4-dimensional i-handles only (i = 1, 2, 3). A

homomorphism FW,s : HF (Y1, t1) → HF (Y2, t2) can be given as follows (forsimplicity we drop the spinc stucture from the notation): define FW as thecomposition FW3 FW2 FW1 , where the homomorphisms FW1 and FW3 arestandard maps, since the cobordisms W1 and W3 depend only on Y1 andY2, and the number of 1-handles (3-handles) involved in the cobordism. Forexample, W1 is a cobordism between Y1 and Y ′

1 = Y1#k(S1 × S2) (where k

is the number of 1-handles in W1), and so FW1,s is a map

HF(Y1, s|Y1

)→ HF

(Y ′

1 , s|Y ′

1

)= HF

(Y1, s|Y1

)⊗ HF

(#k(S

1 × S2), t0)

sending x ∈ HF(Y1, s|Y1

)to x⊗ θk where θk is the highest degree element

in HF(#k(S

1×S2), t0) ∼= H∗(T

k; Z2). Similar formula describes FW3 . Thecobordism W2, 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


section. Counting specific holomorphic triangles in Symg(Σg) with appro-priate boundary conditions (in a similar spirit as ∂ was defined) we getFW2 . As before, a long and tedious proof shows that FW,s is independent ofthe choices made (i.e., the decomposition of W , the chosen almost-complexstructure, the base point, etc.).

Theorem 14.2.7 (Ozsvath–Szabo, [137]). The resulting map FW,s dependsonly on the oriented 4-dimensional spinc cobordism (W, s) and is indepen-dent of the choices made throughout the definition.

Once again, FW,s 6= 0 holds only for finitely many spinc structures s ∈Spinc(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 + |⟨c1(s), [Σ]

⟩ | or

with g(Σ) = 0 and [Σ]2 ≥ 0 then FW,s = 0.

(For the detailed proof of a special case of this theorem see [101].) Simi-lar ideas result a variety of Ozsvath–Szabo 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 themonce the 3-manifold is given by a surgery diagram. The key step in suchcomputations is the application of an appropriate surgery exact sequence,which relates Ozsvath–Szabo homologies of three 3-manifolds we get bydoing surgeries on some knots. As we will see, the scheme does not producethe Ozsvath–Szabo homology group of the 3-manifold given by surgerydirectly, but rather gives it as part of several exact sequences. In addition,maps in the sequences are usually induced by cobordisms, hence exactnessprovides information about the maps as well. Below we give the mostimportant surgery exact sequence proved for the HF -theory.

To state the theorems, let us assume that Y is a given 3-manifold witha knot K ⊂ Y in it. Fix a framing f on K and suppose that Y1 is theresult of an integral surgery on K with the given framing. Let X1 denote

14.3. Surgery triangles 245

the resulting cobordism. Suppose that Y2 is the result of a surgery along Kwith framing we get by adding a right twist to the framing f fixed on K.Equivalently, Y2 can be given by doing surgery on K with framing f and(−1)-surgery on a normal circle N to K. This alternative viewpoint alsoprovides a cobordism X2 from Y1 to Y2 given by the second surgery. Let t

be a fixed spinc structure on Y − νK, and let t(Y ), t(Y1) and t(Y2) denotethe set of extensions of t to Y, Y1 and Y2, resp. Denote the homomorphismHF

(Y, t(Y )

)→ HF

(Y1, t(Y1)

)induced by the cobordism given by the

first surgery on K by F1. Here F1 is the sum of FX1,s for all spinc structures ∈ Spinc(X1) extending elements of t(Y ) and t(Y1). We define F2 for thecobordism X2 in a similar fashion.

Exercise 14.3.1. Perform a surgery along a (−1)-framed normal circle N ′

to N ⊂ Y1 and denote the resulting cobordism from Y2 by X3. Show thatX3 is a cobordism from Y2 to Y . (Hint: Blow down N ′ and put a dot onthe image of N . Finally cancel the resulting 1-handle/2-handle pair, seeFigure 14.6.)

K KK

n n n

N N0−1

−1

Figure 14.6. Identification of a 3-manifold in the surgery exact triangle

Let F3 denote the homomorphism HF(Y2, t(Y2)

)→ HF (Y, t) induced by

the cobordismX3 given by the 2-handle attachment along N ′ as it is given inExercise 14.3.1. Consider the triangle of cobordisms as given by Figure 14.7.

Theorem 14.3.2 (Surgery exact triangle, Ozsvath–Szabo [136]). Underthe above circumstances the surgery triangle induces an exact triangle

HF(Y, t(Y )

)HF

(Y1, t(Y1)

)

HF(Y2, t(Y2)

)

F1

F3 F2

for the corresponding homology groups.


Y1

Y2

X1

X3 X2< >n< >n

K

Y

K

n

NN

−1<−1>

K

−1N

Figure 14.7. Cobordisms in the surgery exact triangle

Remark 14.3.3. With Z2-coefficients the map FW induced by a cobordismW is simply the sum

∑FW,s for all spinc structures extending the fixed

ones on the boundaries of W . With Z-coefficients, however, signs have tobe attached to the various maps FW,s for exactness to hold. For a completeargument see [136].

By summing over all spinc structures on Y −K and denoting

⊕t∈Spinc(Y )HF (Y, t)

by HF (Y ) as usual, we get

Corollary 14.3.4. The triangle

HF (Y ) HF (Y1)

HF (Y2)

F1

F3 F2

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 ⊂ S3 isequal to gs. Then for n ≥ 2gs − 1 > 0

HF(S3n(K)

) ∼= HF(S3

2gs−1(K))⊕ Z

n−2gs+12 .

Proof. For n = 2gs − 1 the proposition obviously holds. The general casenow follows by induction. To see this, consider the surgery triangle forY = S3, and knot K with framing n:

HF (S3) HF(S3n(K)

)

HF(S3n+1(K)

)

F1

F3 F2

Since the first cobordism contains a surface of genus gs with square n,the adjunction formula of Theorem 14.2.8 implies that F1 = 0, henceHF

(S3n+1(K)

) ∼= HF(S3n(K)

)⊕ Z2, concluding the proof.

To see a more complicated example, suppose that Y fibers over S1 with fiberF of genus ≥ 2, and consider the canonical spinc structure tcan ∈ Spinc(Y )induced by the oriented 2-plane field formed by the tangencies of the fibersof Y → S1. Obviously

⟨c1(tcan), [F ]

⟩= χ(F ). The surgery exact triangle

and the adjunction formula together imply

Proposition 14.3.6. Under the above circumstances HF (Y, tcan) ∼=Z2 ⊕ Z2.

The proof of the proposition involves two lemmas, only one of which will beproved below.

Lemma 14.3.7. If Y1, Y2 both fiber over S1 with equal fiber genus ≥ 2then for the canonical spinc structures ti ∈ Spinc(Yi) we have HF (Y1, t1) ∼=HF (Y2, t2).

Proof (sketch). Let mi be the monodromy of the fibration Yi → S1

(i = 1, 2), and factor m1m−12 into the product of k right-handed Dehn twists

along homologically nontrivial simple closed curves. This factorization givesrise to a Lefschetz fibration over the annulus, which is a cobordism betweenY1 and Y2. The proof will proceed by induction on k. By composing thecobordisms it is enough to deal with the case of k = 1. In that case we get


Y2 from Y1 by doing surgery on the vanishing cycle of the singular fiber ofthe Lefschetz fibration over the annulus. Writing down the surgery trianglefor that surgery, we get

HF(Y1, t(Y1)

)HF

(Y2, t(Y2)

)

HF(Y0, t(Y0)

)

F1

F3 F2

The third group vanishes by the adjunction formula of Theorem 14.2.6:Since Y0 is the result of a surgery along the vanishing cycle with coefficient0 relative to the framing induced by the fiber, it contains a surface ofgenus (g − 1) in the homology class of the (old) fiber. Therefore F1 isan isomorphism and it is not hard to see that the nonzero terms belong tothe canonical spinc structures ti.

Now the following lemma (which we give without proof) concludes theargument for Proposition 14.3.6.

Lemma 14.3.8. For S1 × Σg with the canonical spinc structure tcan we

have HF (S1 × Σg, tcan) = Z2 ⊕ Z2.

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∈Spinc(Y ) HF (Y, t) has dimension∣∣H1(Y,Z)

∣∣ . Since for a rational ho-

mology sphere HF (Y, t) never vanishes, being an L-space is equivalent to

HF (Y, t) = Z2 for all spinc structures t ∈ Spinc(Y ). For example, lensspaces 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 ⊂ S3 is a knot of 4-ball genus gs > 0.If there is n > 0 such that S3

n(K) is an L-space then all S3m(K) with

m ≥ min (2gs − 1, n) is an L-space.

Proof. Recall from Proposition 14.3.5 that if S3n(K) is an L-space and

n ≥ 2gs − 1 then S32gs−1(K) is also an L-space. (Use the fact that

|H1

(S3n(K); Z

)| = |n| for all n 6= 0.) In addition, by applying the surgeryexact sequence for Y = S3, the knot K and framing m it is easy to seethat if S3

m(K) is an L-space then so is S3m+1(K) (m ≥ 1). This observation

concludes the proof.

14.4. Contact Ozsvath–Szabo invariants 249

Example 14.3.10. If K denotes the right-handed trefoil knot then S3n(K)

is an L-space for all n ≥ 1. This can be seen by the computation ofProposition 14.3.5 together with the fact that S3

5(K) is a lens space.

Exercises 14.3.11. (a) Extend Proposition 14.3.9 to all rational m withm ≥ min(2gs − 1, n). (Hint: Cf. [101].)

(b) Using the fact that HF (Y ) 6= 0 holds for any 3-manifold (cf. Exam-ple 14.2.5(c)) verify that with K denoting the right-handed trefoil knot,

HF(S3

0(K))

= Z22 holds. (Hint: Use the surgery exact triangle and the

fact that HF(S3

1(K))

= Z2.)

(c) Let Yn denote the circle bundle over the torus T 2 with Euler number

n > 0. Show that HF (Yn) = Z4n2 . (Hint: Apply the surgery exact triangle

induced by the cobordisms of Figure 14.8. Find a torus of self-intersectionn in the coboridsm X and use induction on n. Find another triangle tohandle the case of n = 1.)

14.4. Contact Ozsvath–Szabo invariants

One of the main applications of Heegaard Floer theory is in contact topol-ogy. Contact Ozsvath–Szabo invariants can be fruitfully applied in deter-mining tightness of structures given by contact surgery diagrams, hencethese invariants fit perfectly in the main theme of the present notes. Thedefinition of the invariant of a contact structure given by Ozsvath and Szabois based on a compatible open book decomposition with connected binding.According to Giroux’s result discussed earlier, well-definedness of such aninvariant requires the verification that the quantity does not change underpositive stabilization. The construction of Ozsvath and Szabo goes in thefollowing way: Suppose that a compatible open book decomposition withconnected binding is fixed on (Y, ξ). Then 0-surgery on the binding of thisopen book decomposition produces a fibered 3-manifold YB and a cobordismW between Y and YB . Notice that the contact structure ξ induces a spinc

structure tξ on Y , and YB admits a natural spinc structure tcan induced bythe oriented 2-plane field tangent to the fibers.

Exercise 14.4.1. Show that W admits a unique spinc structure s such thats|Y = tξ and s|YB

= tcan.


n

00

00

n+1

X0 0

Figure 14.8. 3-manifolds in a particular surgery triangle

Turning W upside down to get W , we have a map

FW,s : HF (−YB , tcan) → HF (−Y, tξ),

and HF (−YB , tcan) has been computed to be isomorphic to Z2 ⊕ Z2. Bymaking use of the corresponding homology theory HF+, a nontrivial el-ement h ∈ HF (−YB, tcan) can be distinguished: There is a long ex-

act sequence connecting the related theories HF+(Y, t) and HF (Y, t) forany spinc 3-manifold (Y, t), and for a fibered 3-manifold Y and t = tcan

we have (similarly to Proposition 14.3.6) that HF+(Y, tcan) = Z2. Now

h ∈ HF (−YB , tcan) is the element mapping to the nontrivial element inHF+(−YB , tcan).

Definition 14.4.2. The contact Ozsvath–Szabo invariant c(Y, ξ) of the

contact structure (Y, ξ) is defined to be equal to FW,s(h) ∈ HF (−Y, tξ).


The fundamental theorem concerning c(Y, ξ) is

Theorem 14.4.3 (Ozsvath–Szabo, [140]). The Ozsvath–Szabo homology

element c(Y, ξ) ∈ HF (−Y, tξ) does not depend on the chosen compatibleopen book decomposition, hence is an invariant of the isotopy class of thecontact 3-manifold (Y, ξ).

Remark 14.4.4. The definition given in [140] involves the Ozsvath–Szaboknot invariant of the binding of a compatible open book decomposition —using this definition Ozsvath and Szabo verifies independence of the openbook decomposition and then proves that the two definitions (one relying onsurgery 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 donot discuss the details of the definition here.

The main properties of the invariant c(Y, ξ) are summarized in the followingstatements

Theorem 14.4.5 (Ozsvath–Szabo, [140]; cf. also [100]). If (YK , ξK) is givenas contact (+1)-surgery along the Legendrian knot K ⊂ (Y, ξ) and W is thecorresponding cobordism then by reversing the orientation on W and usingthe 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 spinc 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 linkL ⊂ (S3, ξst) and add K to L. Applying the algorithm of Akbulut andthe first author [7] for L ∪ K we get an open book decomposition of Ycompatible with ξ such that K lies on a page of it. Denote the results ofthe 0-surgeries along the bindings on Y and Y (K) with YB and

(Y (K)

)B

respectively. The cobordism WB of the handle attachment along the knotK gives rise to a map F−WB

: HF (−YB) → HF( −(Y (K)

)B), which

fits into an exact triangle of the type encountered in Lemma 14.3.7. Thesame argument now provides that F−WB

is an isomorphism, resulting in acommutative diagram

HF (−YB) HF(−(Y (K)

)B)

HF (−Y ) HF(− Y (K)

)

F−WB

∼=

F−W

FWYFY (K)


Since the distinguished generator hY ∈ HF (−YB) maps to the distinguished

generator hY (K) ∈ HF(−(Y (K)

)B), the statement of the theorem follows

from the commutativity of the above diagram and the definition of thecontact invariant.

Example 14.4.6. The contact Ozsvath–Szabo invariant of the overtwistedstructure (S3, ξ′) depicted by Figure 11.3 vanishes. This can be verified byapplying the above principle for (S3, ξst) and K as in Figure 11.3. The co-bordism −W inducing the map F−W with the property F−W

(c(S3, ξst)

)=

c(S3, ξ′) contains a sphere of self-intersection (+1), hence F−W = 0, there-fore c(S3, ξ′) = 0 as claimed.

This example can be generalized as

Theorem 14.4.7 (Ozsvath–Szabo, [140]). If (Y, ξ) is overtwisted thenc(Y, ξ) = 0.

Proof. Consider the oriented 2-plane field ξ1 on Y with the property thatthe oriented 2-plane field (Y, ξ1)#(S3, ξ′) is homotopic to the oriented 2-plane field induced by (Y, ξ). (Here ξ′ is the oriented 2-plane field induced bythe contact structure of Example 14.4.6.) By the classification of overtwistedcontact structures, there is a contact structure representing the oriented 2-plane field ξ1. Consequently, the above argument shows that contact (+1)-surgery along the knot of Figure 11.3 located in a Darboux chart of somecontact structure ξ1 on Y provides an overtwisted structure homotopic,hence isotopic to (Y, ξ). Therefore c(Y, ξ) can be given as F−W

(c(Y, ξ1)

).

Since −W contains a 2-sphere of self-intersection (+1), the adjunctionformula provides F−W = 0 and therefore c(Y, ξ) = 0.

Corollary 14.4.8. If c(Y, ξ) 6= 0 for (Y, ξ) and (YK , ξK) is given as contact(−1)-surgery along the Legendrian knot K ⊂ (Y, ξ) then c(YK , ξK) 6= 0,therefore it is tight.

Proof. Let K ′ be a Legendrian push off of K in (Y, ξ), giving rise to aLegendrian knot (also denoted by K ′) in (YK , ξK). By the CancellationLemma 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)-surgerywe have F−W

(c(YK , ξK)

)= c(Y, ξ). If c(Y, ξ) 6= 0, then this shows that

c(YK , ξK) 6= 0. In the light of Theorem 14.4.7 this implies tightness of(YK , ξK).


Proposition 14.4.9 (Ozsvath–Szabo, [140]). For (S3, ξst) the contact

invariant c(S3, ξst) generates HF (S3) = Z2.

Proof (sketch). Recall that (S3, ξst) admits an open book decompositionwith the unknot as binding. Therefore the map defining the invariant fitsinto the exact triangle

HF (−S1 × S2) HF (−S3)

HF (S3)

F

G

Since we know that HF (S3) = Z2 and HF (S1×S2) = Z2⊕Z2, it follows thatG = 0, and F is onto. Now by using a certain grading on Ozsvath–Szabohomologies (cf. [138]) it is not hard to see that the element h ∈ HF (S1×S2)used in the definition of the contact invariant maps into the nonzero elementof HF (S3), concluding the proof.

Lemma 14.4.10. Consider the contact structure ηk on #k(S1×S2) given by

contact (+1)-surgery on the k-component Legendrian unlink. The contactinvariant c

(#k(S

1 × S2), ηk)

does not vanish.

Proof. The lemma will be proved by induction on k. For k = 0 wehave the standard contact 3-sphere (S3, ξst) which has nonzero invariant byProposition 14.4.9. By definition, ηk is given as contact (+1)-surgery along aknot in ηk−1, therefore F−W(c

(#k−1(S

1×S2), ηk−1

)) = c

(#k(S

1×S2), ηk)

for the cobordism we get by the handle attachment. Therefore the injectivityof F−W immediately provides the result. Now writing down the surgeryexact triangle for the above handle attachment, for the Ozsvath–Szabohomology groups we get

HF(#k−1(S

1 × S2))

HF(#k(S

1 × S2))

HF(#k−1(S

1 × S2))

F−W

Since dimZ2 HF(#k(S

1×S2))

= 2k, 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 (#kS

1 × S2, ηk) is tight. It is known that #k(S1 × S2)


carries a unique isotopy class of tight contact structures, which is the Steinfillable boundary of D4 ∪ k 1-handles. In conclusion, (+1)-surgery on thek-component Legendrian unlink produces a contact 3-manifold contacto-morphic to the boundary of the Stein surface we get by attaching k 1-handlesto D4, cf. Exercise 11.2.7.

Exercise 14.4.11. Show that if (Y, ξ) is a Stein fillable contact 3-manifoldthen c(Y, ξ) 6= 0. (Hint: Recall that any Stein fillable contact structure canbe given as Legendrian surgery along a link in

(#k(S

1 × S2), ηk)

for somek. Use Lemma 14.4.10 and Corollary 14.4.8.)

Making use of the Embedding Theorem 12.1.7 of weak symplectic fillingsand the nonvanishing of the mixed Ozsvath–Szabo invariants for closedsymplectic 4-manifolds [137, 141], the above exercise was generalized for aversion of contact invariants in some “twisted coefficient system” as follows:

Proposition 14.4.12 (Ozsvath–Szabo, [143]). If (Y, ξ) is a weakly sym-plectically fillable contact 3-manifold then by using an appropriate twistedcoefficient 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 thesegroups we recall the equivalence between certain words in some mappingclass groups and geometric structures discussed in earlier chapters. Weclose this chapter with some theorems making use of those connections.

15.1. Short introduction

Let Σng,r denote an oriented, connected genus-g surface with n marked points

and r boundary components.

Definition 15.1.1. The mapping class group Γng,r is defined as the quotientof the group of orientation preserving self-diffeomorphisms of Σn

g,r (fixingmarked points and boundaries pointwise) by isotopies (fixing marked pointsand boundaries pointwise). For n = 0 (r = 0, resp.) we use the notationΓg,r (Γng , resp.), and in case n = r = 0 we write Γg. For F = Σg,r we willdenote Γg,r by ΓF .

Simple closed curves in the surface give rise to special mapping classes:

Definition 15.1.2. A right-handed Dehn twist ta : Σng,r → Σn

g,r on an em-bedded simple closed curve a in an oriented surface Σn

g,r is a diffeomorphismobtained by cutting Σn

g,r along a, twisting 360 to the right and regluing.More formally, we identify a regular neighborhood νa of a in Σn

g,r withS1 × I, set ta(θ, t) = (θ + 2πt, t) on νa and smoothly glue into idΣn

g,r−νa. Aleft-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 acurve a we do not need to orient a even though the surface Σn

g,r has to beoriented.

It is well-known that Dehn twists generate Γng,r — in fact we can choosea finite (fairly simple) set of generators, see [171] and Theorem 15.1.12.First we discuss relations which hold in Γng,r. In the following we will usethe usual functional notation for products in Γng,r.

Lemma 15.1.4. If f : Σng,r → Σn

g,r is an orientation preserving diffeomor-phism and a ⊂ Σn

g,r is a simple closed curve then ftaf−1 = tf(a).

Proof. Let a′ = f(a). Since f maps a to a′ we can assume that (up toisotopy) it also maps a neighborhood N of a to a neighborhood N ′ of a′.Let us examine the effect of applying ftaf

−1. The homeomorphism f−1

takes N ′ to N , then ta maps N to N , twisting along a, and finally f takesN back to N ′. Since ta is supported in N , the composite map is supportedin N ′ and is a Dehn twist about a′.

We say that a simple closed curve a ⊂ Σng,r is separating if Σn

g,r − a has twoconnected components — otherwise a is called nonseparating. Lemma 15.1.4together with the classification of 2-manifolds provides

Lemma 15.1.5. Suppose that a and b are nonseparating simple closedcurves in Σn

g,r. Then there is an orientation preserving diffeomorphismf : Σn

g,r → Σng,r which takes a to b. Consequently ta and tb are conjugate

in Γng,r. In particular, if a and b are homologically essential simple closedcurves in a surface with at most one boundary component then ta and tbare conjugate.

Exercise 15.1.6. Verify that if a intersects b transversely in a unique pointthen tatb(a) = b. (Hint: Use Figure 15.1.)

a

b

tat (a)bt (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 ⊂ Σng,r are disjoint then tatb = tbta. If a intersects

b in a unique point then tatbta = tbtatb.

Proof. The commutativity relation tatb = tbta is obvious. To prove the braidrelation tatbta = tbtatb we observe that tatb(a) = b (see Exercise 15.1.6). ByLemma 15.1.4 we get tatbta = tatbtat

−1b t−1

a tatb = ttatb(a)tatb = tbtatb.

Lemma 15.1.8. Let a1, a2, · · · , ak be a chain of curves, i.e., the consecutivecurves intersect once and nonconsecutive curves are disjoint. Let N denotea regular neighborhood of the union of these curves. Then the followingrelations hold:

• The commutativity relation: taitaj

= tajtai

if |i− j| > 1.

• The braid relation: taitajtai

= tajtaitaj

if |i− j| = 1.

• The chain relation: If k is odd then N has two boundary componentsd1 and d2, and (ta1ta2 · · · tak

)k+1 = td1td2 . If k is even then N has one

boundary component d and (ta1ta2 · · · tak)2k+2 = td.

d21d

a

a

a3

2

1

Figure 15.2. Chain relation for k = 3: (ta1ta2

ta3)4 = td1

td2

The next lemma was first observed by Dehn and then rediscovered byJohnson [78] who called it the lantern relation.

Lemma 15.1.9. Let U be a disk with the outer boundary a and with 3inner holes bounded by the curves a1, a2, a3. For 1 ≤ i ≤ 3, let bi be thesimple closed curve in U depicted in Figure 15.3. Then the lantern relation

tata1ta2ta3 = tb1tb2tb3

holds.


3

32

2

1

1 a

b

a

ba

b

a

U

Figure 15.3. The lantern relation

Lemma 15.1.10. If i denotes the hyperelliptic involution (i.e., rotation ofthe standard embedded Σg ⊂ R3 by 180 around the x-axis, see Figure 15.4)and a is a curve in Σg ∩ xy − plane then [i, ta] = 1.

Remark 15.1.11. The idea of the proofs of Lemmas 15.1.8, 15.1.9 and15.1.10 is the following: We split the surface into a union of disks by cuttingalong a finite number of simple closed curves and properly embedded arcs.We prove that the given product of Dehn twists takes each one of thesecurves (arcs, resp.) onto an isotopic curve (arc, resp.). Then the product isisotopic to a homeomorphism pointwise fixed on each curve and arc. ButAlexander’s lemma says that a homeomorphism of a disk fixing its boundaryis isotopic to the identity, relative to boundary. Thus the given product isisotopic to the identity.

Now a presentation of Γg (and Γg,1) can be given using the relationsdescribed above. It turns out that the mapping class groups Γg and Γg,1 are


Figure 15.4. The hyperelliptic involution i

generated by ta0 , . . . , ta2g with curves a0, . . . , a2g depicted in Figure 15.5.Let Aij = [tai

, taj] for all pairs (i, j) with ai ∩ aj = ∅. Let Bi denote

a2g+1a1a g2

a0

a aa2

3 5a4 a6

Figure 15.5. The simple closed curves inducing a generating system

the braid relation taitai+1tai

t−1ai+1

t−1ait−1ai+1

for i = 1, . . . , 2g − 1 and B0 =

ta0ta4ta0t−1a4 t

−1a0 t

−1a4 . Finally C, D and E = [i, ta2g+1 ] denote the appropriate

chain, lantern and hyperelliptic relations, cf. Figures 15.6 and 15.7. Noticethat there are a number of relations of type A and B, but the relationsC,D and E are unique (as shown by the figures). Write all these relationsin terms of the generators ta0 , . . . , ta2g and consider the normally generatedsubgroups R1 = 〈Aij , Bi, C,D〉No and R = 〈Aij , Bi, C,D,E〉No in the freegroup F2g+1 on 2g + 1 letters corresponding to the generators ta0 , . . . , ta2g .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→ F2g+1 → Γg → 1 and

1 → R1 → F2g+1 → Γg,1 → 1

are exact; in other words, the above generators and relations provide apresentation of Γg and Γg,1.


a2

a3a

0a

a1

Figure 15.6. The chain relation in the presentation

a1 a a3 5

a

Figure 15.7. The a-curves in the lantern relation of the presentation


Remark 15.1.13. For g = 2 omit the lantern relation to get the correctresult. If we denote tai

by ai for simplicity, an alternative presentation ofΓ2 can be given by generators a1, a2, a3, a4, a5, the braid and commutativityrelations for them (i.e., aiai+1ai = ai+1aiai+1 and aiaj = ajai for |i−j| ≥ 2),requiring that i = a1a2a3a4a

25a4a3a2a1 is central, i2 = 1, and finally that

(a1a2a3a4a5)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 gluinga disk with marked center to a boundary component) we get an obviouslysurjective map Γng,r → Γn+1

g,r−1. It is easy to see that the Dehn twist ∆ = tδalong a curve δ parallel to the boundary we collapsed becomes trivial. Infact,

1 → Z → Γng,r → Γn+1g,r−1 → 1

turns out to be an exact sequence, where Z is generated by tδ. Forgettingthe marked point we get a map Γng,r → Γn−1

g,r , and now the sequence

1 → π1(Σn−1g,r ) → Γng,r → Γn−1

g,r → 1

is exact (here π1(Σn−1g,r ) is the fundamental group of the (n− 1)-punctured

surface with r boundary components). Using these exact sequences, pre-sentations for all Γng,r can be derived by starting with Wajnryb’s result andknowing presentations for the kernels in the above short exact sequences;for such results see [58]. It follows that Γng,r is generated by finitely manynonseparating Dehn twists plus Dehn twists along boundary-parallel curves.In fact, if g ≥ 2 then for each boundary component of Σn

g,r we can embed alantern relation (as shown in Figure 15.9) in Σn

g,r in such a way that one ofthe boundary curves in the lantern relation is mapped onto that boundarycomponent of Σn

g,r and all the other curves in the lantern relation are non-separating in Σn

g,r. It follows that Γng,r is generated by finitely many Dehntwists along nonseparating curves for g ≥ 2.

Proposition 15.1.14 (Powell, [147]). For g ≥ 3 the commutator subgroup[Γg,Γg] is equal to Γg, i.e., Γg is a perfect group.

Proof. Let a be any nonseparating curve on Σg. For g ≥ 3, there is anembedding of a sphere with 4-holes (one of which is bounded by a) into Σg

where all seven curves in the lantern relation

tata1ta2ta3 = tb1tb2tb3


a

a a

b

bba 2

1 3

2

31

Figure 15.8. Appropriate lantern relation involving a with a nonseparating

a

a

bb

a

a

b

3

2

23

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 ai’s aredisjoint from the bj ’s we have

ta = tb1t−1a1 tb2t

−1a2 tb3t

−1a3 .

By Lemma 15.1.5, on the other hand, there are diffeomorphisms hi suchthat tbi = hitai

h−1i for i = 1, 2, 3. Substituting these expressions into the

relation above we get

ta = h1ta1h−11 t−1

a1 h2ta2h−12 t−1

a2 h3ta3h−13 t−1

a3 = [h1, ta1 ][h2, ta2 ][h3, ta3 ].

We showed that a nonseparating Dehn twist is a product of (three) commu-tators. This finishes the proof since Γg is generated by Dehn twists alongnonseparating 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 acommutator is a commutator.


Remark 15.1.15. In fact, any mapping class group Γng,r is perfect, i.e.,Γng,r/[Γ

ng,r,Γ

ng,r] = 0 for g ≥ 3 (see [81], for example). For g = 1, 2 it is

impossible to embed a lantern relation into Σg with nonseparating boundarycomponents, and hence the above proof breaks down from the beginning.Using the presentations of Γn1,r and Γn2,r, however, one can derive that

• Γn1,0/[Γn1,0,Γ

n1,0] = Z12,

• Γn1,r/[Γn1,r,Γ

n1,r] = Zr for r ≥ 1, and

• Γn2,r/[Γn2,r,Γ

n2,r] = Z10.

Lemma 15.1.16. Any element in Γg can be expressed as a product ofnonseparating right-handed Dehn twists.

Proof. The following is a standard relation in the mapping class group Γg:

(ta1ta2 · · · ta2g )4g+2 = 1,

where the curves ai are depicted in Figure 15.5. We deduce that t−1a1 is

a product of nonseparating right-handed Dehn twists. Therefore any left-handed nonseparating Dehn twist — being conjugate to t−1

a1 — is a productof nonseparating right-handed Dehn twists. This finishes the proof of thelemma combined with the fact that Γg is generated by (right and left-handed) nonseparating Dehn twists.

Exercises 15.1.17. (a) Show that any element in Γg,1 can be expressed asa product of nonseparating right-handed Dehn twists plus left-handed Dehntwists along a boundary-parallel curve. (Hint: Use the same argument asabove with the relation (ta1ta2 · · · ta2g )

4g+2 = tδ in Γg,1 where δ denotes acurve parallel to the boundary.)

(b) Show that a separating right-handed Dehn twist in Γg,1 can be expressedas a product of nonseparating right-handed Dehn twists.


15.2. Mapping class groups and geometric structures

As our earlier discussion indicated, the geometric objects we discussed inthe preceding chapters have counterparts in various mapping class groups.To clarify the situation, below we summarize these relations.

• A product Πki=1[ai, bi] of k commutators in Γg gives a Σg-bundle over

the surface Σk,1 with one boundary component. The mapping classesai and bi specify the monodromy along the obvious free generatingsystem 〈α1, β1, . . . , αk, βk〉 of π1(Σk,1). If Πk

i=1[ai, bi] = 1 in Γg, weget a Σg-bundle X → Σk. (The bundle is uniquely determined by theword once g ≥ 2.) In case Πk

i=1[ai, bi] = 1 holds in Γ1g, the bundle

X → Σk admits a section. In this case Πki=1[ai, bi] = (tδ)

n in Γg,1 forsome n ∈ Z, and it is not hard to see that the self-intersection of thesection given by this word is exactly −n.

• An expression Πki=1ti ∈ Γg with ti right-handed Dehn twists provides a

genus-g Lefschetz fibration X → D2 over the disk with fiber Σg closed.If Πk

i=1ti = 1 in Γg then the fibration closes up to a fibration over thesphere S2 and the closed up manifold is uniquely determined by theword Πk

i=1ti once g ≥ 2. Once again, a lift of the relation Πki=1ti = 1

to Γ1g shows the existence of a section, and its self-intersection is −n

if Πki=1ti = (tδ)

n in Γg,1 for the Dehn twist tδ along the boundary-parallel simple closed curve δ ⊂ Σg,1.

• By combining the above two constructions, a word

w = Πk′i=1tiΠ

kj=1[ai, bi]

gives a Lefschetz fibration over Σk,1 and if w = 1 in Γg we get aLefschetz fibration X → Σk. Sections can be captured in the sameway as above.

• An expression Πni=1ti = tδ1 · · · tδk in Γg,k (where all ti stand for right-

handed Dehn twists and tδi are right-handed Dehn twists along circlesparallel 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 afterblowing these sections down we get a Lefschetz pencil. Conversely, byblowing up the base locus of a Lefschetz pencil we arrive to a Lef-schetz fibration which can be captured (together with the exceptionaldivisors of the blow-ups, which are all sections now) by a relator ofthe above type.

15.3. Some proofs 265

• If we allow the Dehn twists ti to have negative exponents in the previ-ous constructions, we can also encounter achiral Lefschetz fibrationsin this way.

• An element h ∈ Γg,r (r > 0) specifies a 3-manifold equipped withan open book decomposition by considering the mapping cylinderof h and collapsing the boundaries to the core circles. Notice thatthe binding has r components. Through the equivalence discussedin Section 9 the mapping class h ∈ Γg,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 openbook decomposition (and so leaving the contact structure unchanged)we can change g and r.

• Since Πni=1ti ∈ Γg,r gives a Lefschetz fibration with nonclosed fibers

over the disk D2, and these manifolds can be equipped with Steinstructures, a factorization h = Πn

i=1ti in Γg,r into right-handed Dehntwists gives a Stein filling of the contact 3-manifold determined byh ∈ Γg,r. All Stein fillings arise in this manner, although we mightneed to pass to a stabilization of h to recover certain fillings of thecontact 3-manifold specified by h.

15.3. Some proofs

We close this chapter with a few results which show an interesting bridgebetween Lefschetz fibrations, contact structures and mapping class groups.For g ≥ 3, Proposition 15.1.14 shows that Γg is a perfect group, i.e.,every element of Γg is a product of commutators. The minimal numberof commutators one has to use to express an element as a product in agroup is called the commutator length of that element.

Theorem 15.3.1 ([83]). The commutator length of a Dehn twist in Γg(g ≥ 3) is equal to two.

Proof. Consider a sphere X with four holes with boundary componentsa, a1, a2, a3. Since the genus of Σg is at least three, X can be embedded inΣg in such a way that a1, a2, a3, b1, b2, b3 are all nonseparating. The simpleclosed curve a can be chosen either nonseparating or separating boundinga subsurface of arbitrary genus (cf. Figures 15.8 and 15.9). Furthermore,


the complement of a1 ∪ b1 and of a2 ∪ b2 are connected. Hence, there isan orientation preserving diffeomorphism f of Σg such that f(a1) = b2 andf(b1) = a2. Let h be another orientation preserving diffeomorphism of Σg

such that h(a3) = b3. Then the lantern relation combined with the abovechoices implies

ta = tb1t−1a1 tb2t

−1a2 tb3t

−1a3 = tb1t

−1a1 tf(a1)t

−1f(b1)hta3h

−1t−1a3

= tb1t−1a1 fta1f

−1ft−1b1f−1hta3h

−1t−1a3 = [tb1t

−1a1 , f ][h, ta3 ].

Next we show that the commutator length of a Dehn twist is not equal toone. 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. Itis easy to see that χ(X) = 1. Since the fibration is relatively minimal, and soby Proposition 10.3.8 the 4-manifold X is a minimal symplectic 4-manifold,we have the inequality

0 ≤ c21(X) = 3σ(X) + 2χ(X)

which implies that σ(X) ≥ −23 . This gives σ(X) ≥ 0 since σ(X) is an

integer. Recall that the holomorphic Euler characteristic is defined by

χh(X) =1

4

(σ(X) + χ(X)

)

and it is an integer for any closed almost-complex, hence for any closedsymplectic 4-manifold. Rewriting the above equality we get

χh(X) =1

4

(σ(X) + 1

).

Therefore σ(X) = 4χh(X) − 1 and so

c21(X) = 3σ(X) + 2χ(X) = 12χh(X) − 1.

On the other hand, by [155] it follows that

c21(X) ≤ 10χh(X)

since X admits a Lefschetz fibration over T 2. Since the holomorphic Eu-ler characteristic χh(X) is an integer, it follows that χh(X) ≤ 0 imply-ing σ(X) + 1 = 4χh(X) ≤ 0. This last inequality, however, contradictsσ(X) ≥ 0, which has been shown earlier.


Recall from Proposition 15.1.14 that Γg is a perfect group for g ≥ 3.The mapping class group Γg is, however, not uniformly perfect, that is,there is no constant K such that any element of Γg can be written as aproduct of at most K commutators. This statement can be proved byusing the correspondence between certain words in mapping class groupsand Lefschetz fibrations. (For a different proof see [13].)

Theorem 15.3.2 (Endo–Kotschick [36], Korkmaz [82]). Let c ⊂ Σg bea separating simple closed curve. If tnc = Πkn

i=1

[αi(n), βi(n)

]then the

sequence kn cannot be bounded. In conclusion, the mapping class groupΓg is not uniformly perfect.

Proof. Notice that a commutator expression of the type of the theorem givesa relator which gives rise to a Lefschetz fibration Xn → Σkn . Suppose thatkn is bounded, say kn ≤ K. By adding trivial monodromies if necessary,this assumption provides a sequence fn : Xn → ΣK (n ∈ N) of Lefschetzfibrations over the fixed base ΣK . It is easy to see that

χ(Xn) = χ(Σg)χ(ΣK) + n = 4(K − 1)(g − 1) + n,

while by Novikov additivity and the signature calculation for a separatingvanishing cycle (cf. [130]) we get

σ(Xn) = −n+ σ(X − ∪ni=1νf

−1n (qi)

).

On the other hand one can show that

σ(X − ∪ni=1νf

−1n (qi)

)≤ C

for some constant C depending on K and g only. (The points qi denote thecritical values of the Lefschetz fibration fn.) This implies that

c21(Xn) = 3σ(Xn) + 2χ(Xn) ≤ −3n+ 2n + C ′ = −n+ C ′,

where C ′ = 3C+8(K−1)(g−1) and hence for n large enough the expressionc21(Xn) will be negative. This observation contradicts the result of [155]where it is proved that a relatively minimal Lefschetz fibration over abase of positive genus is minimal, hence its c21 invariant is nonnegative,cf. Corollary 10.3.10. The contradiction shows that the sequence kn isunbounded, verifying the statement of the theorem.


Remark 15.3.3. In fact, using the exact sequences in Section 15.1 onecan show that the mapping class group Γng,r is not uniformly perfect. Asdiscussed in [36, 82], the fact that Γng,r is not uniformly perfect has inter-esting corollaries regarding the second bounded cohomology of Γng,r. Also,the proof given above can be refined to get explicit lower bounds for thecommutator lengths for certain elements in Γng,r; for details see [36, 82].

Similar question can be raised for the length of expressions writing a givenelement as product of right-handed Dehn twists. Since by Lemma 15.1.161 ∈ Γg can be written as a nontrivial product of right-handed Dehn twiststhere is no bound for the length of such expression for h ∈ Γg. The situation,however, is different in Γg,r once r > 0.

Theorem 15.3.4 ([7, 157]). If r ≥ 1 then 1 ∈ Γg,r admits no nontrivialfactorization 1 = t1 · · · tn into a product of right-handed Dehn twists.

Proof. Suppose that 1 ∈ Γg,r admits a nontrivial factorization 1 = t1 · · · tninto a product of right-handed Dehn twists. Now cap off all but one of theboundary components with disks to get a relation in Γg,1 where identity isexpressed as a nontrivial product of right-handed Dehn twists. Thus we re-duce the problem to show that 1 ∈ Γg,1 admits no nontrivial factorization1 = t1 · · · tn into a product of right-handed Dehn twists. Clearly we can as-sume that g ≥ 1. Moreover we can assume that all the ti’s are nonseparatingDehn twists since any separating right-handed Dehn twist in Γg,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 Γg,1 can be expressedas a product of nonseparating Dehn twists. Now replace every left-handedDehn twist in this expression by a product of right-handed Dehn twists toconclude that any element in Γg,1 can be expressed as a product of (non-separating) right-handed Dehn twists. We will show that this is impossibleusing contact geometry. For any given g ≥ 1 we can construct a genus-gsurface with one boundary component by plumbing left-handed Hopf bands.This would give us an open book with monodromy φ ∈ Γg,1 whose compat-ible contact structure ξ is overtwisted by construction. But if φ could beexpressed as a product of right-handed Dehn twists then ξ would be Steinfillable, leading to a contradiction. For a different proof see [157].


This observation leads us to

Conjecture 15.3.5. For any mapping class h ∈ Γg,r with r > 0 there is aconstant Ch such that if h = t1 · · · tn factors as a product of right-handedDehn twists in Γg,r then n ≤ Ch.

The affirmative solution of this conjecture would provide a bound for Eulercharacteristics of Stein fillings of fixed open book decompositions — a weakerversion of the statement given in Conjecture 12.3.16.

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Index

achiral Lefschetz fibration, 155adjunction

– equality, 51– formula, 243– inequality, 13, 226

almost-complex structure, 50, 106almost-Kahler structure, 50

basic class, 226Bennequin inequality, 21, 77binding, 131blackboard framing, 29botany, 53boundary connected sum, 26branch set, 150bypass, 183

characteristic foliation, 78, 87classification, 179compatible, 191complex point, 127connected sum, 26

– boundary, 26contact

– 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, 66continued fraction, 35convex, 86

Darboux theorem, 70Dehn

– surgery, 31– contact, 185

– twist, 158, 193, 255destabilization

– negative, 137– positive, 137

Dirac operator, 224dividing set, 86dotted circle, 39, 169

elimination lemma, 82elliptic, 128

– singularity, 80

fibered link, 132fillable

– holomorphically, 201– Stein, 201, 254– strongly symplectically, 201– weakly symplectically, 201

filling– Stein, 230, 265

Floer homology, 241four-ball genus, 18framing, 27

282 Index

– blackboard, 29, 74– contact, 68– Seifert, 29– Thurston–Bennequin, 68

Fredholm map, 225Frobenius theorem, 64front projection, 72

gauge group, 224geography, 53gordian number, 18

handle, 28handlebody, 28

– relative, 28h-cobordism theorem, 11Heegaard

– decomposition, 27, 96– diagram, 30, 236

Hirzebruch signature theorem, 104Hodge star operator, 224holomorphic convex hull, 121holomorphically convex, 121Hopf link, 133hyperbolic, 128

– singularity, 80hyperelliptic involution, 258

Kirby calculus, 38Kodaira 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, 223Liouville vector field, 113longitude, 32

mapping class group, 131, 255– presentation, 259

meridian, 32minimal model, 58monodromy, 131Moser’s method, 55Murasugi sum, 134

neighborhood theorem– contact, 70– Lagrangian, 57– symplectic, 56

nonisolating, 90normal connected sum, 114

ω-concave, 111ω-convex, 111open book decomposition, 96, 131

– binding, 131– compatible, 138, 191– monodromy, 131– page, 131– standard, 133

overtwisted– contact structure, 21, 76– disk, 76

Ozsvath–Szabo invariant, 22– contact, 205, 249

page, 131PALF, 163perfect, 267

– uniformly, 267plumbing, 134plurisubharmonic, 122pseudo-holomorphic submanifold, 51pseudoconvex, 123

rational surgery, 31real point, 127Reeb vector field, 67regular fiber, 155relation

Index 283

– braid, 257– chain, 257– commutativity, 257– hyperelliptic, 258– lantern, 257

relatively minimal, 155Rolfsen twist, 35rotation number, 74

Sard–Smale theorem, 225Seiberg–Witten

– invariant, 223– moduli space, 224

– parametrized, 224– simple type, 226, 228

Seifert– fibered manifold, 45– framing, 29

self-linking number, 82simple

– (ADE) singularity, 218– cover, 150– elliptic singularity, 218– type, 226

singular fiber, 155singularity

– elliptic, 80– hyperbolic, 80

slam-dunk, 35slice genus, 18slope, 91sobering arc, 148spin

– group, 99– structure, 99

– induced, 101spinc

– group, 100– structure, 100, 240

– induced, 101stabilization

– negative, 137– positive, 137

state traversal, 205Stein

– cobordism, 124– domain, 124, 162

– manifold, 121– surface, 122

surface bundle, 264surgery, 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, 51Symplectic Thom conjecture, 227symplectization, 71

3-dimensional invariant, 105Thom conjecture, 232Thurston–Bennequin

– framing, 68– invariant, 166

tight contact structure, 21totally real submanifold, 51transverse knot, 682-plane field, 102twisted coefficient system, 254twisting, 138

unknotting number, 18

vanishing cycle, 156Vitushkin’s conjecture, 17

Weinstein handle, 115Whitney

– disk– holomorphic, 240

– trick, 11writhe, 73

B. Ozbagci and A. I. Stipsicz- Surgery on contact 3-manifolds and Stein surfaces

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