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SOME APPLICATIONS OF THE GIROUX CORRESPONDENCE IN
LOVV-DIMENSIONAL TOPOLOGY

By
Cagri Karakurt

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
I\«‘Iathematics

2010

ABSTRACT

SOME APPLICATIONS OF THE GIROUX CORRESPONDENCE IN
LOW-DIMENSIONAL TOPOLOGY

By
Cagri Karakurt

E. Giroux has showed that the study of contact structures is equivalent to that of
open book decompositions on 3—manifolds. In this thesis, we discuss two applications
of this correspondence in the low-dimensional tOpology.

In the first part, we show that every compact smooth 4-manifold X has a structure
of a Broken Lefschetz Fibration. Furthermore, if b; (X) > 0 then it also has a Broken
Lefschetz Pencil structure with nonempty base locus. This improves a theorem of
Auroux, Donaldson and Katzarkov, and our construction uses only the 4—dimensional
handlebody theory.

In the second part, we study some invariants of contact structures that arise
from the Giroux correspondence. We show that the Ozsvath-Szabé contact invariant
c+(€) E H F+(—Y) of a contact 3—manifold (Y, £) can be calculated combinatorially
if Y is the boundary of a certain type of plumbing X, and 5 is induced by a Stein
structure on X. Our technique uses an algorithm of Ozsvath and Szabé to deter-
mine the Heegaard-Floer homology of such 3—manifolds. We discuss two important
applications of this technique in contact topology. First, we show that it simplifies
the calculation of the Ozsvath-Stipsicz-Szabé obstruction to admitting a planar open
book. Then we define a numerical invariant of contact manifolds that respects a par-
tial ordering induced by Stein cobordisms. We do a sample calculation showing that

the invariant can get infinitely many distinct values.

ACKNOWLEDGMENT

I would like to thank my supervisor Dr. Selnian Akbulut for his excellent guidance,
patience and constant encouragement. The first part of this thesis is a joint work with
him. I am so grateful to him for sharing his expertise with me.

Many thanks to my committee members Dr. Nikolai Ivanov, Dr. Rajesh Kulkarni,
Dr. Michael Shapiro, and Dr. Benjamin Schmidt. I would also like to thank Dr. Ron
Fintushel for teaching me the basics of 4—manifolds. A special thanks goes to Dr.
Matt Hedden for his interestng course on Heegaard-Floer Theory and his mentorship

in the final year of my Ph. D. education.

iii

TABLE OF CONTENTS

List of Tables ................................. v
List of Figures ................................ vi
Introduction ........................... 1
Prelimineries .......................... 9
2.1 Contact Structures ............................ 9
2.2 Legendrian Knots ............................. 11
2.3 Fillings ................................... 13
2.4 Contact Surgery .............................. 15
2.5 Homotopy Invariants ........................... 16
2.6 Open Book Decompositions and the Giroux Correspondence ..... 18

Constructing Broken Lefschetz Fibrations on Four-Manifolds . . 22

3.1 Introduction ................................ 23
3.2 Concave F ibrations and Pencils ..................... 25
3.3 Almost Stein Manifolds .......................... 31
3.4 Destabilizations .............................. 36
3.5 ALF-Almost Stein ............................ 38
3.6 Some ALF Moves and Their Effects ................... 42
3.7 Matching the Open Books ........................ 44
3.8 Repairing Achiral Singularities ...................... 46
3.9 Proof of the Main Theorem ....................... 48
Calculation of the Ozsvath-Szabo Contact Invariant ....... 51
4.1 Heegaard—Floer Homology and Contact Invariant ........... 51
4.2 The Algorithm .............................. 54
4.3 Main Theorem .............................. 60
4.4 Example .................................. 61
4.5 Calculation of a .............................. 65
Bibliography .......................... 71

iv

LIST OF TABLES

, , ) for 10 spine structures ........... 66

4.1 HF+ of M(—4,

toll—-
NIH
ooIH

LIST OF FIGURES

Images in this dissertation are presented in color

2.1 A Legendrian Trefoil ........................... 11
2.2 Legendrian Reidemeister moves ..................... 12
3.1 Nine—P?— .................................. 26
3.2 the passage from flat to concave fibrations ............... 29
3.3 Stabilization of concave fibrations .................... 30

3.4 Negative stabilization of concave fibrations without achiral singularity 31

3.5 Framing fixing in almost Stein manifolds ................ 33
3.6 Repairing defective handles in an almost Stein manifold ........ 35
3.7 Isotopy by finger moves. ......................... 38

3.8 Two ALF 5 corresponding to the Almost Stein manifolds in Figure 3.6 41

3.9 Square—bridge Reidemeister moves .................... 42
3.10 Positive and negative finger moves .................... 43
3.11 Another interpretation of positive finger move ............. 44
3.12 Repairing bad framings .......................... 47

vi

3.13 Alf picture of figure 3.12 ......................... 48

4.1 Plumbing description of Seifert fibered space AI(—4, %, - -- ,%, 31;) . . . 62
4.2 Legendrian handlebody diagram giving .12,_1. ............ 64
4.3 Kn......._. ............................. 65

4.4 A Plumbing graph for Brieskorn homology sphere 2(2, 272. + 1,471 + 3) 68

4.5 Sequence of Blow-ups from Kn to plumbing .............. 70

vii

Chapter 1

Introduction

Mathematics is beautiful! One can appreciate its beauty even more when he/ she sees
results that connect seemingly different areas. In year 2002, E. Giroux announced
such a result in the International Congress of Mathematics, [Gi]. His result estab-
lished the equivalence two objects of different nature that appear on 3—manifolds.
More precisely, he proved that there is a one to one correspondence between con-
tact structures and open book decompositions on a closed oriented 3—manifold up to
certain equivalence relations. The strength of his theorem comes from the fact that
these two live in completely different realms; While contact structures are analytic
objects that are defined by means of smooth 1—forms, open book decompositions are
geometric in nature; They are fibrations over a circle that are defined in the comple-
ment of a link. Some of the remarkable applications of the Giroux correspondence
are the proof of a 20 years old conjecture stated by Harer [H2] and proven by Giroux
and Goodman [CO], the definition of a Floer theoretic invariant of contact structure
given by Ozsvath and Szabo [082], and a symplectic compactification theorem due to
Eliashberg [E4], see also [Et2] and [A02], which was used in the proof of the Property

P conjecture.

The aim of this thesis is to discuss two other applications of Giroux’s correspon-

1

dence. In the first part of the thesis, we discuss constructions of Lefschetz type
fibrations on 4—manifolds. Open books naturally appear in the boundary of these
objects and the Giroux correspondence will be used in an essential way to glue two
such fibrations. In order to state our theorem in a precise form, we first need to recall

some basic definitions and fix conventions:

Let X be an oriented 4-manifold and E be an oriented surface. “"6 say a map 7r :
X —+ E has a Lefschetz singularity at p E X, if after choosing orientation preserving
charts (C2,0) 9) (X. p) and C 9+ 2, 7r is represented by the map (2, w) H zw. If 7r
is represented by the map (z, w) +—+ SID we call p an achiral Lefschetz singularity of
7r. We say p E X is a base point of 7r, if it is represented by the map (2,211) +—> z/w.
We call a circle Z C X fold singularity of 7r, if around each p E Z we can find charts

(IR x R3, IR) ¢—> (X, Z) where 7r is represented by the map
(t, r1,r2,r3) r—> (Lit? + 27% — 13%) (1.1)

Definition 1.0.1. Let X4 be a. compact oriented 4-manifold. A broken Lefschetz
fibratz'on structure on X is a map 7r : X —> 82 which is a submersion in the comple-
ment of disjoint union of finitely many points P and circles C in X, such that P are
Lefschetz singularities of 7r, and C are fold singularities of 7r. When C = 0, we call 7r

a Lefischetz fibr‘ation.

We call 7r a broken Lefschetz pencil, if there is a finite set of base points 3 in X
(base locus) such that the restriction 7r : X — B —> 52 is a broken Lefchetz fibration.

W hen C = 0 we call 7r a Lefschetz pencil.

Furthermore, If we allow achiral Lefschetz singularities among the points of P we
will use the adjective “achiral” in front of all these definitions, e.g. Achiral Broken
Lefschetz F ibratz'on. Sometimes for brevity we will refer them simply by LF, BLF,
ABLF, LP, BLP, ABLP.

V’Vhen these singular fibrations defined on a 4—manifold with boundary one can
describe three boundary behaviors: convex, concave and flat, see definition 3.1.1. By
following a convention due to Akbulut and Ozbagci, we will call a convex Lefschetz
fibration without achiral or fold singularities a Positive Allowable Lefschetz F z'bration,
or PALF for short, if every vanishing cycle is a non-separating curve on fiber. Now,

we can state our first main theorem which was proven jointly with Akbulut [AK]:

Theorem 1.0.2. Every closed smooth oriented 4—manifold X admits a broken Lef-
schetz fibration. Furthermore if b;- (X) > 0 then X also admits a broken Lefschetz

pencil structure (with nonempty base locus).

Remark 1.0.3. Previously by using analytic methods which generalize the technique
of approximately holomorphic sections in [D], Auroux, Donaldson and Katzarkov
proved that every closed smooth oriented 4~manifold with b;- (X) > 0 admits a broken
Lefschetz pencil structure [ADK], so in particular they proved that, after blowing up,
X admits broken Lefschetz fibration structure. More recently, by using the map
singularity theory Baykur [B] and independently Lekili [L] proved that every closed
smooth oriented 4-manifold X admits a. broken Lefschetz fibration. Our proof and [L]
give a stronger result, namely make the “fold singularities” of 7r global, i.e. around
each singular circle Z we may assume that there are charts (.5'1 x R3, S 1) <—> (X, Z)
so that 7r is represented by the map (1.1). Recently, in [GK2] there was an attempt
by Gay and Kirby to give a topological proof of [ADK], which ended up producing a
weaker version of Theorem 1.2 with “achiral broken Lefschetz fibration” conclusion.
Here we prove this theorem by using 4—dimensional handlebody techniques. Also. a
corollary to our proof is that, X can be decomposed as a union of a convex PALF and a
concave BLF glued along their common (open book) boundaries. Lekili described a set
of moves of (achiral) broken fibrations that does not change the ambient 4—manifold,

[L]. One of the moves assert that an achiral singularity can be traded with three

3

fold singularities. After the results of this thesis announced in [AK], Baykur wrote
an appendix [B3] to [L] interpreting Lekili’s moves in terms handlebodies. Therefore
the combination [GK2], [L] and [BB] also gives a handlebody proof of theorem 1.0.2.
Another recent remarkable result about broken fibrations is due to Williams [W]
which says that Lekili’s moves are sufficient to connect any two broken fibrations in

the same homotopy class.

Our proof here proceeds by combining the PALF theory of [A0] with the decom-
position (positron) technique of [AM], and it strongly suggests that linking a certain
4-dimensional handlebody calculus on ALF’s to the Reidemeister moves of the knots
in square bridge position. The philosophy here is that, just as a PALF f is a topo-
logical structure which is a primitive of a Stein manifold W' = |.7-'|, an ALF is weaker
topological structure which is a primitive of an “Almost Stein manifold” (for exam-
ple, while the boundaries of PALF’S are positive open books, the boundaries of ALF’s
are all open books). ALF’s are more flexible objects, which are amenable to handle
calculus. Our argument uses this flexibility. Unlike the alternative proof mentioned

above, the positron technique does not create extra fold singularities.

In the second part of this thesis, we study some invariants contact structures
arising from the Giroux correspondence and their relations. In [OS2], Ozsvath and
Szabc') defined a Heegaard—Floer homology class that reflects certain properties of a
given contact structure using a compatible Open book. In another direction, based
on Giroux’s work, Ozbagci and Etnyre [EtnO] defined an invariant, so-called support
genus, which is the minimal page genus of an open book decomposition compatible
with a fixed contact structure. Previously, Etnyre [Et] had found out that being
supported by a genus zero open book puts some restrictions on intersection forms of
symplectic fillings of a contact structure. His result was later improved by Ozsvath,
Stipsicz and Szabo who showed that the image of the Ozsvath-Szabo contact invariant

in the reduced version of Heegaard-Floer homology is actually an obstruction to be

4

supported by a planar open book. More precisely, they proved the following.

Theorem 1.0.4. (Theorem 1.2 in [088]) Suppose that the contact structure is on
Y is compatible with a planar open book decomposition. Then its contact invariant

c+(€) E HF+(—Y) is contained in Ud - HF+(—Y) for all 0'. E N.

In spite of having useful corollaries, this theorem may not be easy to apply all
the time because it involvolves calculation of the group H F + and identification of
the contact invariant in this group. The former problem can be solved if we restrict
our attention to a certain class of manifolds. In [081], Ozsvath and Szabo gave a
purely combinatorial description of Heegaard-Floer homology groups H F + of some
3—manifolds which are given as the boundary of certain plumbings of disk bundles over
sphere. We will show how to pin down the contact element within this combinatorial
object.

To state our second main results, in what follows we shall assume that G is a
negative definite plumbing tree with at most one bad vertex. Let X (G) and Y(G) be
the 4- and 3—manifolds determined by the plumbing diagram respectively. Denote the
set of all characteristic co-vectors of the lattice H 2(X (G), Z) by Char(G). W'e form
the group K+(G) = (Z7720 x Char(G))/ ~ where the relation N is to be described
in Section 4.2. Recall from [OS2] that the Heegaard-Floer homology group H F + of
any 3-manifold is equipped with an endomorphism U. In [OS3] (see also Section 4.2

below), Ozsvath and Szabé established the following isomorphism.

 

me) ~ . + _ .
Hom (Z>0 x Char(G)oZ/2Z) _ ker(U) C HF ( Y (0)) (1.2)

Recall that if g is a contact structure, its Ozsvath-Szabo contact invariant c+({)
is a homogeneous element in Ker(U) C H F +(—Y(G)). It is also known that 0+ (5)

is non-zero if 5 is induced by a Stein filling. The following proposition pins down the

5

image of contact invariant under the above isomorphism.

Proposition 1.0.5. Let J be a Stein structure on X (G) and E be the induced contact
structure on Y(G). Under the identification described in equation 1.2, the contact

invariant c+(§) is represented by the dual of the first Chern class c1 (J) E H 2(X , Z)

Remark 1.0.6. This proposition can be generalized in several different directions.
First, we may allow the graph G to have two bad vertices. In this case, the group
on the left hand side of equation 1.2 gives only even degree elements in Ker(U) C
H F +(—Y(G)). Second, the graph G which has at most one bad vertex could be
semi-definite implying that b1(Y) = 1, and we use the generalization of the Ozsvath-
Szabc') algorithm given in [R2]. Finally, keeping G positive definite, we may require J
to be an w—tame almost complex structure on X (G) for some symplectic structure
an which restricts positively on the set of complex tangencies of Y(G') (i.e. (X (G), w)
forms a weak filling rather than a Stein filling for the corresponding contact structure

on the boundary).

When combined with Theorem 1.0.4, Proposition 1.0.5 allows us to determine
whether or not certain contact structures admit planar open books. Recall from
[OS3] that the correction term for any spinc structure t of a rational homology 3-
sphere Y is the minimal degree of any non-torsion class in H F +(Y, t) coming from

HF°0(Y, t).

Theorem 1.0.7. Assume the conditions in Proposition 1.0.5. Denote the correc—
tion term of the induced spinc structure t on Y(G) by (1. Also, let (1305) be the

3-dimensional invariant of the contact structure 5. Suppose that we have either

6

d3(€) # —d — 1/2 or rank(HF(j—(—Y(G),t)) > 1 then f can not be supported by a

planar open book.

Note that checking the conditions stated in this theorem is simply a combina-
torial matter, [OSS] (see also Section 4.2 below). Corollary 1.7 of [088], which
holds for arbitrary rational homology 3-spheres, implies the above statement when
(13 74 -cl(€) — 1 / 2. This could be taken as an evidence to conjecture that Theorem

1.0.7 also holds for every rational homology 3-sphere.

Remark 1.0.8. There is another version of Ozsvath-Szabé contact invariant C(é‘)
which lives in EFT—Y) and is related to c+(£) by l(c(§)) = c+(§) where L is the
natural map L : HF(—Y) —> H F +(—Y). The invariant C(fi) can be calculated combi-
natorially as shown in [P] and [BP]. However, for the present applications the usage

of the 0+ is essential.

The techniques of this thesis can also be used to study a natural partial ordering on
contact 3—manifolds up to some equivalence. Following [E2], [EH] and [Ca], we write
0/1351) -_< (Y2,§2) if there is a Stein cobordism from (Ybél) to (Y2,{2). Moreover,
we write (Y1, g1) ~ (Y2, {2) if these contact manifolds satisfy (Y1, {1) j (Y2,§2) and
(Y1, 61) j (Y2,.£2). Clearly, ~ defines an equivalence relation on the set of contact
manifolds, and j is a partial ordering on the equivalence classes. One can define a
numerical invariant of contact manifolds that respects this partial ordering. Namely,

if we let

(yo/,5) = —max {d : c+(§) e Ud . HF+(—Y)}

the naturality properties of the Ozvath-Szabo contact invariant (of. Section 4.1

below) imply that we have 0(Y1,El) g 0(Y2,§2) whenever (Y1, {1) j (Y2,€2). Note

7

that 0 invariant can be infinite. In fact, 0(Y, é) z —00 if (Y, «5) admits a planar open
book by Theorem 1.0.4. Clearly, if two contact manifolds have different a-invariants,
they lie in different ~ equivalence classes. The following theorem tells that there are

infinitely many such equivalence classes.

Theorem 1.0.9. Any negative integer can be realized as the 0 invariant of a contact

manifold.

In fact, we are going to obtain some contact manifolds with distinct o invariants by
doing Legendrian surgery on certain stabilizations of some torus knots. See Theorem
4.5.1 below.

The organization of this thesis is as follows. In Chapter 2, we discuss the basics
of contact geometry and state the Giroux correspondence in a precise form. The
remaining chapters are independent from each other. In Chapter 3, we prove the
existence theorem (i.e. Theorem 1.0.2) of Broken Lefschetz fibrations on 4—manifolds.
Chapter 4 is devoted to the study of the Ozsvath-Szabo’ contact invariant and the

proofs of Proposition 1.0.5, Theorem 1.0.7 and Theorem 1.0.9.

Chapter 2

Prelimineries

2. 1 Contact Structures

In this section we discuss basics of contact topology. For a more detailed discussion,
reader can consult [Ga], or [OS].

Let Y be a closed oriented 3—manifold. A contact structure g on Y is a 2—plane
distribution on the tangent bundle TY which is totally non-integrable. This condition
means that locally one can express 5 as the kernel of a one form a such that a /\ do
is a positive multiple of the volume form. If such a 1—f0rm exists globally, we call g
a co-orientable contact structure. We always assume that our contact structures are

co—orientable. The pair (Y, {) is called a contact manifold.

Equivalence of contact structures can be defined in several ways. One natural
equivalence relation on the set of contact structures is homotopy. Two contact struc-
tures {1 and {2 on a 3—manifold Y are called homotopic if they are homotopic when
regarded as 2—plane fields (i.e. sections of the Grassmannian bundle Gr2(TY) —) Y).
A stronger notion of equivalence is isotopy. Two contact structures {1 and {2 on a
3—manifold Y are called isotopic if there is a diffeomorphism gt : Y —> Y which is

isopic to the identity diffeomorphism such that ¢*(£1) = {2. By a theorem of Gray

9

[Gra], this is equivalent to that there is a homotopy between {1 and {2 through con-
tact structures. Finally, one can talk about equivalence of contact manifolds. Two
contact manifolds (Y1, {1) and (Y2,§2) are said to be contactomorphic if there is a
diffeomorphism q“) : Y1 —> Y2 such that @1451) = 95*(52).

Example 2.1.1. On R3, consider the one form a = dz +rdy. It is easy to see that a
satisfies the non-integrability condition, so the 2—plane field {std = kera is a contact

structure. We refer this as the standard contact structure on R3.

There is a basic dichotomy in the classification of contact structures. To state
this, we need to give a few definitions first. Let (Y, 5) be a contact manifold. An
embedded curve K C Y is called Legendrian if the tangent space T pK of K lies in
the contact plane {p at every point p E K. If K is a Legendrian knot, the contact
structure induces a framing (i.e. a section of the normal bundle) on K, see section 2.2.
If K is in addition null homologous, this framing is determined by the number twists
it makes relative to the framing induced by a Seifert surface. This number is called
the Thurston-Bennequin invariant of the Legendrian knot K, and it is denoted by
tb( K). “Note that the Thurston-Bennequin invariant does not depend on the choice of
a Seifert surface used in its definition. An embedded disk D is said to be overtwisted
if 8D is Legendrian with tb(3D) = 0. A contact manifold is called overtwisted if it
contains an overtwisted disk, otherwise it is called tight. The isotopy classification of
overtwisted contact structures is well-understood thanks to the following theorem of

Eliashberg, [E3].

Theorem 2.1.2. Any homotopy class of 2—plane fields on a 3—manifold can be
represented by an overtwisted contact structure. Two overtwisted contact structures

are isotopic if and only if they are homotopic.

The classification of tight contact structures, however, is in general a difficult

problem. The solution is known only for some special cases.

10

2.2 Legendrian Knots

The aim of this section is to recall some of the basic facts about Legendrian knots
in (R3’€Std). Our plan is to define a special projection and Show how elementary
invariants of Legendrian knots can be determined by studying this projection.

Let L be a Legendrian knot in (R3,§Std). The image of L under the projection
map (33,31, .3) —> (y, 2:) is called the front projection of L. It is easy to show that L
is completely determined by its front projection. Indeed, the Legendrian condition

implies that the :1: coordinate of any point on L can be recovered by the formula

(1::

—E. (2.1)

(E:

Since the a; coordinate of any point on K is finite, this formula rules out vertical
tangencies in a front projection. Instead, we see cusps in front projections as in Figure
2.1. Note that cuspidal singularities disappear when we lift the knot to R3. Another
notable point about front projections is that they allow only certain types of crossing.
Indeed, Equation 2.1 tells that at any crossing the strand with more negative s10pe

must be over the other strand.

[\/

31

Figure 2.1: A Legendrian Trefoil

11

Equivalence of Legendrian knots can be defined in a natural way; we say that
two Legendrian knots are Legendrian isotopic if they are isotopic through Legendrian
knots. In (R3, (Std), any Legendrian isotopy can be broken into the elementary pieces
shown in Figure 2.2. These will be called Legendrian Reidemeister moves. Of course,

any Legendrian isotopy invariant should not be changed under these moves.

gm
<:——~\<

 

Figure 2.2: Legendrian Reidemeister moves

Let L be a Legendrian knot in (R3véstdl' Two classical Legendrian isotopy in-
variants of L can be formulated easily using the front projection of L. We denote
the Thurston-Bennequin invariant of L by tb(L) and the rotation number by rot(L).

These two invariants are given by the following formulae

tb(L) = u:(L)+%c(L) (2.2)
rota) = yam—cam). (2.3)

where w(L),c(L), cd(L), and cd(L) denote the writhe, the total number of cusps, and

12

the number of upward and downward cusps respectively. To define the last two we
need to fix an orientation. So, rot(L) depends on the orientation of L. It is easy to
see that tb(L) and rot(L) do not change under Legendrian Reidemeister moves.
There is another interpretation of the T hurston-Bennequin invariant in terms of
relative framings. Let L be a Legendrian knot in an arbitrary contact manifold. The
contact planes induce a framing (i.e. a homotopy class of nowhere vanishing local
vector field) on L as follows. The tangent bundle TL of the knot is a sub-bundle
the restriction of the contact planes on L. The quotient f / TL is an orientable line
bundle on a circle, so it is trivial. A section of this bundle defines a framing on L. We
call this framing the Thurston-Bennequin framing of L. If L is also null-homologous
there is another canonical framing on L coming from the normal vector field along
the boundary of a Seifert surface. The difference of these two framings is an element
of 7r1(SO(2)) = Z. The invariant tb(L) is the number corresponding to the difference

of Thurston-Bennequinf framing and Seifert framing.

2.3 Fillings

In this section we study several aspects of symplectic fillability in contact topology.

Let X be an oriented 4emanifold possibly with boundary. A symplectic structure
a; is a 2—form which is closed and non-degenerate. Non-degeneracy in this context
means that the exterior product of to with itself is a positive multiple of the volume
form. A vector field '0 on X is called a symplectic dilation if the Lie derivative
of w along 2) is a positive multiple of w. A hypersurface Y in X is called convex
(respectively concave) if locally, there exists a symplectic dilation which is positively
(respectively negatively) transverse to Y.

The simplest way to obtain a symplectic manifold from a contact manifold is the

process so—called symplectization. Let (Y, 5) be a closed contact 3—manifold. Assume

13

the contact structure is given as the kernel of a 1—form a. Let X = Y x [0,1] and
t denote the variable corresponding to [0,1]. The two form wa = d(eta) can easily
be seen to be a symplectic form on X. Note that the vector field at is a symplectic
dilation making the boundary components Y X {1} and Y X {0} convex and concave
respectively.

Let X be a complex manifold, and let J : TX ——) TX be the corresponding
almost complex structure, let J * : T‘*X —> T*X be its dual map. To any function
(,0 : X —> R we can associate a 2—form wt]; := —dJ*d:p and a symmetric 2-tensor
9,59 := w¢(., J.). A function (,9 as above is called strictly plurisubharmonic if 999 is
a Riemannian metric. A compact complex manifold X with non-empty connected
boundary is called a Stein manifold if it admits a strictly plurisubharmonic function
which is constant on its boundary. In this case, the pair (J, to) is called a Stein
structure on X. Stein manifolds come equipped with many extra structures. The
exact form (.099 is non-degenerate, so it defines a symplectic structure. The gradient
vector field V90 of the strictly plurisubharmonic function is a symplectic dilation,
indeed the Lie derivative preserves the symplectic form £V¢(wgp) = wife, and it.
points outward on the boundary. This implies that the set of complex tangencies of
the boundary form a contact structure on the boundary. This construction forms a
model for a type of fillability. A contact manifold (Y, E) is called Stein fillable if there

is a Stein manifold (X, J, ,0) such that Y = 8X and g = ker(1.vg(wes,)).

Example 2.3.1. Let B4 be the unit ball in the complex plane C2. Consider the
function 30(z1, 22) = [.21]2 + ]:2|2. It is easy to show that the associated 2—form (.099 is
the standard symplectic form dzl Adz—1+d22Adz‘2 making (B4, i, go) a Stein manifold.
Writing zk = ark + iyk, k = 1, 2, we see that the 2—plane field ker(:c1dy1 — 311de +
x2dy2 —y2d:r:2) is a contact structure on S3 = BB4. We call this contact structure as
the standard contact structure on S 3. Note that it restricts to the contact. structure

on R3 which has the same name.

14

There are two other notions of fillability whose definition involve only the essentials
of the above construction. A contact manifold (Y, 5) is said to be strongly symplec—
tically fillable if there is a compact symplectic 4—manifold (X, w) and a symplectic
dilation u in the collar of the boundary such that Y = 6X, g = ker(tv(w)) and 1)
points outward on Y. We say (Y, .5) is weakly fillable if there is a compact symplectic

manifold (X,w) such that Y = 0a) and w|€ > 0. The following implications are clear.

Stein Fillable => Strongly F illable => Weakly F illable

The converses of the above implications are not true in general as shown by Eliashberg
[E5], see also [DG2], and Ghiggini [Ghi2]. Fillability is a significant restriction on

contact structures as the following theorem shows.
Theorem 2.3.2. (Eliashberg, Gromov) A weakly fillable contact srtucture is tight.

By slightly changing the definition of the strong fillability, it is possible to define
another notion of filling. A contact manifold is said to be concave fillable if there is a
compact symplectic 4—manifold (X, w) and a symplectic dilation u in the collar of the
boundary such that Y = 8X, 5 = ker(1.rv(w)) and 2) points inward on Y. Admitting a

concave filling is not as restrictive as admitting a convex filling.

Theorem 2.3.3. (Eliashberg, Gay, Etnyre-Honda) Every contact manifold admits a

concave filling

2.4 Contact Surgery

The purpose of this section is to review two types of surgery operations that are
also compatible with contact structures. We will describe these surgery operations
as four-dimensional handle attachments. Let L be a Legendrian knot in a contact

manifold (Y, 5). As we saw in the previous section, the product manifold Y x [0,1]

15

can be equipped with a symplectic structure making one of the boundary components
convex and the other component concave. Attach a four-dimensional two handle to
the convex component along L with framing tb(L) — 1. In [Wei], ’einstein proves
that there is a canonical way of extending the symplectic structure over the handle
keeping the boundary component convex. We denote the new boundary component by
Ytb—1(L)' The resulting 4—manifold is a symplectic cobordism from Y to Ytb—1(L)
where the sympectic cobordisms are directed from the concave components of the
boundary to the convex components. This cobordism can be glued to any type of
symplectic filling of Y on its concave side. The result of this gluing is a symplectic
filling of Ytb—1(L) of the same type. The following theorem indicates this fact. It
is due to Weinstein [Wei] for the symplectic fillings and Eliashberg [E] for the Stein

fillings.
Theorem 2.4.1. If Y is Stein, strong or weakly fillable then so is Ytb—1(L)-

Dually, a 2-handle attachment to the concave component 8Y x [0,1] along a
Legendrian knot L with framing tb(L) + 1 yields a symplectic cobordism from Y to

Ytb1(L)= [DG]. Consequently, we have the following theorem.
Theorem 2.4.2. [DG] If Y is concave fillable then so is Ytb+1(L)'
The tb :l: 1 surgeries are sufficient to generate all contact manifolds.

Theorem 2.4.3. [DC] Any contact manifold can be obtained from (S3,€Std) by

tb :l: 1 contact surgery on a Legendrian link.

2.5 Homotopy Invariants

We are going to mention two homotopy invariants of 2—plane fields of 3—manifolds

in this section. Under favorable circumstances these invariants suffice to determine

16

the homotopy type of a 2—plane field «5. Then we will discuss the calculation of these
invariants for contact structures using the contact surgery presentations. The reader
is advised to turn to [G] for a complete discussion on homotopy invariants of 2—plane

fields on 3—manifolds.

Let £1 and {2 be two (co-)oriented 2—plane fields on a closed oriented 3—manifold
Y. We want to understand the obstruction to make {1 and {2 homotopic. Fix a
trivialization r of the tangent bundle TY. Passing to the unit normals, we may
regard both {1 and {2 as maps Y -> S2. Of course, these maps depend on r. We
consider the problem of making them homotopic. we can homotope {1 to £2 freely on
the l—skeleton Y0) as 7r1 (S 2) = {1}. The obstruction to extend this homotopy to the
2-skeleton lives in H 2(Y, 772(82». W hen H 1 (Y, Z) has no 2—torsion, the obstruction
can be detected by difference of the first. Chern classes of £1 and {2. Here we regard
£1 and {2 as complex line bundles where the almost complex structure is determined

by the 90° rotation counter-clockwise.

When .5 is a contact structure on Y its first Chern class 01(5) can be calculated as
follows. Obtain (Y, E) by attaching 2—handles along a Legendrian link in S3. Let X
be the corresponding 4—manifold. The first homology of Y is generated by the dual
circles 7,: of the Legendrian handles Li, and the relations between these generators
are determined by the intersection form of X. The Poincare dual of c1(E) can be

written as

PDCIK) = Z TOt(L.i)’)'.i

,7

W hen H1 (Y, Z) has no 2—torsion, the set of homotopy classes of 2-p1ane fields with
a fixed Chern class c1 6 H 2(Y, Z) is isomorphic to Z/(d) where d is the divisibility
of Cl. When Cl is torsion, a numerical invariant distinguishes these classes. Assume
that the contact manifold (Y, 5) is the boundary of an almost complex manifold (X, J)

such that J5 = E. In [G], it is proved that every contact manifold can be expressed

17

this way. In this case, we define the 3—dimensional invariant of (Y, E) by

_ C1(Xa J)? — 30m — 2x00

(13(0 — 4 (2-4)

 

where 0(X) and x(X) denote the signature and the Euler characteristic of X respec-
tively. The cup square of cl(X, J) is taken as follows: cl (X, J )2 = (:2/n2 where n is
the minimum positive integer such that ncl (Y. 6) = 0 and c is any relative cohomol—
ogy class in H 2(X , Y) that hits nc1(X, J) under the natural map. It is easy to see
that the 3—dimensional invariant is independent of the choice of the almost complex
manifold X since the right hand side of equation 2.4 is zero for closed almost complex
manifolds.

“7 hen the contact manifold (Y, 6) is obtained from (S3) by Legendrian handle
attachments where all the framings all equal to tb — 1, calculation of d3 is especially
simple. In this case, (Y,£) is naturally the boundary of a Stein 4—manifold (X, J)
by Theorem 3.3.1. The first Chern class of this Stein manifold is represented by the
cycle 2 rot(LfiSi where each Li is the attaching circle of a Legendrian handle and
Si is the 2—cycle in X associated to Li- See [DGS] and also section 3.3 below for the

generalization of this argument in case tb + 1 surgeries present.

2.6 Open Book Decompositions and the Giroux

Correspondence

Let Y be a closed oriented 3—manifold. An open book decomposition of Y is a
pair (B, 7r) where B is a link in Y and 7t is a fibration over a circle defined on the
complement of B over circle. The link B is called the binding and a typical fiber
of it called a page of the the open book decomposition. An abstract open book is

a pair (5,69) where S is a compact surface with non—empty boundary, and a} is a

18

diffeomorphism of S fixing the boundary pointwise. S is called the page and e is
called the monodromy of the open book.

An abstract open book determines a closed 3—manifold Y(S,c;')) as follows. First,
construct the mapping torus of a by identifying (:c, 1) with (tb(r), 0) in the product
manifold S X [0,1]. Since a restricts to the identity diffeomorphism near 6S, the
boundary of the mapping torus is a disjoint union of tori. Fill in each boundary
component with a solid torus in a way that for every y E 03, the circle Cy =
{(y, t) : t 6 [0,1]} for every y 6 BS bounds a disk. Clearly, Y(S-¢) admits an open
book decomposition with pages all diffeomorphic to S. Conversely, An open book
decomposition (B, 7r) determines an abstract open book (S, c3) where S = 7r_1(point)
and (,6 is the first return map of 7r.

In this thesis, we use abstract open books and open book decompositions inter-
changeably, keeping in mind that the passage between these two notions is as in the
paragraph above. Moreover, defining an abstract open book do we only specify the
isotopy class of the monodromy as two isotopic monodromy maps yield diffeomorphic
3—manifolds.

Given an abstract open book (S, a), we can construct its positive stabilization by
adding a 1—handle to S and composing a with a right handed Dehn twist along a curve
that intersects the co—core of the 1—handle geometrically once. Using a left handed
Dehn twist instead of a right handed one, we can define negative stabilizations. The
3—manifold determined by (S, (1)) is unchanged by these two operations.

We say that a contact structure is compatible with an open book decomposition
if the contact structure is given by a l—form a such that a > 0 on the binding and

do: > 0 on pages. In [TW'], Thurston and Wilkhelnker constructed explicitly a contact

structure on the 3—manifold Y(S.<,b) which is compatible with the open book (S, 0).

Theorem 2.6.1. (Thurston-Wilkhelnkemper) Every open book admits a compatible

contact structure.

19

The relation between contact structures and compatible open book investigated

further by Giroux, [Gi].
Theorem 2.6.2. (Giroux)

0 Every open book admits a compatible contact structure.
0 Two contact structures compatible with the same open book are isotopic.

0 Two open book decompositions compatible with the same contact structure

have a common positive stabilization.

Theorems 2.6.1 and 2.6.2 together establish a one to one correspondence between
set of contact structures up to isotopy and the set of open book decompositions up to
isotopy and positive stabilizations on a closed 3—manifold. This has very important
consequences in contact geometry and low—dimensional topology. For example, we
can characterize certain properties of contact structures in terms of the data that
defines a compatible open book.

As stated in 2.6.2, positive stabilizations on an open book do not change the
isotopy class of the isotopy type of the compatible contact structure. Negative stabi-
lizations, on the other hand, change even change the homotopy type. In fact we will
see in section 3.3 that negative stabilizations change the three dimensional invariant.

Moreover, they make the contact structure overtwisted.

Theorem 2.6.3. (Giroux-Goodman) A contact structure is overtwisted if and only
if it is compatible with an open book which is the negative stabilization of another

open book.

There is another criterion of open books which determines whether the corre-
sponding contact structure is Stein Fillable. The criterion uses the topological char-
acterization of Stein manifolds, see Theorem 3.3.1. No such criterion is known for

other types of fillings.

Theorem 2.6.4. A contact structure is Stein fillable if and only if it is compatible
with an open book whose monodromy can be written as a product of right handed

Dehn twists.

Chapter 3

Constructing Broken Lefschetz

Fibrations on Four-Manifolds

The purpose of this chapter is to prove Theorem 1.0.2 saying that every 4—manifold
admits a BLF structure. Our argument roughly goes as follows: we decompose
the given 4—manifold into a convex and a concave fibrations in the sense of section
3.1. The convex fibration possibly has achiral singularities which will be repaired
later. While the construction of the concave piece is essentially due to Gay and
Kirby [GK2], we use Akbulut-Ozbagci algorithm to get the convex fibration [AO].
Though there are other techniques to construct convex ALFS, e.g. [GK2] and [EF],
the usage of the Akbulut-Ozbagci algorithm is essential for the repairing process
of the achiral singularities. Next, we turn to the problem of matching the open
books on the boundaries of the concave and convex pieces. To do that, we use
Eliashberg’s classification of overtwisted contact structures (Theorem 2.1.2)and the
Giroux correspondence (Theorem 2.6.2). Then we show the matching can be done
keeping certain properties of the open books that are necessary in the next step.
Finally, we use an interpretation of Akbulut-Matveyev positron move to get rid of

the achiral singularities keeping boundary open books fixed so that one can glue the

22

two pieces to obtain a globally defined BLF.

This chapter is organized as follows. In section 3.1, we define several boundary
behaviors Lefschetz type fibrations on boundary. In section 3.2, we review the con-
struction and the stabilization of concave fibrations. The next three sections are
devoted to the review of Akbulut-Ozbagci algorithm to construct convex fibrations
and its generalization into a slightly more general setting. we define almost Stein
manifolds in section 3.3. A teclmical lemma recognizing if an open book decompo-
sition has a destabilization is discussed in section 3.4. Then we relate almost Stein
manifolds to convex achiral Lefschetz fibrations in section 3.5. we describe several
moves of achiral Lefschetz fibrations that can be used to change the homotopy type
of the corresponding contact structure on the boundary in section 3.6, and the open
book matching argument is given in section 3.7. An important lemma for repairing
achiral singularities is proved in section 3.8.2. Proof of Theorem 1.0.2 is given in

section 3.9.

3. 1 Introduction

First, we would like to alter the definition of open book decompositions to serve better

our purpose in this section.

Definition 3.1.1. Let. Y3 be a closed oriented 3-manifold. An open book decompo-
sition of Y is a map onto a 2 dimensional disk, 7r : Y —> D2 such that the preimage

of the interior of the disk is a disjoint union of solid tori and the following conditions

hold:
0 it is a fibration over S1 = 0D2 away from these solid tori.

0 Restriction of 7r to each of these tori is given by the projection map S1 x 02 ——>

132.

For all A E S1 the closure of the set PA := {7rT1(r)\) : r 6 (0,1]} is an oriented
surface with non-empty boundary which will be referred as a page of the open book
decomposition. The flow of a pullback of the unit tangent to S1 = 6D2 defines a
diffeomorphism of a fixed page fixing its boundary point wise. This diffeomorphism

is called the monodromy of the open book decomposition.

Now let X be a 4-manifold with non-empty connected boundary. Let f : X —-> E
be a achiral broken Lefschetz fibration where E = S2 or D2. The following is a
description for several boundary behaviors of f. Some of the conditions we list here are

redundant but we put them for contrasting from each other for a better understanding.

Definition 3.1.2. we say
1. f is cancer if

o2=02
o Fibers of f have non-empty boundary.

0 f | 8 X : 8X —+ D2 is an open book decomposition of 3X.

2. f is flat if

.2=02
o Fibers of f are closed.
. f(8X) = 51 = 602.

o f | 8 X : 0X —> S1 is an honest fibration.

3. f is concave if

24

o E = S 2 = D3, U D3 (the upper and lower hemispheres).

o Fibers over Di are closed, while the fibers over D3 have non-empty

boundary components.
. f(oX) C D3.

0 f | 8 X : 8X —> DE is an open book decomposition of 8X.

These definitions are due to Gay and Kirby [GK2]. They also point out that
one can glue a convex fibration and a concave fibration if the boundary open books
match. Moreover, two flat fibrations can be glued together if they agree on the

common boundary.

3.2 Concave Fibrations and Pencils

In this section, we illustrate several constructions of concave fibrations and pencils.

The major part of these constructions is basically due to Gay and Kirby.

The easiest example of a concave fibration is obtained by restricting an Achiral
Broken Lefschetz fibration on a closed manifold to a tubular neighborhood of a regular
fiber union a section. Note that the complement of this set is a convex broken achiral
lefschetz fibration. More generally, if we remove a convex ABLF from a ABLF (or
ABLP) the remaining part is a concave ABLF (or ABLP). The following proposition
uses these ideas to construct an interesting family of concave pencils. It was first
appeared in a draft of [GK2]. Unfortunately, that draft is no longer available. We

give a. proof of the proposition for completeness.

Proposition 3.2.1. Let E be a closed oriented surface. Let N be an oriented D2-
bundle over 2 with positive Euler number. Then, N admits a concave pencil with no

broken, achiral or Lefschetz singularity.

Proof. Let 7r : N —> E be the bundle map . Denote its Euler number by n 2: e(7r) > 0.
Take a generic section 3 : )3 ——> N. It intersects with the zero section at n distinct
points. Let D1, D2, - - - , Dn be pairwise disjoint small disk neighborhoods of images
of these points in E. The section 3 determines a trivialization T of the complementary

disk bundle such that the square part of the following diagram commutes.

N—7r—1(D1U-~UDn)—T—>(E—(D1U~--UDn))X D2 i) 02:01
[a ]P1
Z—(Dlun-an) i—d> 2—(Dlu~--UDn)

We define the pencil map f : N —) 32 = Di U D3 by f(:c) = 172 0 7(3“) for
:r E N — 7r_1(D1 U - - - U Dn), and on each 7r—1(D,~) g D2 x D2 C C2, f is given by
the usual pencil map C2 —> (CP1 ’3—1 SQ. It is an easy exercise to verify that these two
maps agree on their common boundary and together they define a concave pencil on
N.

A simpler explanation for this construction can be given via the following han-
dlebody diagrams. we draw the usual handlebody diagram of N, disk bundle over a
surface 2 of genus g with Euler number n. The diagram contains one O-handle, 2g
1—handles and a 2—handle with framing n which is attached to the diagram in such a
way that the attaching circle passes through each l-handle exactly twice. To visualize
a concave fibration, we blow up N at n distinct points as in the figure.

-------------------
o 5‘ ‘5 p ’. “ o $‘ ‘

 

vo“ ’0 ~ "ss
OOQO 666 ............. 0666
vvvv vvvv V'VV—J
dé>—1
4,—1
0

Figure 3.1: Nincrfl

26

Forgetting the exceptional spheres for a minute, we see a trivial fibration over disk
with fiber genus 9. To this fibration, we attach several 2—handles where the attaching
circles of the 2-handle are sections of this fibration restricted to the boundary. By
the text in the beginning of this section, the result of attaching these handles is a
concave fibration. The boundary open book of this fibration has pages obtained from
E by puncturing at n points. The monodromy of the open book is the product of left
handed Dehn twists along curves parallel to boundary components of a page. The
last two observations can be verified by the discussion in [E0] or by means of an
easy exercise in handle slides: Anti blow up at each exceptional sphere. This creates
n O-framed 2-handles in the handlebody diagram. Replace each one them with a
1-handle. The resulting 4-manifold X admits a convex ALF with the same boundary
open book. It is transparent from the handlebody diagram of X that our claims
about the boundary open book are correct. Note that if we reverse the orientation of

this manifold we get a convex PALF which we can glue to our concave fibration on

 

N tnCP2 to get a Lefschetz fibration on the closed manifold X U N [inf—IE where all
the singularities lie in the convex side. This Lefschetz fibration is actually symplectic
because it admits n sections of self intersection —1. Blowing down these sections
and isolating N, we see that N admits a concave pencil and a compatible symplectic

structure w, making the boundary of N w—concave. El

An alternative construction of concave piece is provided by Gay and Kirby in
[GK2] via round handle attachment. We are going to sketch this construction in
proposition 3.2.2 but before doing that we need to review some basic facts about
round handles and their relation with broken fibrations. For details, reader can

consult [GK2] or [B2].

A round l-handle is a copy of S1 X D1 x D2 which can be attached to a 4-
manifold along S1 X {—1} X D2 U S1 X {1} X D2. Up to diffeomorphism, all the

27

attaching information is contained in a pair of framed links which will be referred as
the attaching circles of a round round 1-handle. Moreover, we may choose the framing
on one of the attaching circles to be 0. Every round l-handle decomposes uniquely
as a union of a l-handle and a 2-handle passing over the l-handle geometrically twice
algebraically zero times. Therefore, in order to draw a round l-handle to a handlebody
diagram, it is enough to draw a l-handle and a 2—handle as above. Attaching circles
of the round 1-handle can be visualized by shrinking the legs of the 1-handle to pair

of points.

Suppose, we have an (A)BLF or (A)BLP with boundary which is one of the three
types listed in definition 3.1.2. We can attach a round 1—handle whose attaching
circles are circles are sections of the fibration restricted to the boundary. Then, the
fibration extends across the round l—handle with a fold singularity on S1 X {0} X {0}.
Resulting object is an (A)BLF or (A)BLP with same type of boundary. Fibers on
the boundary, i.e. pages, gain germs. we can visualize this as follows; Attaching
circles of the round 1-handle intersects each page at a pair of points. We remove two
small disks containing these points from each page and connect the boundary circles
of these disks by a handle which is contained in the round 1-handle. The monodromy
is composed with some power of Dehn twist along the boundary circle of one of the

disks. This power is precisely the framing of round l-handle.

Proposition 3.2.2. Let 2 be a closed oriented surface. The product 2 X D2 admits

a concave Broken Lefschetz Fibration.

Proof. We start with the standard handlebody diagram of Z X D2. W'e attach one
1-handle, two 2-handles: zero framed 61 and —1 framed B2, and a 3-handle as in the
Figure 3.2. In the figure we did not show the whole diagram. O—framed vertical arc
is part of the unique 2-handle of E X D2. The framing on .81 is chosen so that when

we slide it over 52 and cancel [32 with the 1-handle, ,‘31 becomes a O—framed unknot

28

canceling the 3-handle. Therefore we did not change the diffeomorphism type of our
manifold by these handle attachments.
0 U 3handle

r31 |
—_l

 
 

 

Figure 3.2: the passage from flat to concave fibrations

Now, we describe how the fibration structure extends across these handles. The
1-handle and .51 forms a round l-handle which is attached along a bisection of the
boundary fibration. As we remarked above, the fibration extends as a broken fibration
with one fold singularity inside the round handle. This broken fibration is still flat,
i.e. it restricts to a surface bundle over S 1 on boundary. Note that the attaching the
attaching circle of ,32 is a section of this surface bundle. Therefore, attaching 52 turns
the flat broken fibration into a concave broken fibration, creating new disk fibers over
the southern hemisphere of base sphere. Pages are obtained by punching the fibers
on boundary along attaching circle of B2. Finally, we attach the 3-handle in such a
way that its attaching sphere intersects each page along a properly embedded are 7'.
Concave fibration extends across the 3-handle without forming any new singularity.
The fibers over the southern hemisphere gain a 1-handle whereas pages lose a 1-handle

(they are cut open along the arc 7.). [:1

The last topic we would like to discuss in this section is stabilization of concave
fibrations. Recall that positive (resp. negative) stabilization of an open book amounts
to plumbing a left (resp. right) handed Hopf band to its pages and composing its

monodromy by a right handed (resp. left handed) Dehn twist along the core of the

29

Hopf band. If the open book is the boundary of a convex ALF positive (resp. negative)
stabilization can be achieved by creating a canceling 1-2 handle pair where attaching
balls of the l-handle strung on binding and the 2-handle lies on a page with framing
one less (resp. one more) than the framing induced by the page. Stabilization of
concave fibrations is not. that straight forward because 1-handles cannot be attached
this way. Instead we create an extra canceling 2-3 handle pair. Moving the new 2-
handle over the I handle creates a round 1-handle which can be attached to concave
fibration, see Figure 3.3. It is evident from the picture that the resulting concave
fibration has a new broken singularity, and a Lefschetz (resp. achiral) singularity

inside.

U 3handlc

 

Binding

 

 

Figure 3.3: Stabilization of concave fibrations

It is possible to negatively stabilize concave fibrations without introducing achiral
singularity inside. The trick goes as follows; we create another canceling 1-2 handle
pair, where 2-handle has a good (-1) framing. We move bad (+1) framed 2—handle

over the new l-handle and regard them as a round —1-handle, see Figure 3.4.

We formed two fold singularities and a Lefschetz singularity inside of the concave
fibration. One right handed and two left handed Hopf bands are plumbed to the

boundary open book as in Figure 9 of [GK2].

30

U i’rhandle

+1

 

 

 

Figure 3.4: Negative stabilization of concave fibrations without achiral singularity
3.3 Almost Stein Manifolds

Here, we review some basic facts about Stein manifolds of real dimension 4. After-
wards, we will discuss their natural generalizations. For more discussion about Stein

manifolds, reader can consult [OS].

In [E], Eliashberg gave the topological characterization of 4—dimensional Stein

manifolds.

Theorem 3.3.1. (Eliashberg)

Let X = B4 U (l-handles) U (Q-handles) be a 4-dimensional handlebody with one
O-handle and no 3 or 4—handles. Then

1. the standard Stein structure on B4 can be extended over the l-handles so that

the manifold X1 := B4 U (l-handles) is Stein.

2. If each 2-handle is attached to 8X1 along a Legendrian knot with framing one
less than the Thurston-Bennequin framing then the complex structure on X1

can be extended over the 2-handles making X a Stein manifold.

3. The handle decomposition of X is induced by a strictly plurisubharmonic func-

tion.

31

By abuse of language, we call a handle decomposition as in Theorem 3.3.1 as a.
Stein structure without specifying the ahnost complex structure or strictly plurisub—
harmonic function. These two objects should be understood to be constructed as in
the proof in [E]. Now by using the characterization in Theorem 3.3.1 we give the

following definition to generalize the Stein manifolds to a larger category.

Definition 3.3.2. Let X be a compact oriented 4—manifold. X is said to be almost

Stein if it admits a proper Morse function f : X —> IR with the following properties.
1. Critical points of f have index no bigger than 2.
2. Boundary 8X is a level set for f.

3. Attaching circles of 2—handles associated to f are isotopic to some Legendrian
curves in (8X (1), {0) such that the framing on any handle is either one less than
or one greater than the Thurston-Bennequin framing. Here, X (1) is the union
of 0— and 1—handles of X and 50 is the unique Stein fillable contact structure

on its boundary.

In Theorem 3.3.3 below we show that almost Stein manifolds exist in abudence.
The same theorem, stated in different terms was proved in [GK]. In an almost Stein
manifold the 2—handles, which are attached with framing tb — 1, will be called a good

handle, otherwise they will be called defective.

Theorem 3.3.3. Every 4—dimensional handlebody X with without 3 and 4-handles

is an almost Stein manifold with at most one defective handle.

Proof. It is no loss of generality to assume that X has one 0—handle. By the first
part in theorem 3.3.1, X1 = B4 U (1-handles) has a Stein structure. We must show
that 2—handles can be arranged to be attached with correct framings. Let h be a

2-handle in X. First, we isotope the attaching circle K of h to a Legendrian knot.

32

Say, the framing on K is n with respect to the Thurston-Bennequin framing. The
number n can be increased arbitrarily by adding zigzags to the front projection of
K. Decreasing n is tricky. Near a local minimum of front projection of K, we
make Legendrian Reidemeister—1 move, see [G]. This operation does not change the
Thurston—Bennequin framing. Next, we create a pair of canceling 1 and 2—hand1es,
where the 2—handle has framing +1 as in the left hand side of the Figure 3.5. Sliding
h over the new 2—handle, we can decrease its framing. Finally, we observe that the
same canceling pair can be used to fix the framing of all 2-handles, leaving only one

defective handle as desired.

 

 

Figure 3.5: Framing fixing in almost Stein manifolds

Remark 3.3.4. 1. Creating a pair of 1-handle and a +1 framed 2—hand1e corre-
sponds to negative stabilization of the contact structure {0 on 3X1. This creates
an overtwisted disk, and sliding a Legendrian knot over that disk increases its
Thurston-Bennequin invariant. This observation was used by Gay and Kirby to
prove Theorem 3.3.3. Only significant difference is that they apply Lutz twist

to create an overtwisted disk.

2. Yet another proof for Theorem 3.3.3 using [AO] can be given via the correspon-
dence between achiral Lefschetz fibrations and almost Stein structures which

we state in a more precise form in Section 6 below. Harer [H] (see also [EF])

33

proved that every 4—dimensional handlebody X without any 3 or 4—handles ad-
mits an achiral Lefschetz fibration over disk with bounded fibers. This fibration

naturally makes X an almost Stein manifold.

The main theorem in this section asserts that by carving certain properly embed-
ded disk we can transform defective handles into good ones. The following example

provides a model for that transformation.

Example 3.3.5. Let X = D2 x S2. By [E], X does not admit a Stein structure. The
simplest handlebody decomposition of X consists of one 0 and one 2—handle, where
the 2—hand1e is attached to an unknot with zero framing, as in Figure 3.6. The
obvious Legendrian realization of this unknot has front projection with two cusps.
Thurston-Bennequin number of this Legendrian knot is -1 so, the 2—handle is at-
tached with framing tb + 1. An alternative Legendrian realization of the unknot is
drawn next to it. The curve 7 is not a part of handlebody yet. We will Show that
the manifold X = X — D is Stein where D is a properly embedded disk in O—handle
whose boundary is 7.

Consider the Stein manifold on right hand side of Figure 3.6. It is an easy exercise
to show this manifold is diffeomorphic to X: Indeed, we can obtain a handlebody
decomposition of X by putting a dot on 'y (i.e. turning it into a 1-handle) in the
previous figure. Create a canceling 1 and 2—handle pair, where the two handle has
framing —1. Slide the new 1-handle over the 1-handle 7. Next slide two strands
of original 2-handle over the new 2—handle, and then cancel the 1 and 2 pair. The

resulting manifold is the smooth manifold described by the next figure.

Applying the procedure described in the Example 3.3.5 to each defective handle

in an almost Stein manifold we arrive at the following conclusion.

Theorem 3.3.6. Every almost Stein manifold admits a Stein structure in the com-

plement of finitely many properly embedded disks.

34

 

Figure 3.6: Repairing defective handles in an ahnost Stein manifold

By attaching a good handle and homotopically canceling this l-handle created in
the procedure, we see that Steimficatz'on of every almost Stein manifold can be made
within its homotopy class. This was first observed by the first author and Matveyev

in [AM] (e.g. positrons).

Theorem 3.3.7. (Akbulut-N‘Iatveyev) Every almost Stein manifold is homotopy

equivalent to a Stein manifold.

Almost Stein manifolds do not carry as many extra structures as Stein manifolds.
For example, they do not naturally admit a symplectic structure. An almost complex
structure can be defined on an almost Stein manifold, but only in the complement of
finitely many points each contained in a 2-handle. On the other hand, we still have a
contact structure on the boundary defined by means of the set of complex tangencies.
This contact structure will be denoted by 5 X for an almost Stein manifold X.

Handlebody diagram inducing an almost Stein structure on X can easily be con-
verted into a contact surgery diagram in the sense of Ding and Geiges [DC]. We
replace each 1-handle with a circle with dot, a convenient notation first introduced
in [A] (which means that the interior of the disk, which this circle bounds, is pushed
into the 4-manifold interior, then its tubular neighborhood is removed). The only
additional assumption is that we choose the circle to be a Legendrian unknot, and
we put a negative twist to each 2-handle that passes over the 1-handle as in Figure

3.6. The twisting operation is done in order to match the conventions to calculate

35

the relative framing with respect to Thurston-Bennequin framing. W'e convert our
Legendrian handlebody diagram into a contact surgery diagram by erasing the dots
on l-handles and regarding them as tb + l-framed 2-l1andles. Topologically, this op-
eration corresponds to surgering some circles, but it does not affect the boundary
contact manifold.

Homotopy invariants of g X can easily be read off from the front projection of
the Legendrian handlebody diagram inducing the almost Stein structure of X, see
[G]. The Poincare dual of its first Chern class is represented by a chain 2 [CZ-[pi],
where ki is the rotation number of a Legendrian 2-handle and [#27] is the homology
class represented by the meridian of that handle. On 8X, the oriented 2-plane fields
having the same first Chern class C] is a Z/d torsor, where d is the divisibility of
Cl. Each element of this torsor can be distinguished by its 3-dimensiona1 invariant
d3 = (c1(X)2 — 30(X) — 2x(X))/4 + q, provided Cl is torsion. Here q is the number
of defective handles, 01(X) is a cohomology class such that c1(X)(h.) = rot(L) for
every 2—handle h with Legendrian attaching circle L, and the cup square of Cl (X) is
taken in some appropriate sense. Let us call the operation of creating a canceling pair
of a l-handle and a tb + 1 framed 2-handle by negative stabilization. This operation
does not have any topological effect but it does change the almost Stein structure
on X and the contact structure on BX. The key observation here is if we have two
almost Stein manifolds with same first Chern class of the contact structures on the
boundary, then we can match the contact structures up to homotopy by negatively

stabilizing one of the almost Stein manifolds.

3.4 Destabilizations

This section is devoted to a technical lemma. It will be used to determine the mon-

odromy of a (p, q) torus knot but. we believe that the lemma is important on its own

36

as it provides a sufficient condition to destabilize an open book. It is analogous to
subdividing contact cell decompositions as in [Gi] and product disc decompositions

of sutured manifolds defined by Gabai [Gab].

Lemma 3.4.1. Let S be a page of an open book decomposition of the 3-sphere S3.
Suppose K is a non—separating knot on S with self linking number —1 with respect
to the framing induced by S. Then S is isotopic rel K to plumbing of some surface
S with a Hopf band H whose core is K. Moreover, the surface S is isotopic to S cut
along a properly embedded arc 7 that intersects transversely with K at precisely one

point.

Proof. Since S has nonempty boundary and K is non-separating, the are 7 in the
statement exists. Create two parallel copies of K on S, one to the left of K and other
to the right. Call them K1 and K 2 respectix-ely. The union of these two curves is the
boundary of a Hopf band H whose core is K

In the Figure 3.7 below we describe an isotopy of S in which it is apparent that H
is plumbed to the rest of the surface. Starting at a little right of one of the components
of 7 0 BS we perform a finger move along a parallel copy of 7 till we hit K1. Next we
turn right and continue the finger move along K1 until we approach 7 from its left.
Finally, we perform the same isotopy from the other side but this time we turn left
at K 2.

This isotopy isolates H from S. We can write S = H U S where the intersection
H n S is a small neighborhood of the arc 7 0 H in H. This is precisely the definition
of plumbing.

To see why the last statement holds, we cut H along an are from K1 to K2
intersecting with K exactly once. Clearly, we deplumbed H from S. Now, pulling

everything back, the assertion should be clear.

37

 

 

 

 

 

 

 

 

é a 7
.................. ' I-.‘........-....... K
E E as

Figure 3.7: Isotopy by finger moves.

3.5 ALF-Almost Stein

In this section, we are going to review the algorithm of Akbulut and Ozbagci to
establish a one to one correspondence between ALFs and almost Stein manifolds.
Originally, in [A0] authors established this correspondence between PALFs and Stein

manifolds but. their arguments hold in our case as well.

Theorem 3.5.1. (Akbulut—Ozbagci) A 4-manifold is almost. Stein if and only if it

admits an ALF over disk with bounded fibers.

Combining this result with our previous observation, Theorem 3.3.3 we get the

Harer’s theorem [H];

Corollary 3.5.2. Every 4—dimcnsional handlebody without. 3 or 4 handles admits an

ALF over disk with fibers having non-empty boundary.

Proof. of theorem. 3.5.]:

We will prove almost Stein implies ALF part only. First, we prove a special case
where X has no l-handles. By assumption, X is obtained by attaching several 2-
handles along Legendrian link L in S3 with standard contact structure. In the front.
projection of L, we replace all local maxima and minima with corners and then turn
the picture 45° counter clockwise to get the square bridge diagram of L. This diagram
allows us to find a (p. q) torus knot K and a Seifert surface 2 of K with the following

properties:

38

1. Complement of K fibers over S1 with fibers 23, forming an Open book decoru-

position of S3, which supports the standard contact structure of S3.

2. The open book decomposition in item (1) extends to a PALF of 4—ball B4 with

(I? — 1)(q - 1) vanishing cycles.

3. The link L can be put on E such that the surface framing agrees with the

Thurston-Bennequin framing.

Once we achieve to find such K and 2, PALF of B4 in item (2) extends across
the 2—handles as an ALF. Fibers are still diffeomorphic to E but the monodromy is
composed with product of Dehn twists along components of L. Dehn twists are right

handed for good handles and left handed for defective ones.

The original construction of K and Z is due to Lyon [Ly]. Without loss of general-
ity, we may assume that L is contained in the cube [—1, 1] X [0, 1] x [0, 1] C R3 C 33.
Say, L has 1) horizontal and (1 vertical segments. Choose numbers 1 > a1 > (12 >
---a.p > 0 , and O < b1< b2 < < bq < 1, and asmall constant 5 > 0. For the
i-th horizontal segment of L, we put the plate {1} X [0, 1] X [ai - 5, (ti + E]. For the
j-th vertical segment, we put the plate {—1} x [bj — 6, bj + 5] x [0,1]. We connect
horizontal segments to vertical ones by a band making quarter twist. The result is a

Seifert. surface of a (p, q) torus knot with minimal genus.

Clearly, we can put L on the surface 2 constructed in this way. It is an easy
exercise to verify that page framing on each component of L is its writhe minus the
number of south-west corners. This is exactly the recipe for Thurston-Bennequin
framing. If we project 2 to y-z plane, we get a (p, q)—grid. This grid has (p—1)(q — 1)
holes in it. Take a loop enclosing one of these holes inside the grid and lift it to E. This
curve has self linking number —1 with respect to the page framing. Hence, we can

destabilize the Hopf band around the curve as in lemma 3.4.1, leaving (p— 1) (q ~—1)—1

39

holes in the grid. If we continue to destabilize, we can verify (2) by explicitly finding

the monodromy curves.

Finally, we saw that the monodromy of (p, q) torus knot is a product of right
handed Dehn twists. Therefore, the contact structure it supports is Stein fillable.
Eliashberg [E2] proved that every such contact structure on S3 is isotopic to the

standard one.

Remark 3.5.3. One problem in contact topology is to find an explicit open book
decomposition supporting a given contact structure starting from its contact surgery
diagram. As noticed by Stipsicz [St], the algorithm we described above provides an
open book decomposition supporting a contact structure given by a surgery diagram
if the framing coefficients :l:1. Ozbagci extended this result for the case of rational
surgery in [O]. The idea of destabilizing the unnecessary Hopf bands in this algorithm
is used by Arikan in [Ar] to find a smaller genus open book supporting the given

contact structure.

we continue the proof by considering the general case. First, convert all the 1-
handles into circles with dots as described at the end of section 3.3. Ignoring the
dots for a minute, we apply our algorithm to this new Legendrian link we obtained
to put it on a page of a (p, q) torus knot. Each component of the Legendrian link is a
non-separating curve on a page. By an isotopy we can put those components which
represent l-handles in such a position that each one of them intersects every page
exactly once. We push the disks bounded by these circles inside of 4-ball, each one of
them is realized as a section, then scoop out tubular neighborhoods of these sections.
The last operation creates holes on the fibers. Of course, the number of these holes

is the same as the number of l-handles. El

As an illustration, we apply the algorithm described in the proof to get ALF

pictures of the last two Legendrian handlebody diagrams in Figure 3.6. In Figure

40

3.8 we put both diagrams on Seifert surface of a 7 x 7 torus link. The black curves
represent l-handlcs. That is why they are clasped to the binding. The monodromy of
the ALF on the top picture, is a left handed Dehn twist along the red curve composed
with the monodromy of the toms knot; whereas the monodromy of the. PALF on the
bottom picture, is a right handed Dehn twist along the blue curve composed with
the monodromy of the torus link. The importance of this example is that it provides
an interpretation of Akbulut-lt'latveyev positron move which concerns (almost) Stein
manifolds in terms of (achiral) Lefschetz fibrations. We will return to this point in
section 3.8.

J12
nnunnrfin

:immmmmjfi
3LE‘IETMW
WWW
I‘E‘IKUTFKW
immmmmmrfi
WT
c—L

[Tl1 I12I‘I3FI4 l—I5
li‘frs‘amrmrsmfi
:LHWWW
LLWWJ
flukmmfi'firsfis‘fi
SLSUTMW‘QTMTKWLJ
6imm‘s—TG'TELEME

Figure 3.8: Two ALFs corresponding to the Almost Stein manifolds in Figure 3.6

41

3.6 Some ALF Moves and Their Effects

In this section, we are going to describe a set of moves that produce new ALFs out
of a given one. we will describe the moves by means of a handlebody diagram in
which 2-handles are drawn in square-bridge position. The corresponding ALF should
be understood to be constructed as in Section 3.5. It is our hope that these moves

can be extended to result in a uniqueness theorem for Lefschetz type fibrations.

Move 0:(Refining) Our first move is increasing the number of grids on fibers
constructed as in section 3.5. It amounts to positively stabilizing the initial ALF
several times. Most of the forthcoming moves require refining before we perform
them. We are going to omit saying that we apply this move when it is clear from the

content what sort of refining must be done.

Move l:(Isotopy) Any sort of isotopy in the category of square-bridge knots is
applicable provided that it does not change the Thurston—Bennequin framing. This

includes square-bridge Reidemeister moves which we show in Figure 3.9.

_l

l] E[_

I

F

l

[ I
| I
Figure 3.9: Square-bridge Reidemeister moves

42

Move 2:(Stabilization) We include both positive and negative stabilizations in
our list of moves. As mentioned above, increasing the grid is a special type of posi-
tive stabilization. In general, we indicate positive (resp. negative) stabilizations by
creating a pair of canceling l-handle and a 2-handle where legs of the l-handle strung

on the binding and the 2—handle has framing tb — 1 (resp. tb + 1).

Move 3:(Finger Move) Let K be an attaching circle of a 2-handle lying on a page
with framing either one less one more than the framing induced by the pages. Near K,
we stabilize our ALF and make K run over the I handle of the stabilization without
changing its framing see Figure 3.10. This move is called positive or negative finger
move depending on what type of stabilization we made. Performing one positive and
one negative finger move consecutively does not change the total space of the ALF.
However, it adds or subtracts 2 to the number obtained by evaluating the first Chern
class of the boundary contact structure on the meridian of K depending upon the
orientation (see [BF] Lemma 10). Positive finger move has a nice interpretation in
terms of square-bridge knots if the grid is refined enough. Given any horizontal or
vertical segment of K, we create a small rectangle on top of it as in figure 3.11, then
install this to our ALF. Clearly, this adds page framing one negative twist as we
traverse the rectangle. Since the monodromy of a (p, q) torus links is a product of
right handed Dehn twists only, there is no similar interpretation for left finger move

in our language.

 

'-'
K :l:I

 

 

 

Figure 3.10: Positive and negative finger moves

43

_.._l—l_

Figure 3.11: Another interpretation of positive finger move

 

3.7 Matching the Open Books

In this section, we are going to prove a technical lemma to match open books of a

convex ALF and a concave BLF.

Lemma 3.7.1. Let X = X1 UX2 where X1 is a convex ALF and X 2 is a concave BLF
constructed as in either proposition 3.2.1 or 3.2.2. Assume that the first homology
group of 8X1 = 8X2 has no 2—torsion. Then Lefschetz type fibrations on X1 and
X2 can be modified such that the boundary open books match. Moreover, we can
arrange that every achiral vanishing cycle on the new convex ALF links twice with

an unknot on a page.

Proof. Denote the induced open books on 8X1 and 0X2 be ob 1 and ob 2. The
contact structures supported by these two open books will be denoted by £1 and 52.
By the Giroux correspondence, see Theorem 2.6.2, we can match the open books up
to positive stabilization if we can match the associated contact structures.

At this point, we would like to explain what we mean by matching the open books.
Boundaries of convex and concave sides are identified by an orientation reversing
diffeomorphism f : 0X2 —> 8X1. We apply Giroux’s theorem to the open books
f *0b 1 and dig to conclude that they have common positive stabilization if and only
if the contact structures f *52 and {2 are isotopic. Note that stabilizing ob 2 requires
a round handle attachment as in section 3.2. Consequently, new broken singularities
formed in concave side. In addition, positive stabilization of f * ob 1 means that we
need to negatively stabilize ob 1, so, new achiral singularities are formed in convex
side.

Now, we consider the problem of matching the contact structures f *51 and {2.

44

Before matching them up to isotopy, we study a more elementary problem; namely,
matching them up to homotopy. To simplify the notation we drop f* from f *51
and f *0b 1. As explained in [G], two oriented 2-plane distributions on a 3-manifold
are homotopic if and only if they have the same Chern class and the 3-dimensional

invariant (13, in case the first homology has no 2-torsion.

We can make the Chern class of {1 equal to 01(52) using techniques of Section 3.6.
The Poincare dual of Cl (£1) is represented by a cycle 2 let-[7i] where 7,: is a meridian
of a 2-handle. The handle could be good or defective but we know that its attaching
circle is in square—bridge position. The number kz' is the number of up corners minus
the number of down corners. we may increase or decrease this number by 2 applying
a positive and a negative finger moves consecutively. It can be shown that in our
examples 01(62) 2 0. Therefore we are done with matching Chern classes provided
that 01051) has correct parity. This is true because c1(§) is even for an arbitrary
oriented 2-plane distribution on a 3-manifold M. To see this, fix a trivialization of
the tangent bundle. Unit normal vector field to 6 defines a map 9 : M —> S2 such

that 5 = g*TSQ. So cl(§) = g*c1(TS2) = 2g*PD([point]).

To make the 3-dimensional invariants equal, we negatively stabilize one of ob 1
and ob 2 enough number of times. If we want to avoid achiral singularities in concave
side, we need to form an extra round handle singularity and perform two positive

stabilizations on ob 2.

Finally, we apply a theorem due to Eliashberg [E3], which says that two over—
twisted contact structures on a 3-manifold are homotopic if and only if they are
isotopic. To guarantee that both {1 and £2 are overtwisted we negatively stabilize

both their open books once more.

3.8 Repairing Achiral Singularities

This section is devoted to a procedure to get rid of achiral singularities of a convex
ALF without changing the total space and the boundary open book. This procedure
is an adaptation of the Akbulut-Matveyev positron move [AM] to the Lefschetz type

fibrations. The proof the main result is evident. in the following example.

Example 3.8.1. we first describe our initial ALF. Consider 4—ball B4 fibering over
disk with disk fibers. This fibration induces an open book decomposition of the
boundary 084 = 53 = R3 U {30}. To visualize this open book, we indicate the
binding as an unknot drawn as a rectangle in R3. Then, the meridian of this rectangle
is a section of a fiber bundle over circle with disk fibers. Next, we negatively stabilize
this open book to get a convex ALF with one defective handle. Recall that negative
stabilization amounts to creating a pair of cancelng 1 and 2-handles where the feet
of 1—handles strung on the binding and the attaching circle of the 2-handle is on a
page with framing one more than the surface framing. The main theorem requires
a defective handle to pass twice over a l-handle. This is why we change the picture
by isotopy and carve 4—ball in such a way that the 2 handle satisfies the assumption
of the main theorem. The square-bridge position of the final handlebody is given in
Figure 3.12 on the left. A compatible ALF is constructed as in proof of Theorem
3.5.1 with the exception that the 1-handle of the canceling pair is indicated as a pair
of 3-balls, on top of Figure 3.13. Now we perform a left twist along the l—handle to
repair the ALF. The square-bridge diagram and the corresponding PALF is shown on
the right of Figure 3.12 and on the bottom of Figure 3.13. Note that two diagrams
in Figure 3.13 differ by a Dehn twist along the curve labeled as a.

By applying this twist diffeomorphism to every defective handle, we obtain our

PALFication lemma:

Lemma 3.8.2. Suppose that a 2-handlebody X admits an convex ALF constructed

46

as in the proof of theorem 3.5.1, possibly with some extra stabilizations. Assume
further that every framed link representing an achiral vanishing cycle passes over a
2-handle twice. Then, X also admits a PALF and there is a self diffeomorphism
tp (isotopic to identity) of X which is also a fibration equivalence outside of achiral

singular points.

 

r__|tb+1 Q?I_1tb—l
U . l \7 . I

| T‘.—| | | r I
.__ L—li

Figure 3.12: Repairing bad framings

 

 

 

 

47

9 11 m 17 11 H
i. immmmrfi
‘ “WWW
immmsfi
WWW
Lsrwrsflwwrfi
flmmmml

slams

 

54% m 17 1‘1 11 LL
‘-. imwsrsfi

mmmm
Kareem
immunsmmnfi
l | | :3

“TQWCTKWLJW 1

Figure 3.13: Alf picture of figure 3.12

3.9 Proof of the Main Theorem

Proof. of Theorem. 1.0. 2: Our first task is to find a concave filn'ation inside of X. If
b2 (X) > 0 a neighbor hood of a surface with pasitive self intersection can be equipped
with a concave pencil by Proposition 3.2.]. Otherwise, we choose any self intersection

zero surface in X and find a concave Broken Lefschetz Fibration on its neighborhood

48

as in Proposition 3.2.2. This concave piece constructed in either way will be called
X 2. Let X1 be the complement X2. By Lemma 13 in [GK] we may assume that X1
has no 3 or 4—handles. In section 3.3, we proved that every such manifold is “Almost
Stein”, see Theorem 3.3.3. Thus by applying the Akbulut-Ozbagci algorithm which
was described in section 3.5.1, we can find a convex Achiral Lefschetz Fibration on
X1 —-> D2 with bounded fibers. More specifically we get an ALF f with total space
IFI = X1-

Having decomposed X = X1 U X 2 as a convex ALF union concave BLF our next
aim is. to match the open books on the common boundary of these two submanifolds.
In lemma 3.7.1 above, we described how techniques of [GK2] and [BF] can be modified
to achieve this aim. Our version allows us to get the property that every achiral
vanishing cycle (i.e. the frame knot representing it) links twice with an unknot
sitting on a page of the open book (coming from 8?). This property will be crucial
in the next step of our construction. This is the only non-explicit step in our proof
because our arguments depend on Eliashberg’s classification of overtwisted contact
structures and Giroux’s correspondence between contact structures and open book
decompositions. During the application of the Giroux correspondence we need to
stabilize both sides. Recall from section 3.2, we create more fold singularities when
we positively stabilize the concave side. In order to use Eliashberg’s theorem we need
to negatively stabilize both sides in the first place making sure that the boundary
contact structures are both overtwisted. We can negatively stabilize the concave piece
without creating achiral singularities at the cost of two extra positive stabilizations.
This way we collect all the achiral singularities in the convex side and all the fold

singularities in the concave side.

In order to apply the “positron move” introduced in [AM], we isotope the unknots
mentioned in the previous paragraph to meridians of the binding of the common open

book on 8X1 = 8X2 and carve the disks bounded by these curves from X1. This

49

amounts to attaching a 2—handle to the concave side X 2 along a section. Yet another
“convex-concave decomposition” of X is obtained this way. Moreover we did not lose
the property that boundary open books match. Note that we would not be able to
apply positron move if there was an achiral singularity in the concave side as we can
carve only the convex fibrations and attach handles along sections only to concave
fibrations. Now that we have all the defective handles go over a 1-handle twice, all
the bad (tb+ 1) framings can be can be turned into a good (tb— 1) framing by twisting
these handles, see lemma 3.8.2. The twisting operation induces a self diffeomorphism
{D and a handle decomposition of the convex side X1. The handlebody decomposition
is naturally related to a PALF of X1 and '9’; sends nonsingular fibers of the previous
ALF into fibers of this PALF. In particular, ’9’) induces a diffeomorphism (isotopic to
identity) preserving the open books on the boundary 0X1. This means that we can

glue our new PALF to the concave side and the result is still diffeomorphic to X. Cl
An easy corollary of this proof is:

Corollary 3.9.1. Every closed smooth oriented 4-manifold X admits a BLF (or BLP
if b; > 0), by decomposing as a convex PALF and a concave BLF (or BLP) glued

along the common (open book) boundary.

50

Chapter 4

Calculation of the Ozsvath-Szabo

Contact Invariant

This chapter is organized as follows. In Section 4.1, basic properties of Heegaard-Floer
homology and and contact invariant are briefly reviewed. Section 4.2 is devoted to
the algorithm of Ozsvath and Szabo’ to determine the generators of Heegaard-Floer
homology of 3-manifolds given by plumbing diagrams. Remarks given at the end
of the section allow us to find relations easily by combinatorial means. we prove
Proposition 1.0.5 and Theorem 1.0.7 in Section 4.3. A sample calculation is done in

Section 4.4. Theorem 1.0.9 is proved in Section 4.5

4.1 Heegaard—Floer Homology and Contact Invari-

ant

Let Y be a closed oriented 3—manifold and t be a spinc structure on Y. In [084] and

[035], Ozsvath and Szabé define four versions of Heegaard-Floer homology groups

51

I7? (Y, t), H F +(Y, t), H F _(Y, t),and H F 00(Y, t). These groups are all smooth in-
variants of (Y t). When Y is a rational homology sphere, they admit absolute @-
gradings. The groups H F +, H F _, and H F 00 are also Z[U] modules where multi-
plication by U decreases degree by 2. Any spinc cobordism (X ,5) between (Y1.t1)

and (Y2, t2) induces a homomorphism well defined up to sign

F3235 : HF°(Y1,t1) —> HF°(Y2,t2)

Here HFO stands for any one of Pfi, HF+, HF”, or HFOO. We work with Z/2Z
coefficients in order to avoid sign ambiguities. Also, we drop the spine structure from

the notation when we direct sum over all spinC structures.

Given any contact structure 5 on Y, Ozsvath and Szabo associate an element
C(5) E 1:777 (—Y) which is an invariant of isotopy class of 5 [082]. In this paper we
are interested in the image 0+ (5) of (3(5) in H F + (Y) under the natural map. We list

some of the properties of this element below.

1. c+(5) lies in the summand H F +( —-Y, t) where t is the spinc structure induced

by 5.
2. c+(5) = 0 if 5 is overtwisted.
3. c+(5) 75 0 if 5 is Stein fillable.
4. c+(5) E Ker(U).

5. c+(5) is homogeneous. W'hen Y is a rational homology sphere, it has degree

—d3 (5) —- 1 / 2, where (13 (5) is the 3-dimensional invariant of 5.

6. c+(5) is natural under Stein cobordisms: If W is a compact Stein manifold,

alV = Y, U —Y, and 5’ and 5 are the induced contact structures, we can regard

52

IV as a cobordism from -Y' to -—Y and the induced map satisfies F [i.(c+(5’)) =

c+(£>.

The contact invariant c+ (5) is studied by Plainenevskaya in [P]. The following
result will be used later in this paper when we prove our main theorem. We state it
in a slightly more general form than in [P] but Plamanevskaya‘s proof is valid for our

case as well.

Theorem 4.1.1. (Theorem 4 in [P]) Let X be a smooth compact 4-manifold with
boundary Y = 8X. Let J be a Stein structure on X that induce a spine structure 51
on X and contact structure 51 on Y. Let 52 be another spinc structure on X that
does not necessarily come from a Stein structure. Suppose that 51 lY = 52]}; , but
the spinc structures 51, 52 are not isomorphic. We puncture X and regard it as a

cobordism from Y to 53. Then

1. F;,,,<c+<<1>) = o

2. F;:51(c+(51)) is a. generator of HFJ(S3).

Note that in this theorem if the spinc structures 51lY and Szly are not the same

then conclusion (1) follows trivially.

Remark 4.1.2. Theorem 4.1.1 was later generalized by Ghiggini in [Ghi] where
he requires J to be only an w-tame almost complex structure for some symplectic
structure to on X (G) that gives a strong filling for the boundary contact structure. In
this paper, we work with rational homology Spheres. For these manifolds, any weak

filling can be perturbed into a strong filling, [OO].

53

4.2 The Algorithm

Let G be a weighted graph. For every vertex 'v of G, let m(-v) and (1(1?) denote the
weight of v and the number of edges connected to 1) respectively. A vertex v is said
to be a bad vertex: if m(2.') + (1(1?) > O. Enumerating all vertices of G, one can form
the intersection matria: whose ith diagonal entry is m(vz-) and i — jth entry is 1 if
there is an edge between vi and oj, and is 0 otherwise. Throughout, we assume that

G satisfies the following conditions.

1. G is a connected tree.
2. The intersection matrix of G is negative definite.

3. G has at most one bad vertex.

There is a 4-manifold X(G) obtained by plumbing together disk bundles Di, i =
1 . - - |G| over sphere where Di is plumbed to Dj whenever there is an edge connecting
v,- to vj. Let Y(G) be the boundary of X (G) In [081], Ozsvath and Szabo give a
purely combinatorial description of Heegaard-Floer homology group H F +(-Y(G)).

Here we repeat their description for completeness.

The second homology H 2(X (G ), Z) is a free module generated by vertices of G.
Let Char(G) be the set all characteristic(co)vect0rs of this module, i.e. every element
K of Char(G) satisfies (K, v) = m('v) (Mod 2) for every vertex 12. Let 7+ be the
graded algebra Z/2Z[U, U _1] /UZ/ZZ[U] where the formal variable U has degree —-—2.
Form the set IHI+(G) C Hom(Char(G), T+) where any element (,0 of lHI+(G) satisfies
the following property; If K is a characteristic vector, v is a vertex, and n is an integer
such that

(K, '1?) + m(-v) = 2n,

54

we have

(1,472+an + 2PD(U)) = Umo(K) if n > 0,

0 I'

U'mem’ + 2PD(v)) = Um—"auo ifn > 0.

The set of spinc structures on Y(G) gives rise to a natural splitting for lHl+(G).
For. if t is a spinC structure on Y(G), one can consider the subset Chart(Y(G))
consisting of those characteristic vectors whose restriction on Y(G’) are t. The set
lHl+(G'. t) is the set of all maps in lHl+(G) with support Chart. It is easy to see that
n+(c) = gamma, t) .

The group lHl+(G) is graded in the following way. An element c5 6 llil+(G) is said
to be homogeneous of degree d if for every characteristic vector K with tb(K) 75 O,

(MK) 6 7+ is a homogeneous element with

K2 + [Cl _

degwo) — 4

d.

We are ready to describe the isomorphism relating lHI+(G) to the Heegaard-Floer
homology of Y(G). Fix a spinC structure t on —Y(G). Let K be a characteristic
vector on Chart(G). Puncture X (G) and regard it as a cobordism form —Y(G) to

S 3. It is known that X (G) and K induce a homomorphism

FX(G),K : HF+(—Y(G),t) —+ HF+(53) : 7+.

Nowthe map T+ : HF+(—Y(G). t) —-) lHl+(G’, t) is defined by the rule T+(5)(K) =

FX(G),K(€)

Theorem 4.2.1. (Theorem 2.1 in [081]) T+ is a U-equivariant isomorphism pre-

serving the absolute Q—grading.

To simplify the calculations, we work with the dual of lHI+(G). Let lK.+ be the
subset of Z20 x Char(G)/ ~, where the equivalence relation ~ is defined as follows.
Denote a typical element of 220 x Char(G) by U m (8) K. Let o be a vertex and n be
an integer such that

272 = (K, v) + m(v)

then we have

Um‘t'” e (K + 2PD(v)) ~ Um e (K) if n 2 0

01'

Um e (K + 2PD(v) ~ U‘m—n a K if n < 0.

Define a pairing K+(G) x lHl+(G) —> Z by (e, Um @K) —> (Umg‘)(K))0 where ()0
denotes the projection to 0th grade in 7+. It is possible to show that this pairing is
well defined and non-degenerate and hence it defines an isomorphism between lI-Il+ (G)
and Hom(K+(G), Z). Using the duality map and isomorphism T+ one can identify
ker UT"+1 C HF+(—Y(G)) as a quotient of K+(G) for every n 2 0.

Lemma 4.2.2. (Lemma 2.3 in [081]) Let Bn denote the set of characteristic vectors
Bn = {K E Char(G) : V’U E G, [(K, 0)] S —m('v) + 2n}. The quotient map induces a
surjection from n

U Ui ® Biz—i

i=0

onto the quotient space
IK+(G)
z>n x Char(G)'

 

In turn, we have an identification

 

II6(0) m+1 +
Hom (Z>n x Char(G)’Z/2Z) _ kerb C H (G,Z). (4.1)

56

3

.rLMA‘AA‘ .

One should regard the above isomorphism as one between Z/2Z[U] modules where

the U action on the left hand side of equation 4.1 is defined by the following relation.

1 if UP (a K ~ U"+1 a K’
U.(UP e K)*(U" e K’) = (4.2)
0 if UP a K 74 Ur+1 e K’

Where (Up (2) K)* denotes the dual of Up 8) K.

Lemma 4.2.2 gives us a finite model for ker U "+1 for every n 2 0. It is known
that these groups stabilize to give H F +. Therefore, one can understand H F +' by
studying the quotients K+(G)/Zn x Char(G) for all n 2 0. The first quotient is well
understood thanks to an algorithm of Ozsvath and SZabo . Below, we describe the

algorithm and discuss a possible extension.

A characteristic vector K is called an initial vector if for every vertex v, we have

m.(v) + 2 S (K, v) s —m(v) (4.3)

Start with an initial vector K0. Form a sequence (K0, K1, - - - , Kn) of character-
istic vectors as follows: Ki+1 is obtained from K,- by adding 2PD(v) where v is a

vertex with (Ki, v) = —m(v). The terminal vector Kn satisfies one of the following.

1. m(v) g (Kn, v) S —m(z-') -— 2 for all v.

2. (Kn,v) > —7n(v) for some v.

The sequence (K0, K1, . - - , K n) is called a full path, and characteristic vector Kn
is called the terminal vector of the full path. We say that a full path is called good

if its terminal vector satisfies property (1) above and it is bad if the terminal vector

57

fl

I—.‘ ‘ —'. -.

satisfies (2). We list some of the properties of full paths, the reader can consult

[081](especially proposition 3.1 in [081]) for proofs.

0 Two characteristic vectors in B0 are equivalent in K+ (G) if and only if there
is a full path containing both of them where the set B0 is defined as in lemma

4.2.2.

o If an initial vector K0 has a good full path then any other full path starting

with K0 is good.

0 If K0 and K6 are initial vectors having good full paths and K0 75 K6 then
K0 76 K6 in K+(G).

o A terminal vector Kn of a bad full path is equivalent to U m (8» K, in 1K+ (G)
for some m > O and K, E Char(G). A terminal vector of good full path can

not be equivalent to such an element of lIll+(G).

Note that these properties allow us to find the generators of ker U; They are simply
the initial vectors having good full paths. In other words, we know the generators of
the lowest grade subgroup of H F +(—Y(G)). Recall from [083] that the lowest degree
(1(Y, t) of non-torsion elements in H F +(Y, t) is called the correction term for a spine
manifold (Y, t). The algorithm above provides us an efficient method to calculate the
correction term (l(—Y(G), t) for any spinc structure t (see Corollary 1.5 of [081])

K2 + lGl

d(—Y(G), t) = min — 4

(4.4)

where the minimum is taken over all characteristic vectors admitting good full paths

which induce the spinc structure 1:.

The whole group lHl+(G) :3 H F +(—Y(G )) is determined by the relations amongst

the generators of Ker(U). Given two characteristic vectors Ki, Kj admitting good

58

full paths and inducing the same spinc structure on Y(G), a relation between K1
and K2 is a pair of integers (n,m) satisfying U n 8) K1 ~ U m 8) K2. If the non
negative integers (mm) are minimal with that property, we call the corresponding
relation minimal. Here we describe a systematic method to find relations. Say K
is a characteristic vector and n is a positive integer. we want to understand the
equivalence class in K+ (G) containing U" 8) K. We define three operations that do

not change this equivalence class.

(R1) U" @K’ —+ U" 69 K where K, is obtained from K by applying the algorithm
to find full paths.

(R2) Un®K -—> Un—1®(K+2PD(v)) where v is a vertex with (K, v)+m(v) = —2

(R3) Un®K ——> Un+1®(K+2PD(v)) where v is a vertex with (Kn, v)+m(v) = 2

Now assume that K is a characteristic initial vector which admits a good full path.
In order to find particular representatives with small U -depths for the equivalence
class containing U n <8) K we apply R1 then apply R2 if possible else R3. Then we
repeat the same procedure till it terminates at an element U T (8 K I . We call the
vector part of this element as a root vector (the exponent m is determined by n and
degrees of K and K’). A Root vector is not unique, it depends upon choices we made
along the way; like the choice of the vertex at which we apply R2 or R3 is applied.
However, the set of root vectors is a finite set which can be found easily and it can be
used to establish relations amongst the generators of Ker(U). This simple observation

will be useful when we do our calculations.

Proposition 4.2.3. Let K1 and K2 be two characteristic initial vectors admitting
good full paths. Suppose n and m are non-negative integers such that the root vector
sets of U n 83 K1 and U m <8 K2 intersect non trivially. Then we have U” (8 K1 ~
U m (8) K2.

Proof. Follows from the definitions. El

59

4.3 Main Theorem

Proof of Proposition 1.0.5. Let 5 be the canonical spine structure and 5’ be any
other spinc structure on X (G). Note that Cl (5) 74 c1(5’). Recall that the isomorphism

KW?) ‘
z>0 x Char(G)

 

Ker(U) 2 Hom ( ,Z/2Z is given by means of the pairing

KW?)
z>0 x Char(G)

 

P: Ker(U) x Hom( ,Z/2Z) —> Z/2Z

Pm. L) = (F;(G),L(a))0

In view of this observation, it is enough to show the following two equations hold.

(F;(G,,(ce)))o = 1 (4.5)
(F;(G)5,(c(€))o = 0 (4.6)

These are simply the conclusions of Theorem 4.1.1.

Proof of Theorem 1.0.7. By Theorem 1.0.4 and Proposition 1.0.5, it is enough
to show that K 95 Im(Uk) for some k E N. Let {K1, K2,- -- , Kr} be the set of charac-
teristic initial vectors admitting good paths such that deg(Kz-) S K and Ki|Y(G) = t
for all i = 1 - - ~r. Basic properties of contact invariant imply that this set is not
empty if one of the assumptions is satisfied. It is known that on any rational homol-
ogy sphere and for any spinc structure, the Heegaard-Floer homology decomposes as

HF+ = T+ EB HFred . 80, one can find integers n0, n1, - ~ ,nr such that

URO e) K ~ U711 ®K1~---U"7‘ e Kr.

60

Moreover, by choosing these numbers large enough, we can guarantee that the dual
of U710 (X) K is the unique generator of the degree —d3(5) — 1/2 + 212.0 subspace of
HF+(—Y). Then by equation 4.2, K ¢ Im(U"0).

4.4 Example

In this section, we shall illustrate an application Theorem 1.0.7 and show that certain
Stein fillable contact structures do not admit planar open books. Obstructions to
being supported by planar open books were known to exist before. The first such ob-
struction was found by Etnyre. It puts restrictions on intersection forms of symplectic

fillings of planar open books.

Theorem 4.4.1. (Theorem 4.1 in [Et]) If X is a symplectic filling of a contact 3-
manifold (Y, 5) which is compatible with a planar open book decomposition then
bi(X) = b% (X) = 0, the boundary of X is connected and the intersection form Q X

can be embedded into a matrix which is diagonalizable over integers.

Ozsvath SZabé and Stipsicz gave another obstruction. It is a consequence of Theorem

1.0.4 above though its statement has no reference to Floer homology.

Theorem 4.4.2. (Corollary 1.5 of [088]) Suppose that the contact 3-manifold (Y, 5)
with Cl (3(5)) = 0 admits a Stein filling (X, J) such that Cl (X, J) 74 0. Then 5 is not

supported by a planar open book decomposition.
Yet another criterion is stated in [088]. It partially implies Theorem 1.0.7 above.

Theorem 4.4.3. (Corollary 1.7 of [088]) Suppose that Y is a rational homology

3—sphere. The number of homotopy classes of 2-plane fields which admit contact

61

structures which are both symplectically fillable and compatible with planar open
book decompositions is bounded above by the number of elements in H1 (Y; Z). More
precisely, each spinc structure 5 is represented by at most one such 2—plane field, and
moreover, the Hopf invariant of the corresponding 2—plane field must coincide with

the correction term d(—Y. .9).

Below, we give examples of non-planar Stein fillable contact structures on a Seifert
fibered space. Non-planarity of some of our examples do not follow from Theorem

4.4.1, Theorem 4.4.2 or Theorem 4.4.3.

Example 4.4.4. Consider the star shaped plumbing graph consisting of eight vertices
where the central vertex has weight —4, a neighboring vertex has weight -—3 and all
the others are of weight —2 (See figure 4.1). The boundary 3—manifold Y is the Seifert

fibered space

NIH

[\D

co
v

The reason why we have so many self intersection —2 spheres is that we want to
avoid L-spaces where the obstruction to admit planar open book does not exist. For

topological characterization of L-spaces among Seifert fibered spaces see [L8].

-3
-4

Figure 4.1: Plumbing description of Seifert fibered space ll/I(—4, %, - .. , %, :1?)

It can be shown that the corresponding intersection form is negative definite,and
it has determinant 128‘. Moreover it can be embedded into a symmetric matrix which

is diagonalizable over integers. To see this, index the vertices so that the central one

62

comes first and the weight —3 vertex is the last. Let e1, 82, - .. , 911 be a basis for
R11 such that e,- -e,-_ = —1 for all i = 1, 2, - . - , 11. The embedding is defined by the

following set of equations

'1’1 —-> —e1 — e2 — 83 — e4
v2 —-> 62 — (:7
13 —> 62 + 67
v4 —) e3 — 88
v5 —-> 63 + e8
135 —-) e4 - 69
v7 —) e4 + e9

1.78 --> 61+ 610 + 611

First, we calculate H F +(—Y, t) for every spinC‘ structure t. For similar calcula-
tions, see [D], [R1], [R2], and Section 3.2 of [081]. Write any characteristic vector K
in the form K = [(K, v1), - - - , (K, '08)]. There are 768 characteristic vectors satisfying
equality 4.3, and 138 of them have good full paths. When we distribute these to spinc
structures of Y, we see that for 10 spinc structures Ker(U) has rank 2, and the rank
is 1 for the rest. In Table 4.1 below, we provide a table that shows H F + for these
spine structures. Characteristic vectors on left are the generators of Ker(U). Ones

that lie in the same box induce the same spinc structure on boundary.

Next, we consider the obvious Stein structures that arise from the handlebody
diagram associated to G. Following Eliashberg, we isotope the attaching circles of
2-handles into Legendrian position so that their framing become one less than the

Thurston-Bennequin framing. For —2 framed unknots, there is unique way to do

63

that. For the other unknots which correspond to v1 and 1,28 take Legendrian isotopes
with rotation numbers i and j respectively where i = —2,0, 2, and j = —1,1. Call
the resulting Stein structure as Ji, j and the induced contact structure by 511]” See
figure 4.2 for a picture of J2,_1. The curve on left corresponds v1 and the other
represents 228. They are both oriented counter clockwise. We omit the other unknots
linking to v1 in order not to complicate the picture. Note that the first Chern class of

Ji.j is given by the characteristic. vector Ki. j = [i,0, 0, 0, 0,0, 0,j]. It is easy to verify
. 2 ,
K 2.: j + [G]
4
contact structures €:l:2.:t1 do not admit planar open books. By the algorithm given

that d3(€i’j) + 1/2 = degree(K,i,j) = . According to Theorem 1.0.7, the
in [Et1102] these contact structures do admit genus one open books, so their support
genus are all one. One can not use Theorem 4.4.2 directly to get this conclusion
because the Chern classes of the corresponding spinc structures are all of order 4.
Though Theorem 4.4.3 also implies our conclusion for 521 and 5_2,_1, it doesn‘t
apply to 5271 or 5_2:1. 80 the latter two are the contact structures we promised at

the beginning of the example.

Figure 4.2: Legendrian handlebody diagram giving J2__1.

Remark 4.4.5. The main result of [Etn0] implies that the support genera of plumb—
ings with at most two bad vertices are at most one. On the other hand the algorithm
of Ozsvath and Szabo does not work if the number of bad vertices is greater than two.
Therefore, the techniques used in this paper do not seem to be pushed further to find

an example of a contact structure with support germs strictly greater than one. We

64

are planning to turn this problem in a future project using a different approach.

4.5 Calculation of o

In this section we shall prove Theorem 1.0.9 by calculating explicitly the o invariant
of a family of contact 3—manifolds. 0ur argument is based on a previous work of

Rustamov, [R1].

For every positive integer n, consider the contact manifold (Yn, 5n) obtained from
(83,5Std) by doing Legendrian surgery on the (2,2n + 1) torus knot Ln stabilized
2n. — 1 times as in figure 4.3. Observe that the Thurston-Bennequin invariant of Ln
is zero so the topological surgery coefficient is negative one. In fact, the 3-manifold

Yn is the Brieskorn homology sphere 2(2, 2n + 1,471 + 3).

 

crossings

n—l

3

v

Figure 4.3: Kn

Theorem 4.5.1. 0(Yn, 5n) 2: pn — 1 where pn is the nth element of the sequence

1,1.2.2,3.3,---.

Clearly, Theorem 4.5.1 implies Theorem 1.0.9. Another immediate application of

Theorem 4.5.1 is that (Yn,5n) can not be supported by a planar open book. This

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Spinc Characteristic Vectors ' Degree Relation H F +(—Y)
[2,0,0,0,0,0,—1] 7/8 , _ , T 7331 7
1 [—2,0,0,0,0,0,0,3] 7/8 U®51“U®I‘2 ‘3 -g
[—2,0,0,0,0,0,1] 7/8 . _ 7- 7392 7
2 [0,0,0,0,0,0,0.3] 7/8 U®A1‘U®K2 ‘3 -g
[—2,0,0,0,0.0,—1] —1/8 , _ 2 7- 15eZ1
3 [0.0,0,0,0,0,0,1] 15/8 U®K1“‘U ®K2 "8- g
[2,07070,0,071] —1/8 ,, — 2 T 15®Z1
4 [o,0,0.0.0,0,0,—1] 15/8 (”Ml—U ®K2 -7; 3
[—2,0, ~-, 2 ,n ,0,—l] 3/4
5+j j+2 U®I{1=U®K2 T_%EBZ_%
[0,0, , 2, ,0,1] 3/4
j+2
3:0,... 5
Spinc Root Vectors
[2303'°°a01 —4909”'707_3]
1 V
'I.
i=2,~-,7
[0,0, 0, 0, 0, 0, 0, —5],
2 [0,0,~- ,0, —-4,0,--- ,0,1],
‘5’“
i
i=2,-~,7
3 [0701”. 303,—.4301H' 107—1]:
2
i=2,--- ,7
4 [—2, 0, 0, 0, 0, 0, 0, —3]
[2,0,..., _4,..., -2,...,0,-1]
5+j 2‘. j+2
i=2,---,7
j=0,...,5
+ 1 1 - c
Table 4.1: HF of All(—4,§,---,§,§) for 10 spin structures.

66

 

was first pointed out in [088]. Finally, combining this theorem with the fact that
o invariant respects the partial ordering coming from Stein cobordisms we have the

following corollary.

Corollary 4.5.2. There is no Stein cobordism from ( ”71,571) to (Ym, 5m) if n > m+1.
In particular, one can not obtain (Ym,5m) from (Yn,5n) via Thurston—Bennequin

minus one (tb — 1) surgery on a Legendrian link.

The above corollary should be compared to a classical result of Ding and Geiges
in [D0], which was stated in Theorem 2.4.3 above, where it was proved that any two
contact manifolds can be obtained from each other via a sequence of (tb— 1) or (tb+ 1)
contact surgeries. In fact, one can always choose such a sequence which contains at
most one tb + 1 surgery [DGS]. Therefore, the corollary tells us that the existence of
(tb + 1) surgery is essential even though the contact manifolds in both ends are Stein
fillable.

Proof of Theorem 4.5.1. Let V be the 4-manifold obtained by attaching a
Weinstein 2-handle to a 4-ball along Ln. Eliashberg’s theorem [E] tells that V admits
a Stein structure. Let 5 be the canonical spinc structure on V, and denote the
homology class determined by the 2-handle (for some orientation of Ln) by h. The

way we stabilize Ln ensures that.

(31 (5)0?) = r0t(Ln) = i1 (4.7)

Where rot(Ln) stands for the rotation number of Ln. Note that the sign of the
rotation number depends on how we orient Ln. Next, V is blown-up n + 2 times, and
we do the handleslides indicated in figure 4.5. we see that the resulting 4—manifold is
given by the plumbing graph G in figure 4.4. The manifold X (G) is no longer Stein
but it does admit a symplectic structure. Let 5’ be the canonical spinc structure on

this symplectic manifold. Let 8i denote the homology class of the ith exceptional

67

sphere. We have

c1(s’)(e,-) =1 i=1,2,--- ,n+2. (4.8)

—2 —1 -3 —2 —2 —2

o—I—o—o—o— -- —0
H—d
—4n—3 n—1

Figure 4.4: A Plumbing graph for Brieskorn homology sphere 2(2, 2n + 1, 4n + 3)

Order the vertices of G so that first four are the ones with weight —1, —2, —3 and
—4n — 3 respectively, and all the remaining ones corresponding to —2s on right are
ordered according to the distance from the root. starting with the closest one. In [R1],

Rustamov proves that

, p
HF+(—Yn)= "3e @( (an_ _z eaqz l nfl.)

where qu = i(i + 1) and Z70”: [U] / U'rZ[U] and U7."1 lies in degree 16. More

precisely, he shows that KerU C H F + (—Yn,) is generated by the characteristic vectors

K, = (1,0,—1,—4n—3+2i), i=1,2,-~ ,2n.

He also proves that the minimal relations are given as follows:

Upz' e K,- ~ v’Pi+Qn-—i e K,,_+1 (4.9)
UPi. 8 KM,- ~ Upi+qn—i a Kn (4.10)
where i = 1,2, - . - ,n. Note that the characteristic vectors Kn and Kn+1 are in the

68

bottom level, and their degree is zero.

Our aim is to pin down the contact invariant c+(5n) in H F +(-Yn). Note that
proposition 1.0.5 in the stated form can not be applied directly as it concerns Stein
fillings of plumbed manifolds. However, as indicated in remark 1.0.6 it is also true
for strong symplectic fillings. The only difference in the proof is that one uses Ghig-
gini’s generalization [Ghi] of Plamenevskaya’s theorem [P]. Alternatively, one can use
the blow-up formula and handleslides invariance for this particular case to see that.
equations 4.5 and 4.6 hold. In any case, we see that the contact invariant c+(5n.) is
represented by the first Chern class c1(5’) of the canonical spinc structure. In figure
4.5, we keep track of the homology classes in order to pin down the first Chern class.
By equations 4.7 and 4.8, we have c1(5’) = Kn or Kn+1 depending on the orientation

of Ln, but the contact invariant is in the image of UPn—l in any case.

69

h’ h—281—282—"'—28n

mg)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r——\) . Q) 61
2n + 1 Q 31:)“; 9:) 627 Handlcslidcs
)? L'ben’
r l .g—U
h—2e1—2eg—---—2e,,
_e"—182 — e1?
51‘ ----- >00. 12 ..
83 — 82 JJ
lsotopy
h—281—282—"'—28n
( 1
[an _ 611—1 83 _ 82]
2K 0 ----- D1? )2 Blow-
2 e4— 61 up
\ 81 eJez J \h’ — 281 — 282 _ ' ' ' _ 2811 _ err-H _ en+2

 

 

 

 

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70

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