. 1‘7. . . ‘
4 ¢ 114‘...

‘2‘ M20: U . hn‘ammu
. 2...... , c.
“Nat . ,
5? LI.

n.

«mung...

#1“.
p.

..
...3. \ .. I.
v. 3.31"»

t. a. T

IN... a.
.l r 5
“$12.42...
.... 1‘25?!» 21......
.r .e

:wna‘a‘ £3
. I ...-11.1. K
v..- fixsnon. .

....

1-4::
fimfifii
nu... .
u...

3., e... 5.x .t p
.5... (:5. .... ..1?..i...
.-. or? .v.‘ stick“...
I ... ...!ludultliilt

. . 1.1.6...“ .31..

‘ 3.99% 1....

...

 

&.

 

This is to certify that the
dissertation entitled

SYMPLECTIC STRUCTURES, LEFSCHETZ FIBRATIONS,
AND THEIR GENERALIZATIONS
ON SMOOTH FOUR-MANIFOLDS

presented by

REFIK INANC BAYKUR

     
 
  

 

 

   
   

 

3 l
>- .53 3..
I CO “,3 has been accepted towards fulfillment
E, g g of the requirements for the

D)...
g :5 5 degree m MATHEMATICS

E ”103,44
v 63 W

 

Major Professor’s Signature
Date

MSU is an afiinnative-action, equal-opportunity employer

 

 

.-.-o-I-C-I-D-I-Q-o-l-e-o-I-.-.-.-.-.-

- -.-v-l-I-o-I-O-I--l-o-u-l-l-v- -..

- ".-.-.-.-.-.-.—.--o—.--—---.—-

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p:/ClRC/DateDue.indd-p.1

Symplectic Structures, Lefschetz F ibrations

and Their Generalizations on Smooth Four-manifolds
By

Refik Inang Baykur

A DISSERTATION

Submitted to
h'Iichigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics

2007

ABSTRACT

Symplectic Structures, Lefsohetz Fibrations

and Their Generalizations on Smooth Four-manifolds
By

Refik Inant; Baykur

In this thesis, we study symplectic structures. Lelschetz fibrat ions. and their var—
ious generalizations on smooth 4-Illt-illift)I(.IS along with the e-issociated (smooth) III-
\i'aria..1'1ts. Our results will be presented in separate chapters as follows:

In Chapter 3. we outline a graieral constructitm scheme to obtain minimal sym-
plectic structures on siniply—conneeted 4—manifolds with small Euler characteristics.

Using this scheme, we illustrate how to obtain minimal syml‘flectic 4-manifolds horne-

 

omorphio to CH’3#(2A7 + llm2 for k = I....,4, or to 3CP2#(21 + SKIP“). for
l = ‘2. . . . ,6. Secondly. for each of these homer)morphism types with N = 1. we
show how to produce an infinite family of pairwise nonditIeonn)rphic nonsymplectic
4-111anifolds belonging to it. In particular, we prove. that there are infinitely many
irreducible nonsymplectic smooth structures on CP2#3@—H33.

In Chapter 4, we study the «l—manifolds with nontriyial Seiberg-\\'itten inyz-iriants
which are equipped with near-syniplectic broken Lefsclietz fibrations. \Ve first study
the topology of these fibrations and describe simple presentations of them. “1% then

)royide several exam )les usiiw handlebody diaorains. \X'e define a near-sym )lectic
O s ('5 .

operation that generalizes the symplectic fiber sum operation, together with its effect
on the Seiberg-Witten invariants and Perutz’s Lagrangian matching invariants. These
techniques are then used to obtain several results on near—symplectic manifolds with
non-trivial invariants.

In Chapter 5, we show that every closed oriented smooth 4-manifold can be de—
composed into two codimension zero submanifolds (one with reversed orientation) so
that both pieces are exact. Kahler manifolds with strictly pseudoconvex boundaries
and that induced contact structures on the common boundary are isotopic. Mean-
while, matching pairs of Lefschetz fibrations with bounded fibers are offered as the
geometric counterpart of these structures. we also provide a simple topological proof
of the existence of folded symplectic forms on 4-manifolds. Finally in the Addendum,

we provide answers to two open questions stated by David Gay and Rob Kirby.

To my family who have become my friends and my friends who have become my family

iv

ACKNOWLEDGMENTS

Acknowledgmcnts. I would like to thank to my advisor Ron Fintushel for his
support and for various helpful conversations. I am grateful to the fellow geometry
and topology graduate students and faculty at Michigan State, with whom I have had
the pleasure to engage in many mathematically stimulating interactions. Especially
many thanks to Adam Knapp, Lawrence Roberts, John l\rIcCarthy, Tom Parker and
Ron Fintushel for this matter. I would also like to thank to Firat Arikan for his help
with drawing several figures appearing herein. Cagri Karakurt and Lawrence Roberts
for their interest. in the material contained in Chapter 4 and for their feedback which
improved the presentatirm of the same chapter. Lastly I would like to express my
gratitute to my friends and family. who made me feel their support at all times even

from thousands of miles away.

TABLE OF CONTENTS

 

 

LIST OF FIGURES ............................. viii
1 Introduction ................................ 1
2 General background ........................... 7
2.0.1 Topology of smooth 4-111anifolds ..................... 7
2.0.2 Symplectic structures ........................... 8
2.0.3 Lefschetz fibrations ............................. 11
2.0.4 Seiberg-VVitten invariants ......................... 14
3 New symplectic 4-manifolds ...................... 17
3.1 Background .................................. 17
3.1.1 Generalized fiber sum ........................... 17
3.1.2 Minimality ................................. 18
3.1.3 Surgery along Lagrangian tori ....................... 20
3.1.4 Surgeries and Seiberg—VVitten invariants ................. 21
3.2 Constructing small exotic symplectic 4-manifolds ............. 22
3.2.1 The construction scheme for odd blow-ups of CIP’2 and 3C1?” ..... 22
3.2.2 Twist knots and Luttinger surgeries ................... 25
3.3 Minimal symplectic 4—111a11ifolds with (1+ = 1 ............... 30
3.3.1 A new description of a minimal s11:11plecti( Etl) ............ 30
3.3.2 A 111 11 construction of a minimal s\ 111plert1c (CF #TCP P2 ........ 32
3.3.3 A new const1uttion of a minimal sy mplec tic (DIP #5CIP P2 ........ 34
3.3.4 .A minimal s1 mph. (tit. CllD2 #3C_IP) in te 1111s of Luttinger surgeries 30
3.4 Minimal symplectic 4—111auifolds with 11+ > 1 ............... 38
3.5 Infinite families of nousymplectic irreducible smooth structures ...... 41
3.5.1 An infinite family of irreducible smooth structures on CP2#55CTDQ 43
3.5.2 Infinite families of irreducible CIID3#(2A' + UC—PB for I; = 2. 3. 4 45
4 Near-symplectic 4-manifolds ...................... 48
4.1 Background .................................. 48
4.1.1 Near—syml1lectic structures ......................... 48
4.1.2 Local models ................................ 4.0
4.1.3 Broken Lefschetz fil‘11‘z'1tions ........................ 51
4.1.4 Lagrangian 111atcl'1ii'1g invariants ...................... 55
4.2 'I‘opology of broken Lefschetz librations ................... 58

vi

4.2.1 Round 1-handles .............................. 59

4.2.2 Round 2-handles .............................. 61
4.2.3 Simplified broken Lefschetz fibrations ................... 63
4.2.4 Examples .................................. 67
4.3 Some near-symplectic operations ...................... 73
4.3.1 Broken fiber sum . . . . . . ........................ 74
4.3.2 Button addition ............................... 80
4.4 Applications to near-symplectic 4-manifolds with non-trivial invariants . 83
5 Folded-symplectic 4-manifolds ..................... 89
5.1 Background .................................. 89
5.1.1 Achiral Lefschetz fibrations and PALFs ................. 89
5.1.2 Open book decompositions ........................ 90
5.1.3 Contact structures and compatibility .................. 91
5.1.4 Stein manifolds ............................... 93
5.2 Simple folded-symplectic structures ..................... 95
5.3 Existence of folded-symplectic structures on closed oriented 4-111anifolds . 100
5.4 Kahler decompostion theorem ........................ 106
5.5 Folded-Kahler structures and folded Lefschetz fibrations ......... 113
5.6 Addendum: Interactions between the two generalizations ......... 120
5.6.1 General syn11‘1lectic structures on broken achiral Lefchetz fibrations . . 121
5.6.2 From achiral to broken I_.efscl1e.tz fibrations ................ 123
5.6.3 Comments on describing invariants on general 4—111a1iifolds ....... 128
BIBLIOGRAPHY .............................. 131

3.1

3.2
3.3

3.4
4.1

4.2

4.3
4.4

4.6
4.7

4.8
4.9
4.10

5.1

LIST OF FIGURES

The 4-torus c x (d x a x b). The neighborhood of a fiber chosen in the 3-torus
d x a x b at c = 0.5 is given by fat slices parallel to a x b face, which get
thinner while c gets closer to 0.5 d: e and disappear when 0 < c < 0.5 — 26
or 0.5 + 26 < c < 1. The neighborhood of the torus section is given by a
cylindrical neighborhood in the direction of d lying in the 3-torus times c.
Neighborhoods of the tori c X {1 and c x b are drawn similarly. ......

The 3-torus d x a x b at c = 0.5 .......................

The first diagram depicts the three loops 0., b. d that generate the 711(T3). The

curves (1 = dod‘1 and b are freely homotopic to the two extra curves given
in the second diagram. The third diagram is obtained from the second via
two slam-dunk operations; wheras the last diagram is obtained after Rolfsen
twists. . . . .......... . . . . . . ..............

The face of cube where we can see the Lagrangian pushoff of ('1

A general odd round 1—ha.11dle (left), and an even round 2-handle attachment
to a genus two Lefschetz fibration over a disk (right). Red handles make up
the 101md I- handle. ..... . . . . . . ......... . . . .

Left: an even round 2 —handle attachment to D2 x T2. Right: an odd round
2-handle attachn‘ient to an elliptic Lefschetz fibration over a disk with two
.Lefschetz singularities. Red handles make up the round 2-Iiandle. . . .

The step fibration on S2 x $9 #S1 x S3. . . . . . . . . ...... . . . .

A family of near-symlectic BLFs over S2 (left). and the diaglam after the
handle slides and cancelations (right). . . . . . .......... . .

A near—symplectic BLF for an S2 bundle over S2 with Euler class lat. On the
right: l-handles sare replaced by dotted circles. . . . . . ......

Identifying the. total space of the BLF in Figiue 4.5 . . . . . . . . .

A nea1-sy111pl(.1(.1tic BLF on 2C? #2C—ll1’2. The 111111111 2— handle separates the
sphere fiber 011 the higher side into two splieies on the low e1 side.

A broken Lefschetz fibration on 8‘. . . ..... . . ......

A broken Lefschetz fibration on S1 x S3 # 5'1 x Sl ...... . ......

Vanishing cycles 111 the. Matsumoto fib1ation. . ..............

. . . . . . —,1) . . ‘)
On the left: 0-surger1' along the bmdmg yu'lds a t1‘1v1al 5‘0f'1brat1on over D~

v.)
on each piece which make 11p S2 X S“. ()n the lig:ht sin 11(111 along the
new binding yields a cusp neighborhood on both sides.0

viii

26
26

28
29

60

62
68

69

‘I ‘1 ‘4 ‘J
[\3

CDCADM

104

5.2

5.3

5.4
5.5

5.6
5.7

The achiral Lefschetz fibration on the second associated model. The total space
is shown to be CIP2#@2. .........................
New monodromies from old ones. On the left: 111 is given by solid arcs, and 112
by dotted ones. On the right: Solid arcs are the positive arcs representing
11+ , whereas conjugated dotted arcs are negative, providing a representation
of 1.1.- after closing the base back to $2 ...................
Replacing a negative Lefschetz singularity by a round singularity: f (left) and
f’ (right) ..................................
Neighborhood of a negative nodal fiber which has a simple nonseparating van-
ishing cycle. ................................
Consecutive 2-handle slides in the blown-up neighborhood of a negative node.
After an isotopy, we obtain 3. Kirby diagram of a round l-handle attachment
to a product neighborhood of a fiber with one less genus. .........

Images in this dissertation are, presented in color.

ix

105

108

124

CHAPTER 1

Introduction

The world of smooth 4-manifolds has been explored using both analytical and topo-
logical tools. Several special structures which are the subject of complex, Riemannian,
or symplectic geometries have been extensively used in this research, often in conjunc-
tion with the gauge theoretic smooth invariants of Donaldson to Seiberg—VVitten. On
the other hand. Kirby calculus and various surgery techniques have become classical
tools to attack problems of 4-1nanifold topology. The last. decade. has witnessed a
novel advance due to the intense. collobaration of these two forces. During this period
most attention has been given to svmplectic 4~manifolds. This was mostly due to
work of Taubes which related the Seiberg—VVitten invariants (SW7) to enumerative ge-
ometry [7.5], Donaldson‘s work which provided a description of svmplectic 4—manifolds
in terms of Lefschetz liln‘atious/pmicils [1.7], and tinallv various surgeries introduced
by Fintushel and Stern, Compf, and several others (for example [38, '27, 28]).

An elegant bridge in this story was established by the work of Donaldson and
Smith [19]. who defined an invariant which, roughly speaking, associates a. Gromov
count to sections. of liberwise symmetric products correspomling to nicely embedded
multisections of a. given svmplectic Lefschetz pencil. This invariant was shown to be
equivalent. to SW later by Usher. Remarkably, the most recent Slll(.)()l’-l1 4-manifold

i1'1variant of all. Heegaard—Floer invariants of ()zsvath and Szabo were slmwn to com-

pute nontrivially on syrnplectic 4-manifolds by using cobordisms that arise from
underlying Lefschetz pencils/fibrations.

In a nutshell, this thesis work focuses on extending the territory of smooth 4-
manifolds that similar techniques can be employed in the alliance of these forces,
through generalizations of symplectic structures and Lefschetz fibrations. The major
problem we have in mind is determining the number of smooth structures on a given
4-manifold.

In the past few years a lot of interest has been gathered around constructing smooth
4-manifolds which are homeomorphic but not ditteomorphic to the projective plane
(CllD2 blown-up at n points (71 < 10) as well as to the connected sum of three copies
of CH” blown-up at m points (m < 20). These manifolds are "small” in the sense
that they have small Euler characteristics. whereas the construction of exotic smooth
structures gets harder when the. manifolds get smaller. The most recent history
can be split into two periods. The. first period was opened by J. Park's paper [60]
which used the ratitmal blowdown technicpie of Fintushel and Stern [27]. and several
constructions of small exotic manifolds relied on an artful use of rational blowdown
techniques combined with improved knot surgery tricks [71, 30. 62]. More recently.
Akhmedov’s construction in [3] triggered the hope that using building blocks with
nontrivial fundamental groups could succeed in obtaining exotic smooth structures on
simply—connected J-manifolds. These techniques were. initially espoused by Fintushel
and Stern in [29] and later discussed in [70] and in [‘25]. The common theme in
the recent constructions ([3. 6. ll). 25]) is the manipulations to kill the funtlamental
group. These constitute the content of Chapter 3. and appeared in a joint article of
the author with Akhmedov and Park [5]

Chapter 3 begins with an outline a general recipe to obtain small minimal svmplec-

tic ~l-manifolds and to fit all the recent constrm-tions in [3. (j. 10] in this construction

scheme (Section 3.2). In particular, we aim to show that seemingly different examples
are closely related through a sequence of Luttinger surgeries. The second goal is to cal-
culate the basic classes and the Seiberg-Witten invariants of these small 4-manifolds.
Using these calculations we show for instance how to obtain infinite families of pair-
wise nondiffeomorphic manifolds in the homeomorphism type of CIP2#(2k + DEF),
for k = 1,. . . ,4 or or to 3CIP’2#(21 + $611.52, for l: 2,. . .,6. (Sections 3.5 and 3.4).
We distinguish the diffeomorphism types of these 4-manifolds by comparing their SW
invariants. Each of our families has exactly one symplectic member.

Recent research suggests the next target beyond the realm of symplectic topology
to be the near-symplectic manifolds i.e. manifolds which admit a kind of singular
symplectic form that is singular along an embedded l-manifold. These are precisely
the closed oriented smooth 4-manifolds with b+ > 0. Taubes’ program [77, 78] aims
to obtain SW invariants as generalized Gromov invariants in this setting. It has mo-
tivated several parallel ideas. In [9]. Auroux. Donaldson, and Katzarkov defined a
generalization of Lefschetz tibrations, which we here call "brokcn Lefschetz fibrations’,
and showed that they are to near-syniplectic 4-manifolds what Lefschetz fibrations
are to symplectic 4-manifolds. Perutz combined these. approachrs to define. an invari-
ant [64. P2]. called Lagrangian. matching zin-zitrriorizt (LM). He conjectured that. it is
equivalent to SW. This invarieamt generalizes the Donaldson-Smith construction [19]
to near-symplectic broken Lefschetz fibrations by ctmsideriug pairs of sections over
a splitting base that ‘match’ by satisfying certain Lagrangian boundary conditions
which arise from the zero locus of the near—symplectic form. The very nature of LM
invariants requires the study of broken Lefschetz librations. These topics constitute
the content of Chapter 4.

The point. of view we take is to consider 4-i'nanifolds with nontrivial Seil.)erg-V\7itten

invariants as an intermediate class that lies in between near-symplectic 4-i'nanifolds

and the symplectic ones. (When the manifolds in consideration have (3+ = 1, we
always take the SW invariant computed in the Taubes’ chamber of a symplectic or
near-symplectic form.) Thus our work in Chapter 4 runs in two veins. We first
study the topology of near-symplectic broken Lefschetz fibrations, describe simplified
representations of them via Kirby diagrams and monodromies, and provide several
examples (Section 4.2). Importantly. all possible round handle cobordisms that arise
in this context are described in this section. Having the conjectural equivalence in
mind. we define new operations on near-symplectic broken Lefschetz fibrations, and
investigate their effect on both LM and SW invariants (Section 4.3). The broken
fiber sum. operation introduced in this section generalizes the symplectic fiber sum
construction to the near-symplectic setting (Theorem 43.1)}

'We use these techniques to obtain various results regarding near-symplectic 4—
manifolds with nontrivial Seiberg-Witten invariants (Section 4.4). Let (X.w) be a
near-symplectic 4—manifold with zero locus Z . Taubes has proved that. if X has
nonzero SW. then there is a. .l-holomorphic curve (.' in X with homological bound-
ary Z. where J is an almost complex structure conqmtible with to in the complement
of Z [78]. In Theorem 4.4.1 we show that the converse of this statement cannot be
true. and that an analogous result holds for the Ll\-l invariants. This is natural and
expected. since. it suggests that the moduli space that one would like to consider here
can be nonempty while the count is zero. Another question we address is the behavior
of near-symplectic 4-manifolds with nontrivial SW invariants under the symplectic
fiber sum operation. Although the symplectic fiber sum operation preserves the class
of symplectic 4-1'nanifolds. we show that it does not. preserve the. class of SW nontriv-
ial near-symplectic 4-manifolds (Theorem 4.4.2). In a comparison with symplectic
Lefschetz tibrations , we determine the constraints on the self-intersection of sections

of near-sYIuplectic broken Lefschct libratious on manifolds with nontrivial SW" invari—

ants (Theorem 4.4.4), and we describe the near-symplectic broken Lefschetz fibrations
on knot surgered E (n) (Proposition 4.4.5).

Further extension of these ideas takes us out of the usual range of SW invariants,
and requires a new setting. (For instance to work with S4 or 3‘ x S3 which have
b+ = 0.) In Chapter 5 we search for ‘nice’ additional structures on general closed
simply-connected oriented 4—manifolds. The results of this chapter, except for the
Addendum, are contained in the article [11].

One possible strategy for understanding oriented smooth 4-manifolds is to break
them up into more tractable classes of manifolds in a controlled manner. Situated in
the intersection of complex, symplectic and Riemannian geometries, Kahler manifolds
are the best known candidates to be pieces of such a decomposition. The main
theorem of Chapter 5 (Theorem 5.4.2) shows that this can be achieved for any closed
oriented smooth 4—mz—mifold X. we decompose X into two exact Kahler manifolds
with strictly pseudoconvex boundaries, up to orientation. such that contact structures
on the common boundary induced by the maximal complex distributions are isotopic.

This decomposition gives rise to a globally defined '2-forrn on X. which
we call a. (nicely folded-Krihler structure. and it belongs to a larger family of
2-forms: folded-Sinai)lectic structures. Cmmas da Silva showed in [13] that any closed
smooth oriented 4-manifold can be equipped with a. folded-symplectic form. by using
a version of the lr—principle defined for folding maps by Eliashberg. In Section 5.2. we
introduce a way to construct some simple examples of folded-symplcctic 4—manifolds.
Afterwards we reprove the existence fact by constructing a. folded-symplectic form to
for a. given harnllebodv decomrmsition of X. essentially by means of simple handle
calculus and contact topology (Theorem 5.3.1). The main ingredient there is achiral
Lefschetz fibrations. and recent work of Etnyre and Fuller [‘23] will play a. key role in

our construction.

Next, we switch gears, and using several results on compact Stein surfaces and
Lefschetz fibrations with bounded fibers (mainly [44], [20], [39], [50], [2]) we prove
the aforementioned decomposition theorem. In fact we obtain a stronger result, as the
pieces of this decomposition are actually Stein manifolds with strictly pseudoconvex
boundaries. It was first shown by Akbulut and Matveyev in [1] that any closed
oriented smooth 4-manifold X can be decomposed into Stein pieces, but there was
no particular information one could use to argue for matching the induced contact.
structures on the separating hypersurface. Our proof follows an alternative way via
open book decompositions. and we conclude that the Stein structures can be chosen
to agree on the common contact boundary.

In Section 5.5. we introduce folded-Kahler structures. and discuss some properties
they enjoy, after showing that all closed oriented smooth 4-manifolds admit them
(Theorem 5.5.2). This improves the folded-symplectic existence. result, and indeed
ljoth structures “Y’(1)HSIFHCI are shcnvn t()I)e eqiuvvdein (Hi the syunlflecturlevel
The collectimi of these discussions yield us to define folded Lefschetz fibreflons which
are. roughly speaking. pairs of positive and negative Lefschetz fibrations over disks
with bounded fibers which agree on the common l)(,)llll(li.ll‘y through induced open
book decompositit)ns. \X’e prove that any nicer folded-Kiihler 4-1nanif(_)ld, possibly
after aiicnientatnin prescrxiin;(lHIeointnqihisni.auln1n>;(tn1n)atfl)kaf1d(kxl Imuschetz
fibrations (Proposition 5.5.6).

h1[34] [knid Chn'rnuifhdil{nln'rnoved Hun any chmed snuuuh ofienhul=4—
manifold can be equipped with a broken achiral Lefschetz IllH‘ztllUIl. In the Addendum
(Sectnin 513)\ve use our resulhsin CH1apters=laind 5 to estalnnditlu?c1nrespcnulnng
symplectic generalization in this setting (Section 5.6.1). and show a way to avoid achi—
rality (Section 5.6.2) in such a. crmstruction. These provide answers to two questions

asked in [34].

CHAPTER 2

General background

In this preliminary chapter we review several definitions, notations and facts that are
used in the later chapters but not contained in the background material given there.
Thus this review is not intended to be complete. For the details or proofs of the

quoted facts. the reader can turn to [40] and [53].

2.0.1 Topology of smooth 4-manifolds

Let. X be a closed smooth oriented 4—manifold. \X’e denote the same 4-manifold
with the opposite orientation by ——X . yet. sometimes use the notation X for standard
manifolds such as CW“).

The intersection form on X is the symmetric bilinear form

()X : ll;(X;Z)/Tor x II3(X:Z)/'l‘or —-> Z

V

defined by (0. I3) r——> oU/3[X]. It is unimodular on such X. and is diagonalizal)le over
the rationals. The rank of the maximal positive eigenspace of QX is denoted by b+
and that. of the negative eigenspace by b‘ . The sitjrimzturc of X is then runilerstood to
be the. signature of this nondegenerate form, namely (7(X) : N — I) . Finally X is

said to be of even type if every diagonalization of QA- has even diagonal entries. and

‘

odd type otherwise. It turns out that these algebraic topological invariants completely

classify the homeomorphism type of such 4-manifolds:

Theorem 2.0.1 (Freedman [31]; Donaldson [15]) The homeomorphism type of
a simply-connected closed smooth oriented 4-manz’fold X is captured by Qx , which

in turn is determined by the Euler characteristic e(X), the signature 0(X) and the

type.

On the other hand, there are infinitely many simply-connected 4-manifolds each
of which admits infinitely many distinct smooth structures (see for example [28]).
By contrast. however, there are no complete invariants to classify the diffeomorphism
types.

Herein the notation n'2X1#nX2 is used to express the connected sum of m copies
of X1 and 11 copies of X2. We say X is reducible if it can be written as a connected
sum X = X1#X2. where neither X, is a, luunotopy 4—sphere. (_)therwise. it is called
trrcducziblr-B. XX'e view the blow-up of X at a point I E X as the result topological
operation described by taking out a 4-ball around .17 and gluing in the connilen‘ient
of a. regular i'leighborhood of the exceptional sphere in W2, so to obtain X#W2.
Com-'ersely. if X contains an creept-zlonal sphere. i.e. a smoothly embedded sphere S
of self-intersectitm —1 . then a tubular neighborhoml of S can be replaced by a 4-ball
to obtain a new closed smooth oriented 4-1nanifold I" with X = Yahtfiz. The latter
operation is called blowing-douvn. A 4—manifold X is called minimal if it does not
contain any exceptiornil spheres. Irreducibility or minimality are not. aspects of the

underlying lmnmomorphism type of X. but of its smooth type.

2.0.2 Symplectic structures

A symplectic structure on a smooth 2n-dimensional oriented manifold X is a closed

2—form a: such that a)" > 0. The pair (X.w) is called a. sylnplcchc nunnfo/d. A

8

diffeomorphism o : X1 —> X; is called a symplectomorphism between (X 1,w1) and
(X2,w2) if <b*(w2) = wl. Two symplectic forms w and w’ on a fixed manifold X
are said to be deformation equivalent if there exists a family of symplectic forms an,
t E [0.1], on X, with wo =w and w, =w’.

The Euclidean space R2” with coordinates 1‘],y[,.. ..I,,.y,, admits a canonical
symplectic form mo = (1171 /\ dyl + + day, /\ dyn. Darboux’s theorem states that
every symplectic 2n-manifold (X .w) is locally symplectomorphic to (R2".wo) [53].
It follows that symplectic manifolds do not have local invariants.

Symplectic structures are closely related to complex structures on the tangent.
bundle. An almost comp/er structure on X is a. smooth. fiberwise linear map J :
TX —* TX covering 1'th such that J'2 = —idy~x. The pair (X,J) is called an
almost complex manifold. An almost complex structure J is said to be compatible
with u) if y(u.r) :2 w'(/I,l..,lll). u. c E TX. is a Riemannian metric on X. For any
Riemannian metric g on (XM') there exists a. compatible almost complex structure
given by ./(r) z: \/2[a| lg lu,‘(l’. -) for all non-zero c E TX. l\loremcr the space of
compatible almost complex structures for a fixml w' is contractible [53]. Therefore.
there is a unique class c,(X.w') associated to (X55) by taking the first chern class
of any compatible almost complex structure on the bundle TX. This class‘ minus
Poincare dual Ky- = —I’D(c1(.X'.w') is called the canonical class.

A complex structure on X” induces an almost-complex structure given by
J: = is. A symplectic manifold (Xur) together with a Riemannian metric g and a
compatible J :2 filed—lg“ 1.: is called [villi/er if ./ arises in this way from a complex
structure on X. Such a: is called a Ix'o'hlcr .str'mrfurc and y a Kali/('1' mctric. Often
times we simply say (X. to) is a Ix'rilzlcr manifold: where it is implicitly assumed that
there exist such compatible 9 and J on X.

Let us once again restrict our attention to closed smooth oriented 4-manifolds.

9

An orientable 4—manifold X admits an almost complext structures if and only if
there is a characteristic class h E H 2(X ) which satisfies 12.2 = 30(X) + 2e(X). More
interestingly, if X is complex, then it is Kahler if and only if b1(X ) is even. On the
other hand the existence of complex and symplectic structures on 4-manifolds are
much more involved and subtler problems to which algebraic topology can not fully
answer. Once again, these are aspects of the smooth structure on X . Also note that
no two of the families of closed 4-manifolds admitting complex, Kahler, symplectic
or almost complex structures coincide (cf. [40]).

There are three types of surfaces of great importance in symplectic 4-manifolds.
Let. (X to) be a symplectic 4-manifold, and let 2 be an embedded surface in X.
Then X is called a symplectic surface in X if (Ewlg) is a. syn‘iplectic manifold. On
the other extreme if col); E 0. then S is called Lagrangian.

Similarly let (X .1) be. an almost complex 4-manifold, and E be a possibly im-
iriersed surface. in X. Then X is called a J —holom017)hic curve or a pseudoholomor-
pliic curve in (X .1) provider] (2.111;) is almost complex. i.e. when .l is a. bundle
endomorpliism on T3 U TX .

If U.) is a. symplectic structure on X compatible with .I . then every pseudol'iolo-
morphic curve is symplectic. Conversely. for every symplectic submanifold of (X. to).
one. can choose a. compatible almost complex structure that makes it. pseudoholomor—
phic [53]. If E is an embedded surface in (.X.u.. J). the self-intersection. and genus

of Z: are related through the formulae:

—,\(2) : (2)2 + [\rx ~ $(t/2c atljimctio'n. equality),
when S is symplectic or pseudoht)lomorphic: and

-\(Z) = [W

when E is Lagrangian.

1t)

If X is the blow-up of a symplectic 4-manifold (Y,w) at a point y E Y, then
it can be equipped with a symplectic form w' which agrees with w away from the
exceptional sphere E in X = Y#W2. Furthermore, the total transform of any
symplectic surface containing y will also be symplectic in (X .w’). Conversely, if S is
an exceptional symplectic sphere in ( X , w’ ) , then it can be blown-down symplectically,
that is Y = X \ i\*"(S) U D4 = Y admits a symplectic form w which agrees with w’
on X \ N(S), where N(S) is a tubular neighborhood of S in X. Any symplectic
surface that intersects S in X will also descend to a symplectic surface in Y. ([53])

We finish with a classical theorem of Thurston which not only provides us with
a plethora of examples of closed symplectic 4-manifolds. but. also moth-"ates several

other results proved and / or used in this work:

Theorem 2.0.2 (Thurston [80]) Let f : X —> B be. an. F-bumlte where the fiber
F is a closed Riemann surface and the. base B is a compact Riemann surface. If F
is nonzero in [12(X; IR). then. X can be equipped with. a. symplectic form to such that

all fibers are symplcctic.

2.0.3 Lefschetz fibrations

A smooth Lefschetz film-alum. on an oriented ~l-manifold X. possibly with boundary.
is a smooth map f: X ——> X. where E is a compact oriented sm'face. such that f
is a. submersion everywhere but. at finitely many points (7 2: {1)1.. . . .1),,} contained
in the. interior of X. and conforming to local models: (i) f(:1. .23) = 21 around each
regular point. and (ii) f(.:1, :3) = :13; around each Lefschetz critical point p, E (l;
both given by orientatitm preserving charts on X and E. The preimage of a regular
value is a. Riemann surface F, called the 'I'cgula'r fiber. whereas and the singular
fibers containing the Lefschetz critical points locally have the model of a complex

nodal singularity around those. points. In a handlebody of X. these singularities are

11

obtained by attaching 2-handles to regular fibers with framing —1 with respect to
the framing induced by the fiber. The. attaching circles of these 2-handles are called
vanishing cycles.

A Lefschetz pencil is a map f : X \ {b}, . . . .b,,.,} —+ S2, such that around any
base point b.- it has a local model f (21. 22) = 21/32 , preserving the orientations, and
that f is a Lefschetz fibration elsewhere. By convention. B 2 {b1, . . . ,bm} is always
non-empty and called the base locus, and C = {p1, . . . .pn} is called the critical
locus. There is an obvious link between these two definitions. In a Lefschetz pencil,
the closures of the fibers of the map f cut the 4-manifold X into a family of closed
surfaces all passing through the b. —-—locally like complex lines through a point in
(C2. Blowing up all the points in the bsae locus, the map f extends to the entire
manifold and we obtain a Lefschetz fibration f : {X ——> SQ. with each exceptional
sphere appearing as a. section.

If F is a regular fiber of a Lefschetz fibration f : X —-> S, then F c——> X —f+ Z
induce an exact sequence 7r,(F) ——> 7n(X) ——> 7r](Z) ——> rad/7) —+ 0. It follows that.
if the base space is simply connected. then each fiber of f is connected and carries
771(X). If a. fiber is not t__ro1mected. then 7r1(X) maps to a finite—index subgroup
of 7n($), and passing to the corresponding finite cover i of Z. we obtain a new
Lefschetz fibration f : X —+ E with connected fibers. Thus without loss of generality
we can assume that the fibers are connected. and in this case the genus of a generic
fiber will be called the genus of the Lefschetz pencil or fibration.

Given a. compact oriented genus g surface F’ with in boundary components and
r marked points on it. the mapping class group of F is defined as the group of orien-
tation preserving self-diffeonmrphisms of F fixing marked points and t)!“ pointwise.
modulo isotopies of F fixing marked points and (7F pointwise. It can be shown that

this group is gt—‘nerated by positive (right handed) and negative (left handed) Delm

twists. Importantly, isotopy type of a surface bundle over S1 with fiber closed ori-
ented surface F is determined by the return map of a flow transverse to the fibers,
which can be identified with an element of a mapping class group F, called the
monodmmy of this fibration.

Let f : X ——> D2 be a Lefschetz fibration, where the regular fiber F is an oriented
genus 9 surface with m boundary components, and suppose all critical points of the
fibration lie on various fibers. Select a regular value 0 in the interior of D2, an
identification of f‘1(0) E F , and a collection of arcs a1, - -— ,ak in the interior of
D2 with each a,- connecting O to a distinct critical" value, and all disjoint except at
O. we index the critical values as well, so that each are a, is connected to a critical
value y, and that they appear in a counterclockwise order around the point 0. Now
if we take a regular neighborhood of each arc away from remaining critical points and
consider the union of these, we obtain a disk V and an F -bundle over 0V 2 S1 . The
monodromy of this fibration is an element of the mapping class group of F, which is
called the global monodromy of the Lefschetz fibration f. We call the ordered set of
arcs {£11. - - - .ak} a. it"prcsenfaf-ion of the Lefschetz fibration f. It is well—known that
this data gives a handlebody description of X . and vice versa.

The next two tlworems establish a beautiful connection between the main concepts

of the last. two sections:

Theorem 2.0.3 (Donaldson [17]) If (X, a.) is a closed symplectic 4 -man1'.fol(l with

u) integral. then it admits a Lefschetz pencil urllh symplectic ‘I‘egular fibers.

Theorem 2.0.4 (Gompf; see [40]) If f : X —+ Z is a. Lefschetz filtration such. that
the hmnologg class of the regular fiber F is nonzero in. Hg(X: R) then X (nlm-zfits a.
deformation. class of symplectic structures with. respect to which the fibers are sym-
plectic. illo'reoucr. such a. symplectic form can. be chosen so that any prescribed finite

set of sections are also symplectic.

113

The proof of the latter theorem generalizes Thurston’s construction to Lefchetz fi-
brations, using the local models around singular points (see Proposition 5.2.2 for

complete details).

2.0.4 Seiberg-Witten invariants

We now review the basics of Seiberg-Witten invariant (cf. [85]). The Seiberg- W itten
invariant of a smooth closed oriented 4-manifold X is an integer valued function
which is defined on the set of SpinC structures on X . If we assume that H1(X; Z)
has no 2-torsion, then there is a one-to—one correspondence between the set of Spin"
structures on X and the set of characteristic elements of H 2(X ;Z) as follows: To
each SpinC structure 5 on X corresponds a. bundle of positive spinors ill/5+ over
'X. Let. ((5) = c101?) E [12(X;Z). Then each ((5) is a characteristic element of
llZ(X';Z); i.e. (71m?) reduces mod 2 to 11:2(X).

' In this setup we can view the Selberg- Wittcn invariant as an integer valued function
8“} : {Ir 6 llg(X:Z) | PDUV) E 11'2(X) (mod 2)} —> Z,

where PD(A‘) denotes the Poincare dual of k. The. Sciberg-W’itten inwrriant SW’X
is a. ("liffeomorphism invariant. when b; (X) > i or when b; (X) = 1 and b; (X) S 9
(see Remark 2.0.5 for the b4r = 1 case). llts overall sign ('lepcnds on our choice of an
orientation of H”(X: 1R.) ti< (let HflX: IR) Xi (let [11(X: R).

If SXX'XW) 74 0. then we call U (and its Pcuncare dual PDM) E l/2(X:Z)) a.
basic class of X. lt was shown in [74] that the canonical class Ky = —c1(X.a‘) of a.

symplectic 4-n'1anifold (X,w) is a. basic class when b+(X) > 1 with SX’X’XUx'X) = 1.

It can be shown that, if d is a basic. class, then so is —z_3 with
8va (_ a) = (—1)<“<-">+”<“’>’I”*" 3va (a),

where e(X) is the Euler characteristic and 0(X) is the signature of X. \X'e say that

14

X is of simple type if every basic class fl of X satisfies
[32 = 2e(X) + 30(X).

It was shown in [75] that symplectic 4—rnanifolds with b; > 1 are of simple type. Let
E C X be an embedded surface of genus 9(2) > 0. If X is of simple type and [3' is
a basic class of X, we have the following (generalized) adjunction inequality (cf. text

[581):
fies) 211(2) — 2 2 {212 + IH- [EH- (2-1)

Remark 2.0.5 When b+( X) = l. the (small-perturbation) Scibcrq-Witten invariant
SVVx,H(K) E Z is defined for every positively oriented element H E Hi(X;R) and
every etc-intent ,1 E C (X) such that /l - ll gé 0. We say that 1] determines a chamber.
It is knoum that if SXX’XHO # 0 for some H E 11;“:(X; R). then (l(.4) 2 0. The wall-
crossing formula prescribes the dependence of SW” x1101) on the ('rhoice of the chamber
(that of H): if H. H’ E Ili(X;R) and A 6 C(X) satisfy H - H’ > 0 and d(A) Z 0,

th en.

SXX's-ntrl) = Squt-i)
r

0 if A - H and. A ~ [1' have the same sign.

+< arse-v zifA . n > o and A - H’ < 0,

 

poet’s” zifxf - I! <0 and ,1 - 11' > 0.

\
These facts imply that SXX'x-Ht'l) is indepmulent of II in the case h“(X) S 9 [7:1].

‘5'” ”"7 simply ((11le about the Sctberg-ll‘lttcn invariant of X in this case.
The Seiln—arg-XXHtten invariant of X with b+(X) 2 1 can I”. fm'mulated as a map
“in Spin‘t-X") —+ MA) = ZIU] s; .\*1—1‘(.\'; 2).

15

where A(X) is the graded abelian group with deg(U) = 2, and SW x (5) is homo—
geneous of degree (l(5). In this formulation, SW x(5) is the fundamental homology
class of the Seiberg-W’itten moduli space in the ambient configuration space 3L5,

under isomorphisms
H.( R-yszZ) = II.(BSX:Z) 2’ A(X).

where 9x = Map(X, .8") is the gauge group. Evaluating SW x on monomials U“ '2)
ll /\ - - - /\ 1;, of degree (1(5), we obtain a map to Z as above.

We finish with some important results regarding the SW invariants. According to
Taubes [75], the SW invariant of a symplectic 4-manifold (X ,w) can be computed
as a Gromov invariant (Gr) emunerating embedded pseudo-holomorphic curves and
their unramitied coverings with respect to a generic J compatible with w. When
we have a symplectic broken Lefschetz pencil (XI) of high enough degree. there.
is another invariant called the Dcmaldson-Smith invariant (DS) associated it. which
counts nicely embedded pseut‘loln)lomoprhic mutilisections within a chosen homology
class [19. (59]. In [82]. Usher proves that D8 and Cr counts agree when the degree of
the pencil is high enough. Hence. under mild assumptions. Cr and DS invariants are
seen to be independent. of the symplectic structure or the Lefschetz pencil that. are
chosen. yielding equivalent smooth invariants.

Last two results to add are as follows:

If X = X1#X2 with h+(X,) > t). i = 1.2. then SXX'X E 0. (SW i-‘anishcing theorem
for connectul sums.)
If X = X#‘C—llw. then every basic class 3 of X is of the. form ,3 = ‘3 i E. where

[12(X: Z) is identified with [12(X1Z) '4? I12(@2:Z). 1.3 is a basic class of X and E

 

6‘7

. v - - ‘ ‘) 7-" r V
1s the. class of the exceptional sphere generating H"( IP’“: 2:). (lhe blow—up formula

for SW invariants.)

10'

CHAPTER 3

New symplectic 4-manifolds

3. 1 Background

3.1.1 Generalized fiber sum

Assume that two 4—manifolds X1 and X2 each contain a closed embedded genus g
surface E: C X; such that the normal bundles VF, have. opposite Euler numbers.
i.e. [RV = —[F2]2. Then there exists a fiber-orientation reversing isomorphism
between the two normal bundles. If we canonically identify each 11F} with a. tubular
neighborhood N, of F,. then there exits an orientation rcwersing dieromorphism
(.9 : X1 \ F] —+ X3 \ [’2 which turns each punctured normal disk inside out. Then we
can define the generalist—1dfiber sum of (X1. F1) and (X3. 172) as X1 \ NI Um X2 \ .‘\72
by identifying 0N, via o. “e denote this operation by X = X1#@X2.

Note that the differmiorphism type of X is determined by the embeddings of F,
together with the choice of 0 up to ‘fiber preserving isotopy" (of the corresponding
fiber bundle isomtn'phisms between VB). \Vhen [RV = [[72]2 = 0. the map a
can be taken as an orientation preserving sell-tlillcomorphism of F times a complex
conjugation on the punctured unit disk D2 \ {H}. In this simpler case. the operation

is called fihcr sum. Finally note that generalized fiber sum operation can be. defined

17

in higher dimensions as well (see for example [55] or [38]), but here we are solely
interested in the 4-dimensional case.

The characteristic classes of X can easily be expressed in terms of those of Xi.
We have e(X) = e(X1)+ e(Xg) + e(UNl) — e(Nl) — e(Ng), where (7N1 = —(’)N2 is an
oriented 3-manifold and each N,- deformation retracts to Pi = 29. So e(aNl) = 0
and e(zV’ 1) = e(Ng) = e(Eg). On the other hand. the signature of X can be computed

by using Novikov’s additivity. So we have:

e(X) = e(X1)+ e(Xz) + 4g — 4 . 0(X) = 0(X1)+ 0(X2). (3.1)

In addition the type is odd unless each [4} is characteristic in Xi, i = 1. 2.
Importantly. this (_)1.)era.tion can be performed symplectieally in the following set-

ting:

Theorem 1 (McCarthy and Wolfson [55], Gompf [38], also see Gromov [42] )
Let (X,,.u,) be symplectic 4-nmmfolds and I7, L—> X, be symplectically embedded
genus g > 0 smfuccs. for i = 1.2. If [1",]2 : -—[Fg]2. then X = X1#¢,.\'2 can
be equipped with. a symplectic form u}. il/oreoeer. given. (I.I'f)ff7'(L7"'l:ly small collar
ncighher/mods X, of ()(g\',) m Xi. we can, choose u.) so that ”[A'AS'I = W'1|.\'1\.'§-'1 and

<.t.'[4\-2\‘\~-2 = ('w'2[’\’2‘\‘\72. ‘u'hcrri (t zs some constant.

The last part of the theorem is immediate if we construct the symplectic fiber sum

following Etnyies syml’)lectic cut-and-paste technique [22-1].

3. 1.2 Minimality

Recall that a 4-manifold X is called minimal if it does not contain an embedded

P
I

2-sphere with self-intersection —1. Similarly a symplectic ~i-manifold (X . ax) is called

symplectieally minimal if it does not contain such a symplectic sphere. In both smooth
and symplectic categories we aim to construct minimal 4-manifolds.

A family of minimal 4-manifolds is the products of non-rational Riemann surfaces.
Let 29 denote a closed Riemann surface of genus g > 0. Since the universal cover of
29 is contractible, 29 is acyclic. It follows from the long exact homotopy sequence
of a fibration that any 29 bundle over Eh, with g. h > 0 is acyclic. In particular,
n2(Zg x 2h) 2 0 and hence 29 X 2,, is minimal. So equipped with any symplectic
form, 29 x 23;, is symplectically minimal.

One new ingredient in our constructions that will follow is the following theorem

of Michael Usher:

w Y’ be the symplectic sum. where, the

_._4

Theorem 2 (Usher [83]) Let X = )"#g

genus g of E and S' is strictly positive.

(i) If either l' \ S or i" \ 3' contains an emlu‘dded symplectic sphere of square

—1. then. X is not. minimal.

(ii) If one of the sumnumds. say Y for drfinitcneh‘h‘. admits the structure of an. S2-
bundle over a, surface of genus g such that. E is a section of this .S'Q—bundle,

then. X is minimal if and only if i" is minimal.
(iii) In all other cases, X is minimal.

()ne final comment. is on the close relationship between minimality and irreducibil-

ity when dealing with symplectic 4-manifolds:

Theorem 3 (Hanlilton and Kotschick [43]) .llinimal symplectic I-Inarnifolds

with residual/y finite fundamental groups are irreducible.

Thus silnply-connected minimal symplectic 4-111anifolds are always irreducible: a,

fact that we will use repeatedly in this chapter.

19

3.1.3 Surgery along Lagrangian tori

Let A be a torus of self-intersetion zero inside a 4—manifold X. Choose a framing
of the tubular neighborhood VA of A in X, i.e. a diffeomorphism VA 2 T2 x D2.
Given a simple loop A on A, let S] be a loop on the boundary 8(1/A) g T3 that
is parallel to /\ under the chosen framing. Let HA denote a meridian circle to A in
8(z/A). By the p/q surgery on A with respect to A, or more simply by a (A, A,p/q)

surgery. we mean the closed 4-manifold
XA,A(p/q) = (X \ VA) U; (T2 X D2),

where the gluing difl'eomorphisin 1,9 : T2 X ('31)2 ——> 0(X \ VA) satisifies
ago/12]) = ppm] + (,[si] e 111(U(X \ l/1\);Z).

By Seifert—Van Kampen theorem. one easily concludes that

mtX.\..\(p/q)) = mtX \ 11A) /< [H.xlpl-S'll” = 1)-

In the symplectic case. we. will be. surgering Lagrangian tori. Luttinger surgery
is a. special case of p/(] surgery on a self-intersection zero torus A described in the
preyious subsection. It was first studied in [31] and then in [8] in a more general
setting. Assume that (Xaa) is a symplectic 4—manifold. and that the torus A is a
Lagrangian submanifold of X. From the “’einstein tubular neighborhood theorem.
there is a canonical framing of VS 2' T2 x D2, called the. Lagrangian framing. such
that 7‘2 x {.17} corresponds to a Lagrangian submanifold of X for every .1? E [)2.
Given a simple loop /\ on A. let S] be. a simple loop on (9(1/A) that is parallel
to A under the Lagrangian framing. For any integer m. the (A,/\.1/‘m) Lutt-inge-r
surgery on X will be X,\.A(l/7n). the l/m surgery on A with respect. to /\ and the
Lagrangian framing. Note that our notation is (‘liflerent from the one in [8] wherein

XA‘ ,\(1/ m) is denoted by X(.-’\. A. m).

‘20

Theorem 4 (Auroux, Donaldson and Katzarkov [8]) XA,,\(1/m) possesses a

symplectic form that restricts to the original symplectic form a; on X \ VA.

In this thesis, we will only deal with Luttinger surgeries where m = i1 = l/m,

so there should be no confusion in notation.

Remark 3.1.1 In Section 3.2.2 and Section 3.5, we will also be looking at non-
Luttinger (A, A, —n) surgeries X A, ,\(-—n) for a Lagrangian torus A equipped with the

Lagrangian framing and a positive integer n 2 2.

3.1.4 Surgeries and Seiberg-Witten invariants

In what follows. we will be frequently using the following theorem:

Theorem 5 (Fintushel, Park, Stern [25]) Let X be. a closed oriented smooth 4-
manifold which. contains a n allhmnologous torus A with A a simple loop on A such.
that S] is nullhomologous in X \ VA. If ;X'A,,\(0) has nontrivial Seiberg-Witten

invariant. then the. set
{XAAU/n) [n=1.'2,3,...} (3.2)

contains infinitely many pairwise n.ondiffeorn.owphic 4-manifolds. Furthermore, if
XAAtU) has just one S(:7ll)(-‘7'.(j-l’ltllf’n basic class up to sign. then every pair of 4—

manifolds in. (3.2) are non(lifi‘co-Inmphic.

Here the Seiberg-XVitten invariant, is the small perturbation invariant. whenever the

4—1nanifold has b+ = 1.

Remark 3.1.2 Note. that X : X\)‘(1/()). Let T be the core torus of the (I surgery

.X',\_,\(0). If k0 is a. characteristie element of Hg(.X',\,,\(()):Z) satisfying A70 - [T] =

21

0, then k0 gives rise to unique characteristic elements I: E H2(X;Z) and kn E
H2(XA_,\(1/n);Z). The product formula in [52/ then gives

swxuwngku) = swxa) +71: SVVXA_A(0)(/m + 22m). (3.3)

262

Let us now assume that X A, ,\(0) has only one basic class up to sign and this basic
class is not a multiple of [T]. Under these assumptions. the infinite sum in (3.3) only
contains at most one nonzero summand. If we further assume that X and X‘.\.A(0)
are both symplectic. then the ady‘zinction inequality implies that the only basic class
of X and X‘\.,\(()) is the canonical class up to sign. Under all these assumptions. it

follows that XA‘A(1/n) also has only one basic class up to sign for every n 2 1.

3.2 Constructing small exotic symplectic 4-

manifolds

3.2.1 The construction scheme for odd blow-ups of (CHD2 and
30??

Here we outline a general construction scheme to construct. sin1ply—connected min-
imal symplectic 4—manifolds with small Euler characteristics. This is an incidence
of the "reverse engineering" ([70. 25]) idea applied to certain symplectic manifolds.
Any example using this scheme and l1()lll(.‘(.)lll(,)l‘])lll(' to (Cllwaltnfill—Dz (for n > 0) and
nNCIlefi‘nfll$2 (for In > U) can he distinguished from the latter standard manifolds

by conmaring their symplectic structures or their Seilierg-XVitteu invariants. respec-

 

tively. Recall that. Cllm#ll(CllD2 (for n > 0) are nomniniinal. and inClll‘BtfinCllfi (for
II) > 0) all have vanishing Seilierg-XVitten invariants. unlike the minimal syn‘iplectic

4-Inanifolds that we produce. Our approach will allow us to argue easily how all

‘22

4-manifolds obtained earlier in [6, 10] arise from this construction scheme, and in
particular we show how seemingly different examples rely on the very same idea.

The only building blocks we need are the products of two Riemann surfaces. In
fact... it suffices to consider multiple copies of 5'2 x T2 and T2 X T2, since all the other
product manifolds except for 82 X S2 (which we will not use here) can be obtained by
fiber summing copies of these manifolds appropriately. Note that any such manifold
is a minimal symplectic manifold. Both 82 X T2 and T*1 = T2 X T2 can be equipped
with product symplectic forms where each factor is a symplectic submanifold with
self-intersection zero. Denote the standard generators of 7rI(T") by a, b, c and (1,
so that H2(T4; Z) 2 Z” is generated by the homology classes of two symplectic tori
a X b and c X (I. and four Lagrangian tori a X c, a X (1. b X c and b X (1 with respect
to the product syimiilectic form on T" that we have chosen. The intersection form
splits into three hyperbolic pairs: (1 X b and c X (I. a X c and b X (l, a X (1 and b X c.
Finally. note that all four Lagrangian tori can be pushed off to nearby Lagrangians
in their \V’einstein neighborht)ods so that they lie in the conmlement of small tubular
neighborhoods of the. two chosen symplectic tori T2 X {pi} and {pt} X T2. With a little
abuse of notation (which will be remembered in our later calculations of fundamental
groups). we will still denote these parallel Lagrangian tori with the same letters.

In order to produce an exotic copy of a. target manifold Z , we first perform blow-
ups and symplectic fiber sums to obtain an interinediate manifold X’ . \X’henever a
piece is blown—up. we make sure to fiber sum that piece along a symplectic surface that
intersects each exceptional sphere positively at one point. This allows us to employ
Theorem '2 to conchule that X’ is minimal. \Ve want this intermediate manifold to

satisfy the following two properties:
(I) X’ should have the same signature and Euler characteristic as Z.

(11) If 7' is the rank of the maxinml subspace of [13(X':Z) generated by homo-

‘23

logically essential Lagrangian tori, then we should have T _>_ s = 2b1(X’) =

b2(X’) — 52(2).

l\x‘loreover, we generally desire to have 7r1(X’) = H1(X’; Z) for the reasons that will
become apparent below. However, Surprisingly one can also handle some examples
where 7r1(X’) is not abelian. (See for example the construction of a minimal sym-
plectic 3CP2#5@2 in [6].)

Finally, we carefully perform 5 Luttinger surgeries to kill 7r1(X’) and obtain a
simply-connected syn‘iplectic 4-manifold X. Since signatures of simply-connected
spin 4-manifolds are always divisible by 16, all of our target manifolds among
(Clll’2#nCTP2 (for n. > 0) and mClP’Q#/z@2 (for m > 0) are of odd type. Observ-
ing that Luttinger surgeries do not change neither the Euler characteristics nor the
signature, one concludes that X is homeomorphic to the target manifold.

Note that these surgeries can easily be chosen to obtain a manifold with bl = 0.
However, determining the correct choice of Luttinger surgeries in this last step to kill
the fundamental group completely is a much more subtle problem. This last part. is
certainly the hardest part. of our approach, at. least for the ‘smaller’ constructions.
The reader might want to compare below the complexity of our fimdamental group
calculations for (ClP’2#(2A‘ + I)??? for It 2 1,. . . ,4 as k. gets smaller.

In order to compute and effectively kill the fundamental group of the resulting
manifold X. we will do the Luttinger surgeries in our building blocks as opposed
to doing them in X’. This is doable, since the. Lagrangian tori along which we
perform Luttinger surgeries lie away from the symplectic surfaces that are used in
any symplectic sum constructions. as well as the blow-up regions. In other words,
one can change the order of these. operations while paying extra attention to the m
identifications. Having the an calculations of the pieces in hand. we can use Seifert-

Van Kampen theorem repeatedly to calculate the fundamental group of our exotic

candidate X .

Below, we will work out some concrete examples, where we construct minimal
symplectic 4-manifolds homeomorphic to (CIP’2#(2k + Dwz, for k = 1.. . . ,4, and
3C1P’2#(2l + 3)@2, for l = 2,. . . ,6 (See [5] for l = 1 case). We hope that the reader
will have a better understanding of the recipe we have given here by looking at these
examples. Another essential observation that is repeatedly used in our arguments
below is the interpretation of some manifold pieces used in [3, 6] as coming from
Luttinger surgeries on T“, together with the description of their fundamental groups.
This is proved in the Section 3.2.2. A concise history of earlier constructions will be

given at the beginning of each subsection.

Remark 3.2.1 The building blocks the used in the construction. scheme described here
. 0 , .

do not suffice to get modils for e’ccn. munbcr of blow-ups of CH)“ 07" JCIP’Z. By the

time of u'l‘iiing. finding appropriate models for these manifolds has not been completely

accomplished.

3.2.2 Twist knots and Luttinger surgeries

Let '1"I 2: a X l) X c X (I T—S (c X (l) X (a X h), where we have switched the order of the
symplectic 7'2 components a X b and c X d just to have a comparable notation with
earlier 7r, calculations (say in [6]). Let K" be an n-twist. knot (cf. Figure 3.3). Let
MK" denote the result of performing 0 Dehn surgery on 53 along It". Our goal here
is to show that the -l-manifold S" X MM is obtained from '1‘1 = (c X (I) X (a X b) =
c X ((1 X (1 X b) = S1 X T3 by first performing a Luttinger surgery (c X (i, (7,. — 1) followed
by a surgery (c X b. b. —n). Here. the tori c X (.1 and c X b are Lagrangian and the
second tilde circle factors in T3 are as pictured in Figure 3.2. \\'e use the Lagrangian
framing to trivialize their tubular neighl)orhoods, so when n = 1 the second surgery

is also a Luttinger surgery.

I‘v
CJI

 

 

Figure 3.1: The 4-torus c X (d X a X b). The neighborhood of a fiber chosen in the 3-torus
d x a X b at c = 0.5 is given by fat slices parallel to a X b face, which get thinner while
0 gets closer to 0.5 :t e and disappear when 0 < c < 0.5 — 26 or 0.5 + 26 < c < 1. The
neighborhood of the torus section is given by a. cylindrical neighborhood in the direction
of d lying in the 3-torus times c. Neighborhoods of the tori c X oi and c X b are drawn
similarly.

 

Figure 3.2: The 3—torus d X a x b at c = 0.5

26

The figures should be self-explanatory. We view the 4-torus as the product c X ((1 X
a X b), and excise the tubular neighborhoods of the tori c X (i , c X b and c X (I as shown
in the Figure 3.1. The tubular neighborhood of the torus a. X b appears as a slice in
the 3—torus d X a. X b while we get closer to c = 0.5, and we have the thickest slice
precisely when 0 = 0.5. Note that the normal disks of each Lagrangian tori in their
VVeinstein neighborhoods he completely in T3 and are disjoint. Thus topologically,
the result of these surgeries can be seen as the product of the first .5'1 factor with
the result of Dehn surgeries along (7. and b in T3. Therefore we can restrict our
attention to the effect of these Dehn surgeries in T3 since the diffeomorphisms of the
3-manifolds induce diffeomorphisms between the product 4-manifolds.

The. Kirby calculus die-igran’is in Figure 3.3 show that the result. of these Dehn
surgeries is the manifold MK". where If" is (the mirror of) the n—twist knot. In

particular, note that. for n = 1 we get the trefoil knot K. Thus the. effect of (c X

A...

\

bl). —n) surgery with n ,> 1 as opposed to the Luttinger surgery (c X b, b. —1) is
equivalent to using the non—synmlectic 4-manifold .8" X MK" instead of syniliilectic
81 X MK in our symplectic sum constructions.

Next we describe the effect of these surgeries on 71]. First. it is useful to View
T3 = d X (a X b) as a T2 bundle over S] with fibers given by {pt} X ((1 X b) and
sections given by d X {pt}. The complement of a. fiber union a. section in T3 is the
complement of 3-dimensitma] shaded regions in Figure 3.1.

It. is not too hard to see that the Lagrangian framings. give the following product

decompositions of two boundary 3—tori (compare with [10. 25]):
0(1/(0 X (1)) E“ c X (dud—1) X [(1. (fl), (3.4)
(9(1/(C‘X b)) ”if c X b X [(1'14’1']. (3.5)

The Lagrangian pusholl of l) is represented by b. as a homotopy to b is given by

the “diagonal" path (dotted lines emanating from the lmrizontal boundary cylinder

27

 

 

 

Figure 3.3: The first diagram depicts the three loops a, b, d that generate the 1r1(T3). The
curves it = dad‘1 and b are freely homotopic to the two extra curves given in the second
diagram. The third diagram is obtained from the second via two slam-dunk operations;
wheras the last diagram is obtained after Rolfsen twists.

6(Vb) in Figure 3.2). For decomposition (3.5), it is helpful to View the base point
as the front lower right corner of the cube represented by a dot in Figure 3.2. It is
comparatively more difficult to see that the Lagrangian pushoff of 6 is represented by
dad-1. The Lagrangian pushoff of d is represented by the dotted circle in Figure 3.4
and is seen to be homotopic to the composition a[a‘1,d] = a(a“1dad'1) = dad‘l. For
decomposition (3.4), it is helpful to view the base point as the front upper left corner

of the cube represented by a dot in Figure 3.2. The new relations in 7r1 introduced

28

 

A

a

_.________‘..________.
Q!
<

 

 

 

 

Figure 3.4: The face of cube where we can see the Lagrangian pushoff of d

by the two surgeries are
dad‘l = [db—1]: db‘lcflb, b = [a_1,d]" = (a‘ldad_1)". (3.6)
From now on. let us assume that n = 1. Then the second relation in (3.6) gives
ab : dad—1. (3.7)

Combining (3.7) with the first relation in (3.6) gives ab = dad‘l = db‘ld‘lb, which

can be simplified to a = (lb‘lrfl . Thus we have
a‘1 = dbd‘l. (3.8)

Hence we see that (3.7) and (3.8) give the standard representation of the monodromy
of the T2 = a X b bundle over S" = d that is the O—surgery on 5'3 along the trefoil

KZKl.

3.3 Minimal symplectic 4-manifolds with b+ = 1

3.3.1 A new description of a minimal symplectic E ( 1)

The first example of an exotic smooth structure on the elliptic surface E(1) =
CIP’2#9Ell—D2, and in fact. the first exotic smooth structure on any closed topologi-
cal 4—n'1anifold, was constructed by Donaldson in [16]. Donaldson’s example was the
Dolgachev surface E (1)2‘3. Later on. Friedman showed that {E (1),)“, | gcd(p, q) = 1}
contains infinitely many nondiffeomorphic 4-1nanifolds (cf. [32]). In [28] Fintushel and
Stern have shown that knot surgered manifolds E( 1) K give infinitely many irreducible
smooth structures on E(1) = CP2#9EF2.

Consider 5‘2 X T2 = 82 X (.S" X S") e(piipped with its product symplectic form.
and denote the last two circle factors by I and y. ()ne can take the. union of three
symplectic surfaces ({51} X T2) U (5'2 X {1})U ({sz} X T2) in 82 X T2. and resolve
the two double points symplectically. This yields a genus two syli'iplt.‘('ti(_' surface in
5'2 X T2 with self-intm‘section four. Synwlectically blowing up .92 X T2 along these four
intersection points and taking the proper transform. we obtain a symplectic genus two
surface S in Y = (.92 X T2)#4_C_1F2. Note that the inclusion imlnced hrnnomorphism
from 7r.(E) :: ((1.1).(:. (1 | [(l.’)][('.(/] = 1) into 7r](l') = (17.3] I [Jug] = 1) maps the

generators as follows:

(I H .17. b +—> y. (' +—> .171. die» ffl.

Let us run the same steps in a second copy of 52 X T2 and label every object with
a prime symbol at the. end. That is. l" = (5'2 X T2)#lElF—2. E’ is the same symplectic
germs two surface described above with 771 generators u'. b'. c’, (1’. and finally let .1". y’
denote the generators of the THO"). Let X be the symplectic fiber sum of Y and

Y' along 3 and 2' via a differ)nnn‘phism that extends the orientation-preserving

30

diffeomorphism qb : Z -—> 2’ , described by:

—l

a Ha'b', b+——> (a') .CHC’. dr—wl'.

The Euler characteristic of X can be computed as e(X) 2 4+4 — 2(2 — 2 - 2) = 12,
and the Novikov additivity gives the signature 0(X) = —4 + (—4) = —8, which are
exactly the Euler characteristic and the signature of Z = CP2#9@2. We claim that
X is already simply-connected and thus no Luttinger surgery is needed. Note that
7r1(l’ \ V2) 2 7r1(Y) since a meridian circle of Z bounds a punctured exceptional

sphere from one of the four blowups. Using Seifert—Van Kampen theorem, we see

that
rut") = (It-1511'! [M] = l-r'w'l = 1.-
.r = .r'y'. y = (1’)“. :1,"1 = (1’)“, y“1 = (}/)_1).
We conclude that, .r = .1", y = g’. y = .1"". Thus 1' = .r’g’ implies y = 1, and in

turn .1“ = 1. So 7r1(X) = 1. Hence by Theorem 2.0.1. X is homeomorphic to E(1).
However. X is irreducible by Theorem 3. and therefore X is not diffeomorphic to
E (1). The 4-manifold X we. obtained here. can be shown to be the knot surgered
manifold E(1)K. where the knot K is the trefoil (cf. [29]).

Alternatively we could construct the above manifold in the foll(,)wing way. First
we sym])lectically sum two copies of (5‘2 X T2);£#.l(C—IP2 along 2 and 23’ via a map
that directly identifies the generators a. b. c, (I with (1'. b’. c’. (1’ in that order. Call this
symplectic --l—manifold X ' and observe that while the clniracteristic mnnbers e and
0' are the same as above. this manifold has m(X') = 111(X’: .71.) E” Z2 and 112(X';Z)
has four additional classes that do not occur in X. These classes are as follows.

Inside ((82 X T2)#4EF3) \ NZ, there are cylinders (.‘a and ('b with

UCO : (1 U C. (70;, = I) U (I.

31

Similiarly we obtain cylinders C’ and C], in the second copy of ((S2 X T2)#4C—IP’2) \

(1

V2’ . Thus we can form the following internal sums in X’:
Ea = COUC;, Eb = CbUCg.

These are all tori of self-intersection zero. Let n denote a meridian of Z, and let.
Ra = (7. X p. and Rb : b X [1 be the ‘rim tori’, where F1. and b are suitable parallel
copies of the generators at and 1). Note that [Rn]2 2 [1f,,]2 2 [2”]2 2 SJ) 2 0, and
[Hal ' lzbl = 1 = l/{bl ‘ [Eal'

Observe that these rim tori are in fact Lagrangian. One can show that. the effect
of two Luttinger surgeries (HQ, (7. —1) and (It’b.b. —1) is the same as changing the
gluing map that. we have used in the symplectic sum to the gluing map d) in the first
construction. This second viewpoint is the one that. will fit. in with our construction

of an infinite family of pairwise nondiffeomorphic smooth structures in Section 3.5.

3.3.2 A new construction of a minimal symplectic CP2#7@2

The first example of an exotic CF2#TWQ was constructed by J. Park in [60] by
using rational blowdown (cf. [27]). and the Seiberg-“itten invariant calculation in [59]
shows that. it. is irreducible. Infinitely many exotic cxanmles were. later constructed
by Fintushel and Stern in [30]. All of their constructions use the rational blowdown
technique. Here we construct another irredm-ible symplectic 4—manifold homemorphic
but. not, diffeomorphic to CP3#T@2 using our scheme. and thus without using any
rational blowdown.

\Ve equip T" = '1‘) X '1'2 and S") X T") with their product symplectic forms. The two
orthogonal symplectic tori in T' can be used to obtain a symplectic surface of genus
two with self-intersectitin two. Symplectically blowing-up at these self—intersection

points we obtain a new symplectic surface S of genus two with trivial normal bundle

in Y = T“#2W2. The generators of w1(T4#2@2) are the circles 11, b, c, d, and the

inclusion induced homomorphism from 713(2) to
7r1(Y) = ((1.1). (.211 I [(1,1)] = [(1,(:] = [(1, (1] = [1), (.z] = [1). (1] = [(r, (1] = 1)

is surjective. Indeed the four generators of 7r1(2) are mapped onto (1,1), (.t, (l in 7r1(Y),
respectively.

On the other hand, as in Subsection 3.3.1, we can start. with 5'2 X T2 and get a.
symplectic genus two surface 2’ in Y’ = (S2XT2)#4Ell—i2. Once again 7r1( Y’) = (If. y I
[.1111] = 1) and the generators (1’,b’.(.",d’ of 7r1(£’) are identified with Lyn—1.314.
res1_)ectivel_v.

We take the symplectic sum of Y and Y’ along 2 and 2’ given by a diffeomor-
phism that extends the identity map sending a +—+ (1’. b e» b’. c +—> c’. (l +—> d’ to obtain
an intermediate 4—1'nanifold X’. The Euler characteristic can be computed as (( X ’) =
‘2 + 4 + 4 = 10. and the Novikov additivity gives 0(X’) = -—2 + (—4) = -6. which
are the characteristic numbers of CP2#T@2. Since exceptional spheres intersect E
and 2’ transversally once. we have 7r](Y \ I/Z) % 7r1(Y) and 7t1(Y'\ 1/2') E 7r1(Y').

Using Seifert—V'an Kampen theorem. we. compute that

7n(/') = ((1.1). c. (1.1.1; | [(1.1)] = [(1,(-] = [(1. (1] = [1). c] = [11.11] = [c.d] = 1,

[.r._1/]=l.11=:1.‘,l) = y. (.‘ = .1‘ . (121/1).

Thus 7n(.\") = (1.1} ] [.1311] : l) ’5 Z"). and it follows that 112(X') 2 1‘2 from our Euler
characteristic (.‘alculation above. The four hmnologically essential Lagrangian tori in
'1'”I are also contained in X'. and thus one can see that condition (11) is satisfied.
The two Luttinger surgeries we choose are —1 surgery on (i X (3 along (”1 and
another -1 surgery on 5X 1' along [1. Here. (i and f) are suitable parallel copies of the
generators (1 and b. res[_)ectivelv \Ve claim that the manifold X we obtain after these

two Luttinger surgeries is siniply-connet‘tetl. To prove our claim. we observe that these

33

two Luttinger surgeries could be first made in the T4 piece that we had at the very
beginning. This is because both Lagrangian tori (i X c and (ix (: lie in the complement
of 2. By our observation in Section 3.2.2, the result of these two Luttinger surgeries
in T4 is diffeomorphic to 81 X MK- Observe that 7r1((Sl X MK)#2@2 \ 112) E“

711(31 X MK), which is (cf. [6] and (3.6)-(3.8) in Section 3.2.2)
((1,1), c, (l I [a,b] = [c, a] = [c, b] = [c, (1] = 1, (1011—1 = [(1, b"1]. b =[a'1,d]).

As before, 7r1(((.8‘2 X T2)#4@152)\u2’) E“ 7r1(b'2 X T2) = (17,}; I [.1331] = 1). Therefore

by Seifert-Van Kampcn theorem,

7r1(X) = ((1.1). c, (1.123; I [(1.1)] = [c, (1] = [c, b] = [6, (1] :2 1.
(for/'1 = [(1.171]. I) = [(1—1,(l], [.1',;1/] = 1,

a = .r, I) = y. c :2 I”. (I = y‘l).

Thus .1‘ and y generate the whole group, and by direct substitution we. see that

y‘lJ'y = [1]—4,1} 1] = 1 and y = [1 1.1/1]. The former gives .‘1‘ = 1. and the
latter then yields y = 1. Hence 7r1(X) : 1. Therefore by Theorem 2.0.1, X is
hornemnorphic to (CIPZ#TtClPQ. Since the latter is not irreducible, X is an exotic copy

of it.

3.3.3 A new construction of a minimal symplectic CP2#5@2

The first example of an exotic (CllfiylnytC—llfi was obtained by .1. Park. Stipsicz and
Szabr’) in [6‘2], combining the double node neighborhood surgery technique discovered
by Fintushel and Stern (cf. [30]) with rational blowdown. Fintushel and Stern also
constructed similar examples using the same. techniques in [30]. The first exotic
symplectic. CP2#5@2 was constructed in [3]. Here, we present anotlnrr construction

with a much simpler 7n calculation, using our construction scheme.

:54

As in Subsection 3.3.2, we construct a symplectic surface 2 of genus two with
trivial normal bundle in Y = Tia/flaw. Let us use the same notation for the
fundamental groups as above. Take another copy Y’ = T4#2W2, and denote the
same genus two surface by 2’ , while using the prime notation for all corresponding
fundamental group elements.

We obtain a new manifold X’ by taking the symplectic sum of Y and Y' along

2 and 2' determined by the map g5 : Z —> 2’ that satisfies:
(1 H ('I, I) i—> (I'. (' (—+ (1', (1H 11'. (3.9)

By Seifert-Van Kampen theorem. one can easily verify that 771(X’) E“ Z4 generated
by, say a.b,(1’.b’. The characteristic numbers we get are: e(X’) = 2 + ‘2 + 4 = 8
and (7(X’) = -—2 + (—2) = —4, the characteristic numbers of CP2#5—CH—DZ. Finally
the homologically essential Lagrangian tori in the initial T4 copies can be seen to
be contained in X' with the same properties. Thus 1' 2 8 = 2111(X’) = 113(X’) —
[12(CIP’2#5@—C—ll_)2). so our condition (11) is satisfied.

\Ve 1:)erform the following four Luttinger surgeries on pairwise disjoint Lagrangian

tori:

~

(mart—1). (hX('.f).—l). ((i’x(-’.a’.—1). (fi'xc’.b'.—1).

It is quite simple to see that the resulting symplectic. 4-manifold X satisfies
H,(X:Z) = 0. Using the observe-ition in Section 3.2.2 again. after changing the
order of operations and assuming that we have done the Luttinger surgeries at the
very beginning. we can view X as the fiber sum of two copies of (81 X .ll;\-)#2@l32
along the identical genus two surface 2 where the gluing map switches the symplectic

bases for S as in (3.9). Thus, using St—‘ifert-Van Kampen‘s theorem as above. we can

‘vv
(:1

see that.

7r1(X) = ((1.1). c. d, (1.',b',c',d’ I [a.b] = [6.11] = [c. b] = [c,d] =1,
dad—l = [(1. b‘l], b = [a"l,(l], [a',b’] = [c',a'] = [c',b’] = [c’,d’] =1,
('(1.’((]')—1 = [11,, (I’d—ll, b’ = [((1,')_l,(l'],

a=c',b=d',c=a', (121/).

Now b’ = [((1’)‘1.d’] can be rewritten as (l = [c‘1,b]. Since b and c commute, d 2: 1.
The relations (Ind-1 = [11.11”] and b = [(1‘1.(l] then quickly implies that (1 = 1 and
b = 1. res1_)ectively. Lastly, (1’(1.’((1’)'l = [(1’,(b’)'1] is bcb‘l = [1), (1"1], so 6 = 1 as
well. Since (1.1), Cd generate 7r1(X). we see that. X is simply-connected. By similar
arguments as before. X is an irreducible symplectic 4-manifold that is homeomorphic

but not diffeomorphic to (CIPQ#5—CTP2.

3.3.4 A minimal symplectic CP2#3@I52 in terms of Luttinger
surgeries

The first. irredtuible symplectic smooth structures on CllmfityéihC—llD2 were crmstructml
independently by Akhmedov and D. Park in [(3] and by Baldridge and Kirk in [10].
Shortly after. a more elegant construction appeared in [25].

Let us demonstrate how the construction of an exotic symplectic CP2#3@2 in
[0'] fits into our recipe. We will use three copies of the 4—torus. TI]. T; and T34.
Symplectically fiber sum the first two along the 2-tori (1, X I), and (12 X ()2 of self-
intersection zero. with a gluing map that identifies (11 with (13 and 111 with ()2. Clearly
we get T2 X 532. where the symplectic genus 2 surface. 22 is obtained by gluing together
the orthogonal punctured symplectic tori (1'1 X ([1) \ I)2 in T3 and (1'2 X (13) \ l)“2 in
732‘. Here. 7r.(T2 X 23) has six generators (1.1 : (13. bl 2112, ('1, ('2. (ll and 1/2 with

relations [111.111] 2 1. [(51.1’ll][('-_).(It_,] = 1 and n'ioreover (II and 111 commute with all

36

c, and (1,. The two symplectic tori (13 X ()3 and 03 X (13 in T51 intersect at one point,
which can be smoothened to get a symplectic surface of genus two. Blowing up T;
twice at the self-intersection points of this surface as before, we obtain a symplectic
genus two surface 2’ of self-intersection zero.

Next we take the symplectic fiber sum of Y = T2 X 22 and Y’ = T§#2@TD2 along
the surfaces 22 and 2’ , determined by a map that sends the circles c1.d1.02,d2 to
(13.113.c3.(l3 in the same order. By Seifert-Van Kampen theorem, the fundamental
group of the resulting manifold X’ can be seen to be generated by a1,b1,cl,d1,c2
and (12, which all commute with each other. Thus 7r,l(X’) is isomorphic to Z6. It
is easy to check that. e(X’) : 6 and (7(X’) = —2, which are also the characteristic
numbers of CP2#3TI_1152.

Now we perform six Luttinger surgeries on pairwise disjoint Lagrangian tori:

((11 X Fl.(:1,—1). ((11 X (f1.(i1,—1). ((11 X (map—1),

~ ~

(i)! X (“2.131. -1). (("1Xf“2,(~'2, —1). ((‘1 X (l2.(12, —1).

Afterwards we obtain a symplectic 4-manifold X with 771(X) generated by (11. b1.

(71. (ll. ('2. ([2 with relations:

[[11,1'11—1]: blclbi‘l. [cl—1,111] : ([1, [(12.1)['1] 21121111131,

[(111,113] :11], [(114121] 21111231111, [(21,111] 2113.

and all other commutators are equal to the identity. Since [111.172] 2 [c1172] = 1,
(I, : ["il~”ll also commutes with ('3. Thus (12 = 1. implying (1.1 = b, = 1. The last
identity implies 1:, : (ll : 1. which in turn implies ('2 = 1.

Hence. X is simply-ctu‘mected and since these surgeries do not change the ("111-trac-
teristic numbers. we have it hoineomorphic to CW#3ETE2. Since Y is minimal and

the exce )t'ional s )heres in Y’ intersect S’. '1‘1‘1e11n'em 2 guarantees that X’ is minimal.
h

37

It follows from Theorem 3 that X is an irreducible symplectic 4-manifold which is

not diffeomorphic to (CIP'Z#3CTD2.

3.4 Minimal symplectic 4-manifolds with b+ > 1

Symplectic fiber sum operation can be effectively used to obtain several other new
minimal symplectic 4-manifolds with bigger Euler characteristics from small minimal

symplectic 4-manifolds. Here we will provide a sample result in this direction.

Theorem 6 Let X be a simply-connected minimal symplectic 4-manz'fold. which is
not a sphere bundle over a Riemann surface and such that X contains (1 genus two
symplectic surface of self-intersection zero. Then X can be used to construct simply-

connected irreducible symplectic 4-11111111f01ds Z’ and Z” satisfying:

(1§(Z’).1),(Z’)) (113(X) + 2.115(X) + 4).

(1)312").1—).;1Z”)) = (193(X)+2.b;(X) +6).

Proof: Let. us denote the genus two symplectic surface of self-intersection 0 in X by
232. By our assumptions. the complement X \ US"; does not contain any exceptional
spheres. Take T“ = "I‘2 X T2 equipped with a product symplectic form, with the genus
two symplectic surface that. is obtained from the two orthogonal symplectic tori after
resolving their singularities. After symplectically blowing up T‘l at. two points on this
surface. we get a syn‘iplectic germs two surface 32 of self—intersection 0 in T4#2@2,
and it. is clear that ('1"‘#2W3) \1/2’2 does not. contain any excepticmal spheres either.
Since we also assumed that X was not a sphere bundle over a. Riemann surface, it
follows from Theorems 2 and 3 that the 4~1nanifold Z’ obtained as the symplectic
sum of X with T‘#2(,_‘1132 along 22 and 252 is minimal and hence irreducible.

Next we take 5'2 X T2 with its product. symplectic form. and as before consider the

germs two symplectic surface obtained from two parallel copies of the symplectic torus

38

component and a symplectic sphere component, after symplectically resolving their
intersections. Symplectically blowing up S2 X T2 on four points on this surface, we
get a new symplectic genus 2 surface 2’2’ with self-intersection 0 in (82 X T2)#4@2.
Although this second piece (8'2 X T2)3l9€45C—ll192 is an 52 bundle over a Riemann sur-
face, the surface 2:; cannot be a section of this bundle. Moreover, it is clear that
((S2 X T2)#4W2) \ V23 does not contain any exceptional spheres. Hence, applying
Theorems 2 and 3 again, we see that the 4-manifold Z” obtained as the symplectic
sum of X with (S2 X T2)#4@B2 along 22 and 2’2’ is minimal and irreducible.

It is a straightforward calculation to see that (e(Z’). o(Z')) = (e(X) + 6, 0(X) — 2)

and (e(Z”), (7(Z”)) = ((:(X) + 8. o(X) — 4). Note that the new meridian in X \1/22

 

. v . . . . . 1 ., .
dies after the fiber sum since. the meridian of 2’2 111 T4#2(Cll’“ can be killed along
any one of the two exceptional spheres. The same argument works for the fiber sum
with (32 X '1‘3)#-1C1P'“’. Hence Seifert-Van Kanrpeiis theorem implies that 7rl(Z’) =

711(Z”) = 1. Our claims about bi; and ()2— follow ininnidiately. C]

 

Corollary 3.4.1 There are erotic 33CP2#(2I + 3)le“2. for l = 2.. . .,6, which arc

(Ill 11'1'(—:d11(71l)lc 111111 syrup/(1111‘.

Proof: \Ve observe that each one of the. irreducible symplectic (Clim2#(2k + 1)@2
(I; = 1 ..... 4) we obtained above contains at least one symplectic genus two surface
of self-intersection zero. (Also see Section 3.5 for more detailed description of these
surfaces.) To be precise, let us consider the genus two surface S which is a parallel
copy of the. genus two surface used in the last. symplectic sum in any one of our
constructions. Since these exotic 4—nianifolds are all minimal. they cannot. be the
total space of a sphere bundle over a Riemann surface with any blow-ups in the fibers.
Also they cannot be homeoniorphic to either F X 5'2 or FXS'“) for some Riemann

surface F. because of their intersection forms. 'I‘lierefore we see that assumptions of

Theorem 6 hold. It quickly follows from Theorem 6 that we can obtain irreducible

symplectic 4-manifolds homeomorphic to 3Cll1’2#(2l+ wall—3’2, for l = 2,. . . ,6. [:1

Remark 3.4.2 Using the generic. torus fiber and a sphere section. of self—intersection
—-1 in an elliptic fibration on E(1) = CP2#9@2, one can form a smooth. symplectic
torus T1 of self-intersection +1 in ..E(1) As each. one of our erotic (Cling-tit(2lc+DEF2
for k = 1,. .. .4 contains at least one symplectic torus of self—intersection —1 (these
tori are explicitly described in Section 3.5). we can symplectically fiber sam each,
erotic CP2#(2k + l)(C—llD2 with E(1) along a chosen torus of self-intersection -1
and T1 to obtain. irreducible symplectic 4-manifolds that are homeomorphic but. not

diffeomorphic to 13CP3#(2L' + INC—Hm for k = 1, . . . .4.

The crux of the above construction is that one. can use simple minimal symplec-
tic 4-niaiiifold blocks. possibly with nontrivial fundamental groups, to produce new
simply-connected minimal symplectic 4-inaiiifolds. In a. joint work with A. Aklinie-
dov. S. Baldridge. B. D. Park and P. Kirk. we exploited this basic idea to fill in a
large region of the geography plane [4:. Let us finish thisisection by quoting the. main

theorem from this work:

Theorem 3.4.3 (Akhmedov, Baldridge, Baykur, Kirk, Park [4]) Let a and
6 denote integers satisfying '26 + 30 Z 0, and e + 0 E 0 (mod 4). If. in. addition.
0 g -—‘2,, then there arists a simply ('onnrctcd minimal symplectic 4-‘Inanifold lift”?
signature 0 and Enter charactcristic c. and odd intcrscction. form, except possibly for
(0,6) equal to (—3.T). (—3. 11). (—5.13). or {—7.15). Moreover. for each. integer
k 2 49. there erists a sluiply connected minimal symplectic 4 -manifold Xgiawk with.
(e. o) = (41; + 1,—1), and for each inter/er A.‘ Z 45. there arists a simply conncctcd

minimal symplectic 4—manifold '\"2k+l.2k+l with (c. (I) = (4k + 4. 0).

40

3.5 Infinite families of nonsymplectic irreducible
smooth structures

In this section we will show how to construct an infinite family of pairwise nondiffeo-
morphic 4-nianifolds that are homeomorphic to CIP2#3Ell—’2. The very same idea will
apply to the others, as we will discuss briefly. We begin by describing these families
of 4-manifolds, showing that they all have the same homeomorphism type, and after-
wards we will use the Seiberg-Witten invariants to distinguish their diffeomorphism
types. The SW invariants will be distinguished via Theorem 5.

We first need to choose a null-homologous torus and pcform l/n surgery on it as
in Subsection 3.1.4. We then prove that 7n = 1 for the resulting infinite family of
4—manifolds. To apply Theorem 5 in its full strength, i.e. to obtain a family that
consists of pairwise nondiffet.)inorphic 4-manifolds. we will show that we have exactly
one basic class for .X’,\_A(()). up to sign. for each exotic X that we have constructed.
\Ve will do this check by straightforward ci-ilculations using adjunction inequalities.

In all the (onstriictitms in Section 3.3, we observe that there is a. copy of V =
(S1 X 1111,) \ (FU S) embedded in the exotic X we constructed. where F is the fiber
and S is the section of S' X .sl/K. viewed as a torus bundle over a torus. As shown
in the Section 3.2.2. .8" X M],- is obtained from T" after two Luttinger surgeries,
which are performed in the complement of F U 5'. So we can think of 5'1 X MK as
being obtained in two steps. Let V0 be the conmlement of F U 5' in the intermediate
4-manifold which is obtained from T4 after the first Luttinger surgery. The next
Luttinger surgery. say (L, "g". —1). produces V from H). (In the Section 3.2.2. L = cxb
and A, = b.) This second surgery on L in ls}, gives rise to a nullhomologous torus A
in V. There is a. loop /\ on A so that the U surgery on A with respect to A gives

H, back. As the framing for this surgery must be the nullhmnologous framing. we

41

call it the ‘O-framing’. Note that performing a l / n surgery on A with respect to A
and this O-framing in V is the same as performing an (L, 7. —(n + 1)) surgery in V0
with respect to the Lagrangian framing. We denote the result of such a surgery by
V(n) = VAA(1/n). In this notation, .V(oo) = V0 and we see that V(0) = V.

We know that performing a. —n surgery on L with respect to 'y and the Lagrangian
framing, we obtain V(n — l) = (S1 X MK") \ (F U S), where Kn is the n-twist
knot. It should now be clear that replacing a copy of V in X with V(n — l) =
(8' X MK") \ (FU S) (i.e. ‘using the n-twist knot') has the same effect as performing
a 1/(n — l) surgery in the O-framing on A in V C X . We denote the result of such
a. surgery by Xn = X‘UU/(n — 1)) Clearly, X1 = X. We claim that the family
{Xn l n. = 1.2.3.. ..} are all homeomorphic to X but have pairwise inequivalent

Seiberrr-VVitten invariz-mts. The first claim is royed in the followin ‘ lemma.
0

Lemma 7 Let X" be (he infinite family (:o'rrespon(ling to a fired erotic copy of

le’2#3ClP‘2 that we have constructed above. Then. Xn are all homcmmn'phic to X.

Proof: For a fixed exotic X. 7:1(.\',,) only differs from 7r,(.\") by replacing a single
relation of the form I) = [(17141] by b = [(z’l,(1]"‘ in the presentation of 7r1(X) we.
have, used. Thus one only needs to check that raising the power of the connmltator in
one such relation does not. effect our calculation of 7r. (X) = 1. This is easily verified
in all of our examples. Hence all the fundamental group calculations follow the same
lines and result in the trivial group.

Since X” is differ from X only by surgeries on a. nulllmmologous torus. the charac-
teristic numbers remain the. same. 011 the other hand. since none have new homology
classes. the parity should be the same. By Theorem 2.0.1 again. they all should be

homerunorphic to each other. C]

Below. let X be a. ~1-1nanifold obtained by fiber summing 4-mauifolds Y and l"

along sulmianifolds E C Y and 2’ C Y’. Let :1 C Y and B’ C Y’ be surfaces

42

transversely intersecting E and 2’ positively at one point, respectively. Then we
can form the internal connected sum A#B’ inside the fiber sum X , which is the
closed surface that is the union of punctured surfaces (A\ (A O 112)) C (X \1/23) and
(B’ \ (8’ H 122'» C (Y’ \ 112’). It is not hard to see that the intersection number
between A#B' and E3 = 2’ in X is one, and thus they are both homologically
essential. If all these manifolds and submanifolds are symplectic and the fiber sum
is done symplectically, then A#B’ can be made a symplectic submanifold of X as
well. Also note that, if either A or B’ has self-intersection zero. then their parallel
copies in their tubular neighborhoods can also be used to produce such internal sums

in X .

3.5.1 An infinite family of irreducible smooth structures on
ceasefire?

Let X be the exotic CW#3@2 that we have described in Subsection 3.3.4. We
begin by describing the surfaces that generate 113(X1Z). There is a symplectic torus
T = T2 X {pt} of self—intersectimi zero in Y = T2 X 23 intersecting Z : {pt} x X32
positively at one point. On the other side. in Y' = 'T"#2@2. there is a symplectic
torus Ti of self-intersection zero, and two exceptional spheres E; and E3. each of
which intersects 2’ positively at one point. (There is actmilly another symplectic
torus T2’ in l" satisfying [2'] = [Til + [T5] — [E1] — [E3] in [12(Y'12). but we will
be able to express the homology class that T; induces in X in terms of the four
homology classes l;)elow.)

Hence we have four homologically essential symplectic surfaces: two genus two
s1.1rfaces E : Z’. G = T#Tl’. and two tori R, = T#Ef. 2' = 1.2. Clearly [EV =2
[Of2 2 0, and [HIV = [If-2]? = —1. It is a. straightforward argument. to see that

these span [12(X: Z). and the. corresponding intersection form is isomorphic to that

43

of CP2#3@2. (Note that [T#T§] = [22] — [G] + [R1] + [R2] )

The O-surgery on A with respect to A results in a 4-manifold X0 = X A. A(0) satis-
fying H1(X0: Z) 2‘ Z and H2(X0: Z) 2’ H2(X; Z) {3- Zz, where the new 2—dimensional
homology classes are represented by two Lagrangian tori Ll and L2. Both LJ- have
self-intersection zero. They intersect each other positively at one point, and they do
not intersect with any other class. Thus the adjunction inequality forces this pair to
not appear in any basic class of X0. Denoting the homology classes in X0 that come
from X by the same symbols. let 8 = aiE] + b[G] + Zr MR.) be a basic class of X0.
Since it is a characteristic element, a and I) should be even, and 7‘1 and r2 should
be odd.

Since b:(X0) > 1. applying the (generalized) adjunction inequality for Seiberg—

\=\I'itten basic classes (cf. [58]) to all these surfaces. we conclude the following.
(i) 2 2 0 + If} - [(iH. implying 2 2 lei.
(ii) 2 2 0 + l/3- [EH implying 2 2 II) + :1. r,-|.

(iii) 0 2 —1+|;i- {H.jl. implying 1 2 |a — ml for i: 1.2.

On the. other hand. since X0 is symplectic and bflXn) > 1. X0 is of simple type so

we have :32 = 2e(Xu) +- 30(Xn) = 2e(X) + 30(X) = 2 - 6 + 3(-2) = 6. inmlying:
(iv) 6 = 2(1(() + Z]: I',) — (If.

From (i). we see that a can only be 0 or i2. Howevrr. (iv) implies that a # 0. Let

us take. a = 2. Then by (iv) and (ii) we have

[6+2]? HZ],

which implics that. Z, [‘3 S 2. 'l‘herefore by (iii) we see. that both r, have to be 1.

=4 £8.

 

 

 

Finally by (iv) again. I) = t).

44

Similarly, if we take a = —2, we must have n = r2 = —1 and b = 0. Hence
the only basic classes of X0 are i(2[2] + Stilt-D = ino, where K x0 denotes
the canonical class of X0. By Theorem 5, all X" = XA.,\(1/(n — 1)) are pairwise
nondiffeomorphic.

Moreover, by Remark 3.1.2 we see that X = X1 also has one basic class up to sign.
It is easy to see that this is the canonical class K x : 2[2] + [31] + [R2]. Therefore the
square of the difference of the two basic classes is 4K ‘3 = 24 7E —4, implying that X
is irreducible (and hence minimal) by a. direct application of Seiberg-Witten theory
(cf. [26]). Furthermore. the basic class [in of Xn corresponding to the canonical class

KXO satisfies
SVVX” (13,.) 1‘ S‘Vx(f\ix’) + (Tl - 1)SVVXO(I\’XO) (3.10)
= 1+(n.—1):'n.

Thus every X" with. n. 2 2 is nmrsymplectic. In conclusion. we have proved the

following.

Theorem 8 There is an infinite family of pairurisc nondiflmmorphic 4 --rnam'fol(ls
. . . . -) ‘. '——- . . .
which. are all homeomorph1c to CP"#3CIP3 . All of these mamfolrls are N‘I‘cdaczl)le. and

they possess cractly one basic class. up to Sign. All crccpt for one are nonsymplcctic.

3.5.2 Infinite families of irreducible (CPQ#(2A< + HTTP” for k, =
2, 3, 4

For exotic CP2#5@2 is. the second homology of X0 will be generated by the following

surfaces: two germs two surface of self-intersection zero. 2 = S’ and G = T#T'. four

tori of self-intersection —1. It,- 2 IC,#T’ (i = 1.2) and SJ 2: Tit/27;. (J = 1.2).

and two Lagre-uigian tori L1 and L3 as before. A basic. class of X0 is of the form

,3 = aIZI + bIGI + 21.1",[R4I + Z]. stSj], where a and b are even and r.- and 31 are

odd. The inequalities are:
(i) 2 2 0 + [If - [GII, implying 2 2 IaI.
(ii) 2 2 0 + [(3 - [XI], implying 2 2 lb + :1. r,- + stjl.
(iii) 0 2 —1+I/'3- [R,II implying 12 Ia— ml for 2': 1,2.
(iv) 0 2 —-1 + [13' [SJ-II, implying 12 Ia — sJ-I for j=1.2.

(v) 4 = 2(1(l) + :14“,- + 21.31) - (2217,2 + Z]. .912- .

By (i). a can only take the values 0. i2, where O is ruled out by looking at (v). If
a = 2, then by (iii) and (iv) r, and SJ are either 1 or 3. However, using (ii) and (v)
as before. we see that none of these can be 3. It follows that 1‘, = sj = 1 for all 'i.
and j. and b = —2. The case when a = —2 is similar. and we see that X0 has only
two basic classes :i;(2[ZII — 2ICI + Z,[lt,-I + ZJ-[SJI).

For exotic (ClipzfitflCTQs. the classes are similar except now we have four tori of
the form 5]. For a. basic class :3 = aIZI + bICI + 23:] r,[R,~I + 2:21 SJ[SJI of X0. we
see that the coefficients have the same parity as above. The first four inequalities are
the same (with (iv) holding for j = 1,. . .,4), whereas the last equality (v) coming

from X0 being of simple. type becomes:

(v’) = 2a(h + 2:11}- + 2:21 sJ) — (23:1 1;) + 2:21.55).

Once. again, (i) implies that a is 0 or $2. but by (v’) it cannot be 0. If a = 2. then
by exactly the same. argument as before we see that r, = sj = 1 for all '2'. and j. and
thus I) z —4. The case a = —2 is similar. Therefore the only two basic classes of X0
are :1:(2[EI — 4[(,.;] + Zf’ZIIHJ + 2:2, [53]).

For exotic (Cllw#£)(1—:—lf”2s, the only difference is that the number of tori in total is

eight. Let us denote two of the additional tori as H3 and It, correspomling to say

46

R0 and 23,, where the other two will be denoted by 5'3 and S4 corresponding to Rb
and Eb as described in Subsection 3.3.1.

For a basic class ,8 = aIEI + bICI + 2::1r,[R,I + 22:1 stSj], once again a,b
are even and n. sj are odd. The inequalities (i)—-(iv) remain the same. Finally (v’ )

becomes:

(v”) 0 = 2a(b + 2:173- + 23.2133) ‘(Zl=17'12+ 22:1 312).

As before, a cannot be 0. If a = 2, then by the same argument we see that r, =
sJ- = 1 for all i and j. Thus b = —6, and we get a basic class [5 == 2IZI - 6IG] +
4 4 . - - . - . - -
22:1[11’1] + EFIISJI. For a = —2 it. is easy to check that we get the negative of this

class.

Hence in all three examples X0 has only two basic classes. and therefore by The-
orem 5, all three families {.X',,} consist of pairwise nondiffeornorphic 4-manifolds
hmneomt)rphic to CP2#5@2. CP2#7CTP2. or Clll‘gaééQTTll—Dz. Furthermore, as in the
previous subsection. we see that each family {Xn} consists of 4-rnanifolds with only
one basic class. up to sign. In each family. all but one rnernl‘)er are nonsyrnplectic
as the only nonzero values of the Seiberg—Witten invariant of X" are in. Finally.
each exotic Xn can be seen to be irreducible by a direct Seiberg-Witten argument as

before (cf. [26]).

47

CHAPTER 4

N ear-symplectic 4—manifolds

4.1 Background

4.1.1 Near-symplectic structures

Let-to be a closed 2-form on an oriented smooth 4-manifold X such that W2 2 O,
and 2,, be the set of points where to degenerates. Then u} is called a near-sympler't‘zc
structure on X if it satisfies the following tr‘ansversality condition at every point
I in sz if we use local coordinates on a. i'reighborhood U of .r to identify the

I) , 1 v ‘ . . .
map a; : U ——> A“(1“l/) as a smooth map u; : 1ft4 —> R". then the hnearrzation

 

Du), : IR" -—> R6 at I should have rank three which is in fact. independent of the
chosen charts [9] ln 1')art'icular. Z = Z“. is a smoothly embedrgled 1-rnanifold in X, if
not empty. We then call (\. to) a near—symplectic 4—i'nani’fold, and Z the zero locus
of a).

If a. given 4-n'ianifold X admits a irear-symplectic structure. then it is easy to
see. that l)+(X) > 0. One of the motivations for studying 1lear-symplectic structures
has been the converse oljiservation. Namely. any closed smomh oriented 4-1nemifold
X with h+(X) > 0 can be equipped with a near—symplectic form. which was known

to gauge theory itifficianarlm since early 1980s and a written proof of which was first

18

given by Honda through the analysis of self-dual harmonic 2-forms ([46], also see
[9]). Thus the near-symplectic family is much broader than the symplectic family of
4—manifolds. For instance, connected sums of symplectic 4-manifolds can never be
symplectic. due to the work of Taubes and the vanishing theorem for SW invariants.

However these manifolds would still have b+ > O and therefore are near-symplectic.

Example 4.1.1 Let M 3 be a closed 3—rnanifold and f : M —+ S 1 be a circle valuded
Morse function with only index 1 and 2 critical points. Then the 4-manifold X =
S 1 X .1] can be equipped with a near-symplectic structure. To see this. first note
that. due to a theorem of Calabi there exists a metric g on M which makes df
harmonic. Parametrize the. first 5'1 component by t, and consider the form w =
(fl /\ (If + *(rlt /\ (If). where the Hodge star operation is defined with respect to the
product of the standard metric on SI and g on .11. It is a straightforward check to
verify that of“) 2 0 and that u} vanishes precisely on Z = S] X Crit(f). Finally using
local charts one can see that a; vanishes transversally at every point on Z (also see.
the next subsection).

The attentive reader will realize that some. of the symplectic building blocks ex-
tensively used in the previous chapter are in fact specific examples of this type.
For consider a 3-manifold .11“ obtained from 5'3 after a O-surgery on an arbitrary
knot K. which comes with a circle valued Morse function as l;)efore. (Note that
Z 2’ ll‘(.fil I“: Z) E“ [,\I,,-. 5"] .) Then to defined as above yields a symplectic form 011
X = .8'1 X .l [K if and only if K is fibered so that f can be assumed to have no critical

points: i.e when Z = 0).

4. 1 .2 Local models

Using a. generalized Moser type of argument for harmonic self-dual 2—forms. Honda

showed in [47] that there are exactly two local models around each connected corn-

49

ponent of Zw. To make this statement precise, let us consider the following 10—
cal model: Take R4 with coordinates (t,.r1,;r:2,:r3) and consider the 2-f0rm Q :
thdQ+=i< (thdQ), where Q(l‘1.1'2. 1'3) = r¥—%(I§+1§) and * is the standard Hodge
star operator on MTV. Restrict Q to IR times the unit 3-ball. Define two orienta-
tion preserving affine automorphisms of R4 by 0+(t.a:1.172,r3) = (t + 27r,:r1.172.x3)
and a_(t, 171,1?2. 3:3) = (t + 271', —:r1.I-2. ~—;r3). Since both maps preserve 52, they in—
duce near-symplectic forms mi 011 the quotient spaces Ni = IR X D3/ai. Honda
shows that given any near-symplectic (X . ad) with zero locus Zw, there is a Lipschitz
self-homeomorphism Q on X which is identity on Zw. smooth outside of Zw and
supported in an arbitrarily small neighborhood of Zw, such that around each cir-
cle in Z”. the. form (9“(w) agrees with one of the two local near-symplectic models
. (Ni. mi). For our pin‘poses. we can always replace the I‘iear-symplectic form w with
such a form (The). Herein the zero circles which admit neighborhtmds (37+. age) are
called of even type. and the others of odd type.

On each local model Ni 2’ S" X D3, one can consider fibration-like maps:

Fi: Ni —> S" X I defined by

Fanl’i-J'znlfid Z (L.Q(-"1--‘I")e-"3)) = (Iv-"l2 _

(:3 + 139). (4.1)

In either case. for a fixed t. we observe that. on the complement of 31 X 0 we have
fibrations with fibers composed of two disjoint disks in I )3 for the preimages of points
with Q > 0, whereas the fibers are annuli for Q < 0. In the preimages of (t. 0) we
have conical singularities which amounts to attaching a l—handle with feet on the
two separate disks so to obtain the annulus on the other side. Dually it is a '2-handle.
attachment in the opposite (.lirt-action which s(:'[:)arat.es the annulus into two disks. N ow
if we let t E .8" vary. this ammmts to doing this handle attachment fiberwise as we
pass the middle circle .8'1 X 0 in S‘ X I . The difference between the two local models

1\"+ and N_ manifests itself here. The model is even if and only if for a fixed Q > 0

50

the two disks are switched after one travel in the t direction, and odd otherwise.

Example 4.1.2 Once again let X = S1 X M K for nonfibered K . For simplicity,
assume that f : MK —> S 1 in Example 4.1.1 is injective on its critical points. Then
the preimage of any regular value of f is a Seifert surface of K capped off with a disk,
i.e a closed orientable surface. While passing an index k critical point (k = 1.2),
a k-handle is attached to get one Seifert surface from another. It follows that F :
MK —> 81 is a fibration-like map, where the genera of fibers are increased or decreased
by one at every critical point, depending on k = 1 or k = 2, respectively. When
crossed with S", this yields a fibration-like map F : id X f : X —> T2. The base torus
T2 = S1 X S1 can be parametrized by (1.5) where t traces the outer circle factor and
s traces the base circle. of f. Thus the monodromy of this fibration is trivial in the
t direction and is prescribed by the knot. monodromy in the 5 direction. Choosing
local charts on a tubular neighborhood S1 X D3 of each component of S1 X C r'if( f )
and on the image 5'1 X I , one can see that F is locally the same as 17+ map we have
defined above. This implies that all circles of Zw. where w is the near-symplectic

structure on X described in Example 4.1.2. are even.

4.1.3 Broken Lefschetz fibrations

In [9], Auorux. Donaldson. and Katzarkov defined a generalization of Lefschetz fi-
brations called "singular Lefschetz fibrations", where they allowed the maps to have
singularities along enilgiedded circles (“indefinite quadratic singularities” [9]) which
are subject to the two local models described in the previous subsection (Equation
4.1) in addition to the usual nodal singularities on. the complement. of these. Here
we refer to these fibrations as brokcn Lefschetz fibre—{£0715 as in [(54, 65]. and the new
type of singularities as odd or even round singulumfizics depending on whether the

local model around the singular circle is odd or even. A bro/ten. Lefschetz pencil is de-

51

fined similarly, where we allow round singularities in the complement of the Lefschetz
critical points and the base locus.
Tha main theorem of [9] states that broken Lefschetz fibrations are to near-

syplectic 4-manifolds what Lefschetz fibrations are to symplectic 4-manifolds:

Theorem 4.1.3 (Auroux, Donaldson, Katzarkov [9]) Suppose F is a smooth
1 —dimensional submanifold of a compact oriented 4-manifold X . Then the following

two conditions are equivalent:
0 There is a near-symplectic form an on X, with Z“, = F,

c There is a broken. Lefschetz pencil f on X which has round singularities along
I‘, with the property that there is a class II E 112(X) such that 11(2) > 0 for

every fibcr component 2 of f.

Moreover. the implications in, each direction. can be obtained in. a compatible way.
That is. given. a iii'(1xr-si/iiiplcct-ic form to. a corresponding broken Lefschetz pencil
([3ij can bc obtnincil so that all the fibers arc siniiplcctic on the complmncnt of the
singular locus. C'oni’crscly. from. a broken Lefschetz pencil (BLF) satisfying the. indi-
cated cohomological condition. one constructs a uniqc dcforn'iation class of to which.

is symplectic on the fibers. (1 may from the singiilai‘itics.

As in the Lefschetz fibration case, blowing—up the base locus of a broken Lefschetz
pencil results in a Lefschetz tibration. “hen the BLP supports a near—symplectic
structure. these blow—u])s/downs are understood to be made symplectically. If we
have in hand a broken Lefschetz fibration over a Riemann surface B (which we
will mostly take as B 2 5'2) that satisfies the same cohonnilogical condition in the
statement. of the theorem. then we can construct compatible near-symplectic forms
with respect to which a chosen set of sections are symplectic [9]. From now on we will

refer to such a fibration f on X as o n.caxr-si/niplcctic broken Lefschetz fibi‘ation. and

Q”!
I»;

say that the pair (X, f) is near-symplectic. Implicit in this notation is that the near-
symplectic form on X is chosen from the unique deformation class of near-symplectic
forms compatible with f obtained via Theorem 4.1.3.

Clearly one can define broken Lefschetz fibrations over any Riemann surface. The
Example 4.1.2 gives such an example of a broken fibration (with no Lefschetz singu-
larities) over T2. In general, a broken Lefschetz fibrations over a Riemann surface
can be split into Lefschetz fibrations over surfaces with boundaries, and fibered cobor-
disms between them relating the surface fibrations over the boundary circles. Round
singularities of a broken Lefschetz fibration are contained in these cobordisms. We
study these cobordisms more rigorously in the next subsection, but a brief discussion
of their use beforehand might be helpful. For now. the reader is invited to convince
himself/ herself that. our discussion of the local models around each round singular
circle in the. previous subsection implies that these cobordisms are given by fiberwise
handle attaclunents. all with the same index (either 1 or 2).

If we fiberwise attach l-handles to a fibered 3-manifold H, to obtain a new fibered
3—me-inifold l]. the attz-ichiug region is necessarilly a. bisection so that the handle
attaclunent is cmnpatible with the monodromy of the filn‘ation on Y0. That is, such
a cobordisrn H' is given by a fiberwise l-handle attachment at the. two intersect-ion
points of this bisection with the fibers of lb. The liln'ations on the two ends of H"
uniquely extend to a broken fibration over S" x I. with only one. round singularity
given by the centers of the cores of l-handles attacluid to the fibers of H). Similarly.
we can fiberwise attach 2-haudles to a fibered h; to describe a cobordism to a new
surface libration l'l’ over a circle. This is obtained by a tiberwise Q—handle attachment
along a. curv “,3 on each fiber F... where s parametrizes the base I/{U ~ 1} 2: SI of
the tibration on lo’ . Once again we obtain a broken fibration from this cobordism W’

to 81 x I. with a single. round singularity coi.'i'esp(_)iuling to the centers of the cores

53

of the 2-handles attached fiberwise.

The diffeomorphism types of the fiber components of the fibrations on the two ends
of such cobordisms can easily be deduced from each other by looking at whether the
bisection intersects with only one fiber component or two, or in the other direction to
whether 7,, is a separating curve or not. (for one s and all). Relating the monodromies
of these fibrations is a more elusive issue which we will address later in Subsections
4.2.1 and 4.2.2. However, the types of singular circles that arise from these round
handle attachements is determined easily. In such a cobordism with l-handles, the
type is even if the attachings trace out an oriented link with two components and odd
. if they trace out an oriented knot. On the other hand, a cobordism with 2-handles
gives rise to an even circle if the monodromy of the fibration on YO’ maps '71 to 70

with the same orientation. and is odd otherwise.

Remark 4.1.4 Roughly speaking, such cobordisms with 1-handle attachments in.—
crease the genus of a fiber component. or connect two different fiber components,
whereas cobordisms with. '2-handlc attachments either decrease the genus or discon-
nect. a fiber comprment. In [“9] it was shown that for any given near—symplectic form.
a) on. X. a compatible broken. Lefschetz jib-ration f : /\'#l)@2 —+ 52, where b is
the number of base points. can. in fact be arranged in the following way: The base
SQ breaks into three pieces I), U :l U Uh. where .4 is an. annular neighborhood of the
equator of the base 32 which does not contain the image of any Lefschetz critical
point. D, and 1);, are (lists. so that (i) On. X1 = f“(l.7,) and X}, = f‘l(l),,) we
have genuine Lefschetz jilnalions; and (Ii) The cobordism ll" 2 f"1(A) is given by
only fibcrwise l-handle attachments if one travels from the X, side to X;,, side. We
call these kind of broken Lefschetz fibrations/pen.cils directed. X, the lower side and

Xh the. higher side.

4.1.4 Lagrangian matching invariants

We will now discuss an invariant due to Perutz [63, 64, 65] associated. to any given
(X, f) where X is a closed smooth oriented 4-manifold and f : X ——) B is an injec-
tive near-symplectic broken Lefschetz fibration. The injectivity condition is needed
to guarantee that f maps components of the round singular locus to disjoint circles
on B, which can be achieved by perturbing any given broken Lefschetz fibration.
Perutz’s work generalizes the Donaldson-Smith construction [19] to near-symplectic
broken Lefschetz fibrations. It relies on a count of pseudo-holomorphic sections of
the associated families of syn‘unetric products over a splitting base that ‘match’ by
’ satisfying certain ‘Lagrangian boundary conditions" [641, 65]. This aspect of the con—
struction suggests the name Lagrangian matching invariants for these invariants. The
cmlstruction of Lagrangian matching invariants (LM) are quite tedious. and the reader
is asked to turn to [64. 65] for the. details which will be ignored below.

Ll\.I invariants are designed to be comparable to SW invariants of the underly-
ing 4—inanifold. and were C(.)Il_iP('tt1_ll'€(_l by Perutz [G4] to be equal to SW. When the
round locus is empty. the equality of the Doiialdson-Sniith and Seiberg-Witten in-
variants for sympleetITc Lefschetz fibrations of high degree was proved by Usher [82]
through T aubes" work on the correspondmice between Gr invariants and SW invari-
ants on symplectic 4-1nanifolds. More evidence in this direction were gathered in
[6-5]. including the equality of LM and SW invariants on the near-symplectic family of
J—manifolds S" x UK. for any knot K, described in Example 4.1.2. The conjecture
in particular proposes the LM invariants to be i11(_le11)e1’1dent of the choice of libra-
tions (possibly after imposing some constraints). even though the calculations make
use of the tibration structure. Hence the nature of the invariant requires the study
of near-syinplcctic broken Lefschetz tiln'ations, which will be the main theme of the

next section. whereas the aforementioned conjecture motivates us to look at SW" and

LM invariants simulatenously in the rest of this chapter.
Let f : X —-> 82 he a near-symplectic broken Lefschetz fibration. Let Spin“(X)
denote the l12(X; Z)-torsor of isomorphism classes of Spine-structures on X, and

F E 112(X; Z) he the class of a regular fiber of f. Then
S[,)in“(X)k = {5 E SpinC(X) | (c1(5), F) = 2k, (*)}

where (*) is the condition that. for any connected component. 2 of a regular fiber,

one has (c1(5), [2]) 2 x(}:).

Definition 4.1.5 k E Z is admissible for (X, f) if either (i) the fibers are all con-
nected and k ‘> 0. 07' (ii) ,\/(X3)/2 < l: < —x‘(.\'s)/2 for all regular values s. A

Spine~8t7‘artm‘e 5 is admissible if 5 E SI)lIlC(.Y)A. with k admissible.

Then the L(l..(]'l'(l.n_(/l(lll matching invariant is a map

U Saran—mm). 5HL4V<.\1;)(5)-

A admissible
where A(X) is the graded aheliau group le] $0; i\"ll‘(,X';Z). deg(l") :- 2. The

element Ll\'1(‘\"n)(5) is hmuogeneous of degree
‘ 1 i) . . , .
(l([\/(\H(5)l ":3 1(('1(5)~ — 3(7(.\) — 2(‘(\ )) (4.2)

and derived from a moduli space [(34, (55], whose crmstructiou in turn uses the broken
Lefsvhetz liliration f on X as well as several auxiliary choices. It. is i11\-'a.riaut. under
isotopies of f through liln‘atious of the same type. and equivariaut under automor-

phisms of (X, f).

Remark 4.1.6 The above definition ('07) be grumiilizril in any base sap/are B. after

replacing A(X) by A(X. f) : Z[l’] X; .4\‘Ho1u(l\'fi, Z). 'u'lu’re

It,r = kel‘(7r.: ll,(,\’;2/Z) —> II,(B:ZZ)) c lll(.\’).

In the level of homology, the Spine -structures that the Lagrangian matching invari-
ants are parametrized over correspond to multisections of a near-symplectic broken
Lefchetz fibration (X, f) which have (homological) boundaries equal to the round
locus. (This is analogous to the tautological correspondence between the multisec-
tions (called ‘standard surfaces’) of a symplectic Lefschetz fibration and the sections
of the Hilbert schemes in the construction of Donaldson—Smith invariants [69].) What
follows is a brief review of this:

The ‘Taubes map" TX is a. bijection (as proved in [77])
Spinc(X) —> 6—1([Z]) C 112(X,Z;Z),

where 6: H2(X, Z; Z) —> H1(Z; Z) is the boundary homomorphism. and Z is oriented
by a vector field e such that i f.(i') points into the higher side of [(Z) C 1)’. The map
TX arises from the canonical Spin -st1uctuie 5”", on the almost complex manifold

X \ Z. It is characterized by
75(5) = if if 51(X \ Z) = PD(.-13)-5(.m,. (4.3)

That is (1(5) 2 c1(sm,,) + '2Pl)(i3). Thus a uniltisection in question is obtained by
d : Ti](5) E [12(X. Z: Z).

To finish with we would like to note another aspect of LM invariants established
by Perutz: that they fit in a. 'fibered field theorv’. This is achieved by assigning
symplectic Floer homology groups to 3-1nanifolds fibered over circles, and relative
invariants assigned to Kl-manil‘olds libered over Riemann surfaces with bmmdaries. A
chosen multiscction of a near-symplectic broken Lefschetz fibration of (X, f) restricts
to umltisections of these libratious. which in turn is used to compute. these Floer
homologies. To simplify our discussion here. assume that the base B = S2 splits
as B : Bl U U B". where 15’] and 8,, are disks and the rest are annuli. with

each one cimtaining the image of the singular locus of f in their interiors only. Let

57

X,- = f‘1(B,~) so that X = X1 U.) - - - Us X". Then we get a map
Lil/NJ = Lil/thlxl OLAIXi’vleQO' - 'OLAIXneflxn Z Splllc(X)m1miss,-b(e ——+ A(X)

where Spin“(X)adm,-ss,-b,e 2 Uk admissible Spin“(X);c and Lil/V is a dual map. (See
[64, 65].) Then this map is evaluated on monomials U a C3) 11 /\ - - - /\ lb of degree (1(5)

to obtain a map into Z as in the SW’ setting.

4.2 Topology of broken Lefschetz fibrations

Handlebody diagrams of Lefschetz fibrations over 8'2 are well—11nderstood and proved
to be useful in the study of tt.)pology of smooth 4-manifolds. The reader is advised to
turn to [ill] for the details of this by now classical theory and its several applications.
In this section. we would like to extend these techniques to the. study of broken
Lefschetz fibratious. For this purpose, we will describe and study round handles that.
arise naturally in the context of 4-dimensional broken Lefschetz fibrations thoroughly.

An ‘n-dimensional round lr-handle is topologically 8'1 x D“ x D"‘1‘k. The
first ct)u’iprehensive study of round handles is due to Asimov [7], and more on 4-
dimensional round l—handles can be found in [34]. However. both articles assume
a restriction on the way these handles are attached. Namely. these round handles
.8" x I)" x I)" l k are attached along .8" x 5""‘1 x I)”“"”"‘. As the work of [9]
implicitly suggests. we shall also consider a ditferent type of attachment. To keep
the follmviug discussion simple. let us define this other way of gluing in the case of
4-dimensional round l-— and ‘2— handles only. the ones which interest. us in this
work.

Take a 3-disk bundle over S" with total space .8" x [)3, and look at the splitting
of this bundle into two subbundles of rank 1 and 2. These splittings are. classified by

‘ . v - . ‘) V' ‘)
homotopy classes of mappings from .51 into RED“. Smce EMBED“) : Z3 there are two

(j!
06

possible splittings up to isotopy. These two splittings can be realized by the ones given
in Subsection 4.1.2. Namely, these are determined by the two orientation preserving
self-diffeomorphisms of R3, where one is the identity map, and the other one is given
by (:r1,:r2. 3:3) i—-> (—:r1,ar2, —;1:3). Our second type of round handle attachment arises
from the latter model. To distinguish the two type of round handles, let us denote this
new one by SI§D3 just to emphasize the splitting we consider. Clearly, SI;D3 is
diffeomorphic to S1 x D3. We call the round handles attached in the usual way (as in
[7]) even round handles, whereas the others are called odd round handles ——-apparently
corresponding to the. even and odd local models in Subsection 4.1.2.

Let us describe the attachments in the odd case more explicitly. The attachment
of an odd round l-handle Sl>~<(D1 >< D2) is made along SUNS” x D2). which is
topologically the I)2 neighborhood of a circle (2 S1 >’Z(S0 X 0)). If we restrict our
attention to the. rank 1-bundle (pi-irametrized by $1) over 8’, both even and odd
round l-handles can be seen to have attaching regions given by the restriction of this
bundle. to its boundary (which gives a bisection of the rank l-bundle) times the rank
‘2 bundle. Then the odd and even cases correspond» to this bisection having one or
two components. respectively. Similarly. an odd round ‘2-handle 3152(1)2 x D‘) is
attached along 5" I; (5'1 x D‘). This is topologically a. collar neighborhood of a. Klein
Bottle, whereas in the. even case we wtnild be. gluing along a collar neighborhood of a

torus.

4.2.1 Round l-handles

. . . ~ -. -) .
Expressing the Circle factor of an even round 1—handle b] x D" x D“ as the umon
of a O-handle Io = I)” x I)1 and a 1-handle I]. = I.)1 x I)”, we can express an even

. r . . ’ 1)
round l-handle as the mnon (IUU 1]) X l)‘ x D“) = (1)” x D1 U D1 x D”) x D1 x D“ =

(1)” x D‘) x (D1 x D2) U (D1 x 1)”) x (D1 x D?) E“ (1)“ x DI) x (l)1 x U3) U(Dl x

L,
C

 

U (9;;

Figure 4.1: A general odd round 1-handle (left). and an even round 2-handle attachment
to a genus two Lefschetz fibration over a disk (right). Red handles make up the round
1—handle.

 

D1) X (D0 X D2) ”:V D1 X D3UD2 X D2, a 4-dimensional 1-handle H1 and a 2-handle
H2. Note that we exchange and rewrite the factors simultaneously. It is not too hard
to see that H2 goes over H1 geometrically twice but algebraically zero times.

In the same way, we can realize an odd round 1-handle as the union of a 1-handle
H1 and a 2-handle H2. However this time the underlying splitting implies that H2
goes over H1 both geometrically and algebraically twice.

We are ready to discuss the corresponding Kirby diagrams. Recall that our aim
is to study the round handle attachments to Lefschetz fibrations. Let F denote the
2-handle corresponding to the regular fiber. Both in even and odd cases, the 2-
handle H2 of the round l-handle links F geometrically and algebraically twice and
can attain any framing k. Both ‘ends' of the H2 are allowed to go through any one of
the 1-handles of the fiber before completely wrapping once around F. In addition,
these two ends might twist around each other as in Figure 4.2.1. (Caution! The
“twisting” discussed in [9] is not this one; what corresponds to it is the framing k.)
The difference between even and odd cases only show-up in the way H2 goes through

H1. In Figure 4.2.1 we depict both types of handle attachments.

60

4.2.2 Round 2-handles

The handle decomposition of round 2-handles is analogous to that of round l-handles.
Expressing the circle factor of an even round 2-handle S 1 X D2 X D1 as the union of
a O-handle IO = D0 X D1 and a l-handle 11 = D1 X Do, this time we can express an
even round 2-handle as the union (Io U 11) X D2 X D1 g D2 X 02 U D3 X D1, a 4-
dimensional 2—handle HL’, and a 3-handle Hg through a similar rewrite as before. For
an odd round l-handle we get a similar decomposition. However the splittings once
again imply the difference: the 3-handle goes over the 2-handle geometrically twice
and algebraically zero times in the even case, and both geometrically and algebraically
twice in the odd case. One can also conclude this from the previous subsection since
a. round 2-handle is dual to a. round l-handle. .

\\'e are now ready to discuss the corresponding Kirby diagrams for attaching round
2-handles to Lefschetz fibered 4-manifolds with boundary. Recall that the round 2-
handle attachment. to a surface tibration Yo, over a. circle that bounds a. Lefschetz
fibration is realized as a. fiberwise 2—handlc attachment. The attaching circle of the
2—handle II."2 of a round ‘2—handle is then a simple closed curve 7 on a 1.'e.*g1.11a.r fiber,
which is preserved under the monodromy of this fibration up to isotopy. Since this
at‘tachimint comes from a fil;)erwise handle attachment, H; should have framing zero
with respect to the fiber. As usual. we do not. draw the 3-handle H; of the round
‘2-handle. which is forced to be attached in a. way that. it completes the fiberwise
2—handle attachment. The difference between the even and odd cases is then some-
what. unplic'it; it. is distinguished by the two possible ways that the curve, ", might
be mapped onto itself by a self-diffcomorpliism of the fiber determined by the mon-
odromy. If 7 is mapped onto itself with. the same orientation. we have an even round
2—handle. and an odd round 2-handle if the orientation of 7 is reversed. The reader

can also refer to the relevant monodromy discussion after the proof of Theorem 4.2.3.

(31

0 a 0 a
v U :i-ll ' U 3—l'1

( l
t)
a a».
V v v

 

(O

9 9

Figure 4.2: Left: an even round 2-handle attachment to D2 x T2. Right: an odd round
2-handle attachment to an elliptic Lefschetz fibration over a disk with two Lefschetz singu-
larities. Red handles make up the round 2-handle.

The upshot of using round 2-handles is that one can depict any Lefschetz fibration
over a disk together with a round 2-handle attachment via Kirby diagrams explicitly
as in the Lefschetz case [40]. One first draws the Lefschetz 2-handles following the
monodromy data on a regular diagram of D2 X 29 (where g is the genus of the
fibration) with fiber framings —1 , then attaches Hg with fiber framing 0 and includes
an extra 3-handle. We draw the Kirby diagram with standard l-handles so to match
the fiber framings with the blackboard framings, which can then carefully be changed
to the dotted notation if needed. Importantly. it suffices to study only these type
of diagrams when dealing with broken Lefschetz fibrations on near-symplectic 4-
manifolds, as we will prove in the next section.

To illustrate what we have stated above, let us look at the following two simple
examples in Figure 4.2. Since the first round 2-handle is attached to a trivial fibration,
7 is certainly mapped onto itself with the same orientation, and therefore it is an even

round 2-handle. For the second one. we express the self-diffeomorphism of the 2-torus

62

fiber induced by the monodromy ,u by the matrix:

and the curve 7 by the matrix [1 0V. Thus p. maps 7 to —",v, and this yields an

odd round 2—handle attachment. Both of these examples will be revisited later.

4.2.3 Simplified broken Lefschetz fibrations

The complexity of the topology of broken Lefschetz fibrations lies in round cobor-
disms. Our goal is to establish an existence result of much simpler broken fibrations,

which can be associated to any near-symplectic 4-manifold.

Definition 4.2.1 A simplified broken Lefschetz fibration on, a closed 4-mantfold X
is a broken. Lefschetz fibmtion over 52 with only one round singularity and with all

critical points on the higher side.

Since the total space of the fibration is connected. the “higher side” always consists
of connected fibers. The fibers on the higher side. have higher genus whenever all the
fibers are connected. while in general the term refers to the direction of the fibration.

\Ve. shall need the following lemma:

Lemma 4.2.2 Let X mlnu't a directed broken Lefschetz filn‘atlo-n f : X —+ 8'2. then
there artists a. new broken Lefschetz fibmtion on. f’ : X —+ $2, 'ieherc all the Lefschetz

siiii/alanifics are contahzcd in. the higher sitlc.

Proof: To begin with, we can perturb the ("lirccted fibration so to guarantee that:
it is injective on the circles of the round locus. Thus the fibration can be split into

a Lefschetz fibration over a disk (the lower side), to which we consecutively attach

63

round l-handles, and then we close the fibration by another Lefschetz fibration over
a disk (the higher side).

To simplify our discussion. for the time being assume that the fibers are all con-
nected, so there is the lower genus side X, with regular fiber F,, the round handle
cobordism W, and the higher genus side X,, with regular fiber Fh. Let the genus of
the regular fibers in the lower side be 9. The standard handlebody decomposition of
X, consists of a O-handle, 29 l-handles and some 2-handles one of which corresponds
to the fiber, and the rest to the Lefschetz handles in X, [40]. By our assumption,
H" is composed of ordered round l-handle cobordisms l/V, U ill/2 U - ' . U Wk, where k
is the number of circle components in the round locus. Let us denote the lower side
boundary of W, by 8-1V,- and the higher side by 0+ H}.

Consider X,Ul'l', . which is obtained by adding a round l-handle R, composed of a
l-handle II, and a ‘2—handle H3. The ()(X,U1'V1) 2 0+ M", 2 (1H) is the total space
of a genus g + 1 surface bundle over a circle. “"6! can make sure that the vanishing
cycles of the Lt-‘fschetz ‘2-handles in X, sit. on the fibers of the genus g fibration on
0..\",. Moreover. we can assume that the bisection which is the attaching region of It,
misses these vanishing cycles. This means that H, and Hg do not link with any one of
the Lefschetz 2-lmmlles in X, but only with the 2-handle (‘()I’l‘t"SI’)()ll(llllg to the fiber
and possibly with. some of the l-haudles cm‘responding to the genera of the fiber.
We can rearrange the handlebody prescribed by the broken Lefschetz fibration on
X, U H] by another one where first II, and [[2 are attached to the standard diagram
of I)2 X [7,. and the Lefschetz Q—handles are attached afterwards. Having modified
the diagram this way. now we can assume that the Lefschetz '2—handles are attached
to ()(X, U W,). which can be pulled to (ill?) via the fiber preserving difi'eomorpl’iism
between chili, and 6) HQ. The fiber trainings of these ‘Z-handles remain the. same,

and illt‘I‘t‘fUI't’ they are still Lefschetz.

(54

Inductively, one slides the Lefschetz 2-handles so to have them attached to 8(X, U
W, U W2 U - - - U Wk) = (9(X, U W) = —8Xh. Higher side Xh together with these 2-
handles is equipped with a new Lefschetz fibration of genus g+k (which is the same as
the genus of 15),) over a disk. Hence we obtain a new handlebody decomposition which
describes a new broken Lefschetz fibration on X, with all the Lefschetz singularities
contained in the new higher side. It is left to the reader as an excercise to verify that
a similar line of arguments work when X, has disconnected fibers. EJ

Given a near-symplectic form on a closed 4-manifold X , Perutz [66] and Taubes
[79] independently showed that one can obtain a new near-symplectic form on X in
the same cohomology class but. with connected round locus. The meat of the next

theorem is this observation and the Theorem 4.1.3.

Theorem 4.2.3 On any closed near-sji/mplcctic 4-mtmifold (X. (.0) possibly after T8-
placing a) with. a near-smuplcctic for-m w' 'ur'ith’i'n the some cohomology class, one
can. find a 71ear-symplectic broken Lefschetz pencil. which yields a simplified near-

symplectic h’rolrc'n. Lefschetz filmition on (I. hlom-up (Yuj’) of (.Y.w').

Proof: Replace to with a. near-symplectic of with connected ZWJ. Theorem 4.1.3
shows that. there is a. lntoken Lefschetz pencil compatible with this near-symplectic
form. so it should have only one round handle singularity. Symplectically blow-up the
base points to obtain a near-symplectic broken Lefschetz— fibration f on the blow-up
-\ of X. Apply the above lemma to get a simplified Lefschetz fibration on X, which
also supports the nem-symplcctic struetiu'e since the fibers are unchanged and still
symplectic under the modification described in the proof of Lemma 4.2.2.

The exceptional spheres appear as '2-handles linked to the higher genus fiber com-
ponent, all with framing -1. and not. linking to each other or to any other handle.
The modification in Lemma. 4.2.2 is performed without involving these handles. so

their linkings and framings remain the same. Since these. represent the exceptional

65

spheres, we can symplectically blow them down to obtain a new Lefschetz pencil on
X, with the desired properties. E]

The simplified broken Lefschetz fibrations now can be represented by using the
handlebody diagrams described in Subsection 4.2.2. Examples are given in the next
subsection.

It is no surprise that the monodromy representations of these fibrations are also
simpler than usual. Here we include a brief digression on this topic: Let M apW(Fg)
be the subgroup of M (172(F9) that consists of elements that fix an embedded curve
7. up to isotopy. Then there is a natural homomorphism: 07 from M apa,(Fg) to
M ap(F‘,,_1) or to M (1])(1791) x .\[(1.p(13:,,2) depending on whether 7 is nonseperating
or separating Fg into two closed oriented surfaces of genera g, and 9-2. Define 89
to be the set of pairs (“.7") such that n E .lltq.).,(F,,) and p. E l\'er((;)w). Recall
that when the fiber genus is at least. two, the gluing map that preserves the fibers is
determined uniquely upto isotopy. Hence. given any tuple 01,“) E S = Lil/33 8g, we
can construct a unique. simplified broken Lefschetz fibratimi unless 7 is separating
and there is a g, S 1. Otherwise. one needs to include the data. regarding the gluing
of the low genus pieces carrying genus U or germs 1 librations.

If the fibers are connected. the map (3)., : Ala/)(Fg) —+ ,llup(Fy_1) above factors as
1.72, : d/opU‘b) ——+ .‘l/(I])(I*1,\ N) and (:7 : .ll(1.p(l'—:q\.\") —+ .l’ll'u.])(F,,_1). where N is all
open tubular neighborlmod of 5. awav from the other vanishing cvcles. (The middle
group does not. need to fix the boundaries.) The map 1.: has kernel isomorphic to
Z wthe framing of the ‘2-handle of a round l-handle. \Vhen we have a simplified
BLF. the kernel of g is isonun'pltic to the braid group on F,, ,1 with 2-strands. by
definition. This gives an idea about the cardinality of 8‘. and in turn about the

cardinality of the family of broken Lefschetz. fibrations on smooth 4-manifolds.
Remark 4.2.4 If one has more than one round 2-hmullc involved in a broken, Lef-

66

schetz fibration, we may or may not be able to draw the Kirby diagrams as above. This
is due to the fact that after each round cobordism, we obtain a new fiber, which does
not need to simply ‘sit on the blackboard’. If one draws the diagram from the lower
side; the 2-handle of a round l-handle might link with the l— and 2— handles of
other round l—handles. To have a complete diagram, one would also need to pull the
Lefschetz handles from the higher side to this diagram; but framings of both 2-handles
of round 2-handles and those of the Lefschetz handles coming from the higher side

all together are harder to determine.

4.2.4 Examples

In this subsection we provide examples of simplified broken Lefschetz librat ions. The
examples are chosen to span various types of fibrations: with even round locus. odd
round locus. connected fibers. disconnected fibers (on the lower side). and finally those
which do not support any near-symplectic structure. The. near-symplectic examples

we present here are used in later sections.

Example 4.2.5 The Figure 1.3 describes a near-sytnplectic broken Lefschetz fibra-
tion on 8'2 x 1.9 #51 x 5'3, with lower side genus equal to g and higher side genus
increased by one via an even round l—handle cobordism. We call this liin‘ation the
step fibril/ion for genus g. To identify the total space. first use the O-framed 2-
handle of the round 2—handle to separate the 2-handle correspomling to the fiber.
Then eliminate. the obvious canceling pair. and note that the remaining l-handle
together with the 3-handle of the round 2-handle describes an S" x S3 summand.
As the rest of the diagram gives 5"“) x 29. we see that the total space is as claimed.

In several aspects. the round handle cobordism H' in the step fibration is the
simplest. possible cobordism. Here. not only Hill" are products of Riemann surfaces

3g and 53,,“ with S". but also it itself is the product of 5'1 with a. 3-dimensional

(57

 

be—WECD 9: "Q ‘U..;:::

Figure 4.3: The step fibration on 52 x 29 #31 x 53.

Q
I J 0

J

 

F

(
é

cobordism from 29 to 29“ given by only one handle attachment. We refer to
these type of cobordisms as elementary cobordisms. The round handle cobordisms in
Example 4.1.2 are all elementary.

When 9 = 0 we can obtain a more general family as in Figure 4.4. These describe
broken Lefschetz fibrations obtained from a trivial torus fibration and a trivial sphere
fibration over disks and an elementary round handle cobordism between them. The
fibrations we get are precisely the near-symplectic examples of [9], and historically the
first examples of near-symplectic broken Lefschetz fibrations over 32. After simple
handle slides and cancellations, one ends up getting a diagram of the connected sum
of an S2 bundle over 32 with Euler class k and an S1 x S3. Thus for even k we get

52 x SEQ-#51 x S3 and SQA>ZSQat7tS1 x S3 for odd k.

Example 4.2.6 In Figure 4.5 we describe a family of simplified broken Lefschetz
fibrations with odd round singularity. We claim that for even k the total space is
S2 x S2 and for odd I: it is CIP2#@F’2. In order to verify this we prefer to use

the diagram with dotted notation on the right of the Figure 4.5. Let H2 be the

68

 

 

 

Figure 4.4: A family of near-symlectic BLFs over 5‘2 (left), and the diagram after the
handle slides and cancelations (right).

2-handle of the round 2-handle, given in red and with fiber framing 0. Using H2,
first unlink all the 2-handles from the top l-handle, and cancel this l-handle against
H2. Then slide the +1-framed 2-handle over the —1-framed 2-handle to obtain the
third diagram in the Figure 4.6, and cancel the surviving l-handle against the (—1)-
framed 2-handle. Finally cancel the remaining unlinked O-framed 2-handle against
the 3-handle. The result follows.

For k = 0 this is Perutz’s button example in [64]. Moreover, when k = —1 the
blow-down of this exceptional sphere yields a near-symplectic broken Lefschetz pencil

on C1?” .

All the examples we discussed so far had nonseparating round 2-handles; in other
words, in all examples all the fibers were connected. However separating round 2-
handles arise quite naturally when studying broken fibrations on connected sums of

near—symplectic 4-manifolds, as illustrated in the next example.

Example 4.2.7 Since b+(2CfP2) : 2, there exists a near-symplectic form on this

69

f3
, t
\/
C

 

 

Q0

e 7k

Figure 4.5: A near-symplectic BLF for an 82 bundle over 5'2 with Euler class k. On the
right: l-handles are replaced by dotted circles.

 

 

 

 

 

non-symplectic 4—manifold. We will construct a near-symplectic structure which re—
stricts to a symplectic structure on each (Clip2 summand away from the connected sum
region, through broken Lefschetz fibrations. Take the rational fibrations fi, 2' = 1,2
on two copies of CIP2#EII52, with —1 sections. Consider a fibration f = fl U f2 on the
disjoint union of these two, by simply imagining them ‘on top of each other’. Now in a
regular neigborhood of a fiber of f , introduce a round 1-handle so to connect the dis-
joint sphere fibers. The result is a broken Lefschetz fibration f’ : 2(le1’29‘3é2m2 ——> 52
with two exceptional spheres. Let h be the Poincare dual of the sum of —1 sections.
Then h evaluates positively on each fiber component of this fibration, so there exists
a near-symplectic structure compatible with f’ with respect to which the two —1 sec-
tions are symplectic. Blowing-down these two sections we obtain a near-symplectic
broken Lefschetz fibration on 2(3113’2 with the proposed properties. A diagram of this

fibration is given in Figure 4.7.

Remark 4.2.8 The very same idea can be applied to connected sums of any two near-

70

Ms

ll

22
22

 

% —1
0C) \ U 4-h
b 3-l1

4-h

 

“Q 9
0 00 C

Figure 4.6: Identifying the total space of the BLF in Figure 4.5.

 

 

 

symplectic broken Lefschetz fibrations over the same base, say by connect summing
in the higher genus sides (also see [65]). For the diagrams of such fibrations over
SQ, abstractly, first slide a 2-handle F1 corresponding to a fiber component over the
2-handle F2 corresponding to the other fiber component. Then regard F2 as the 2-
handle of a round 2-handle. and add an extra 3-handle to the union of two fibration
diagrams. This way we obtain. a connected sum model for our (broken) Lefschetz

fibration diagrams.

Using similar techniques, we can also depict diagrams of broken Lefschetz fibrations
which do not necessarilly support near—symplectic structures. We finish with a few

examples of this sort:

Example 4.2.9 As discussed in [9] a modification of g = 0 case in Example 4.2.5,
yields a broken Lefschetz fibration on 8“. This can be realized by gluing the round
cobordism W to the higher side fibration over D2 by twisting the fibration on 6+W =
T3 by a loop of diffeomorphisms of the T2 fiber corresponding to a unit translation

in the direction transverse to the vanishing cycle 7 of the round 2-handle [9]. As a

71

 

 

( .
u 70—] U iii—h } 4-h

( a tWO 4—h’S

9-1
L J

L J

Figure 4.7: A near-symplectic BLF on 2CP2#2W2. The round 2-handle separates the
sphere fiber on the higher side into two spheres on the lower side.

 

 

 

0

@fi if:

x 74)

Figure 4.8: A broken Lefschetz fibration on 84.

 

 

 

 

result of this, the 2-handle corresponding to the S2 fiber of the lower side is pulled
to the blue curve in Figure 4.8. The diagram then can be simplified as before: Use
the 2-handle of the round 2-handle to separate the 2-handle corresponding to the
fiber, and then proceed with the obvious handle cancelations.

It would also be interesting to note the existence of a broken Lefschetz fibration
on #n S1 x S3, for any n _>_ 1, which do not admit achiral Lefschetz fibrations for

n 2 2 [40]. Taking the product of the Hopf fibration S3 —> 52 with 51, we get a

72

 

(Fig—M Q+C 4 3—h’s
(J F It U 2 as“

 

Figure 4.9: A broken Lefschetz fibration on S1 x S3 # S1 x 53.

fibration S1 x S3 —) 5'2 with inossential torus fibers. Then the connected sum model
discussed in the previous example allows us to construct a fibration on any number

of connected sums of S 1 x 53 s. In Figure 4.9 we give a. diagram for the n = 2 case.

4.3 Some near-symplectic operations

We move on to presenting some surgical operations that give new near-symplectic
broken Lefschetz fibrations from old. The first one generalizes the symplectic fiber
sum operation Theorem 1 to the near—symplectic case, which can be set as a fibered
operation. The second operation relies on an idea of Perutz [64], who modifies the
near-symplectic broken Lefschetz fibration on the same 4-manifold. Both can be
performed in general as near-symplectic operations, without any mention of broken

fibrations.

4.3.1 Broken fiber sum

Let (X 2', ft) be broken Lefschetz fibrations, and F1- be chosen regular fibers of genus
g > 0, 2' = 1,2. Choose regular neighborhoods Ni = fi’1(D,-) of E, and without
loss of generality, assume 91 - 92 : k is a non—negative integer. Then we can obtain
a new 4-manifold X = X1 \ Nl U W U X2 \ N2, where W is a composition of 1:
elementary round 2-handle cobordisms. These cobordisms being elementary implies
that the 2-handles of the round 2-handles can all be pushed onto a regular fiber F1.
The resulting manifold is uniquely determined by an unordered tuple of attaching
circles (71, - -- .74.) of the round 2-handles involved in H”, together with the gluing
maps (91 : (3le ——> 8+W and 992 : 8X2 ——+ E)- W' preserving the fibratious. (Recall
that these gluings are unique up to isotopy when the fiber genus is at least two.)
Hence we obtain a new broken Lefschetz fibration (X. f) that extends the. fibrations
(X,- \ N,. le.\,\.r\’_) by standard broken fibrations over the elementary cobordisms.
we say (X._f) is the broken fiber sum of (X1,f1) and (X2.f2) along F1 and F2.

determined by 5.1, - -- ,1“. and 01, 02.

Theorem 4.3.1 If (X, f.) are near-symple(:tzic broken Lefsclu-xtz fibmtions. then
(X. f) is a, n(:(ir-syinplectic broken. Lefschetz fibmtion. Moreover, git-‘07). arbitrarily
small collar nmighborhoods N!- of é)(i’\'l-) in, Xi 1. we can choose to so that. w] XIWI =

W'll.\'1\.\71 and to|_\.__)\.,\=2 = (fngX'QW-é. where C 28 some constant.

Proof: Let k be as almve. Take step fibrations on 82 x 2g #81 x 5'3 described
in Example 4.2.6 with g = _og. 92 + 1......(12 + k = 91. Take the fiber sum 52 x
292 #81 X 5'3 along a high genus fiber with 8'2 x 292+1#S] X S3 along a low genus
fiber. Then take the fiber sum of this new broken libration along a. high genus fiber
with 52 x 2,12” #5” x S3 along a. low genus fiber. and so on. until 9 2 (J2 +k. Denote
this manifold by ll". Since the broken Li'ri'sehetz fibrat ion on W admits a section,

it can be equipped with a near-syn]pleetie structure. Hence the broken fiber sum

74

 

of (X1,f1) and (X2,f2) along F1 and F2 is obtained by fiber summing the former
along F1 with W along a lower side fiber, and the latter along F2 with W along
a. higher side fiber. Using Theorem 1, we can make these fiber sums symplectically,
after possibly rescaling one of the near-symplectic forms an, i = l. 2. It is clear that

when k = 0 this is the usual symplectic fiber sum. E]

Remark 4.3.2 If (XML) for i = 1,2 are Lefschetz fibrat'ions over S2, then one
can. depict the Kirby diagram. of the broken fiber sum (X, f) in terms of these two
by using Lemma 4.2.2. Since the round cobordism in the broken, fiber sum consists of
clemenatry cobordisms. all the 2-handles of the round 2-handles and the Lefschetz

handles of the lower genus fibration can be drawn. on, the higher genus fiber directly.

Remark 4.3.3 Forgetting the fibrations, we can. describe the abore construction. for

. any ncar—syinplcctic (X...o,) containing symplcctically embedded surfaces F,- with
I) 1 _ , . . . . v .

Ff = F; = (l. illorcovcr it." is possible to form a. cobordism snnilar to W in gen..-

cral urhcn Ff = —F._‘,2 # 0 to handle the most geneml situation.

'l‘opological inyariants of X are easily determined. For examine if X,- are. simply—
conuected and at least one of them admits a. section. then using Seifert—Van Kampen
theorem we conclude that X is also siml)l_\«'-c(_)nn(grcted. The Euler characteristic and

signature of X can be expressed in terms of those of X1 and X2 as:

C(.\') =(‘(.\’1)+('.(.\'3)+ 2(_(]1+ _(jg) — 4 . 0'(.\') Z (7(.\'1) ‘i‘ (7(4Y2). (4.4)

where g, is the genus of F, . for i = 1. ‘2. Therefiire the liolomorphic Euler characteris-
tic \;.(X) = \h(X1)+ \;,(X3)— 1 —(y] +.(/2)/'2. It follows that if X, and X2 are all'nost
complex manifolds. then X obtained as their broken fiber sum along F1 and F2 is
almost cmnplex if and only if A' E .(11 +312 E 0 (mod ‘2). Lastly note that the broken

fiber sum operation might introduce second homology classes in X that do not come

‘J

(fl

Figure 4.10: Vanishing cycles in the. Matsumoto fibration.

from X,, in addition to the usual Rim tori. This 1_)11eiiorlieri()11 occurs for instance when
some 7, match with relative disks in X2 \ N2 to form an immersed sphere 51- Then
the torus T‘. which corresponds to a submanifold o. X SI C 0(X2 \ N2) ‘5 F2 X S].

where o, is the dual circle to w, on F2. intersects with S, at one point.

Example 4.3.4 Take. X1 = 5'2 x 'I’3#4C_ll’2 with the Matsunmtxo libration f1 : X; —+
SB. and X2 = 5'2 x 5'2 with the trivial rational fibration f2 : X 2 ——> 82. The former
is a genus two tibrat ion and has the global monodromy: (t31'."_3-_;.3_-5d.4)2 = 1. where the
curves til. 43-2., {33 and 134 are. as shown in Figure 4.10.

If we denote the standard generators of the fundamental group of the. regular
fiber 23 as ((1.!)1.(t2.[)2. then the curves .3,- are base point homotopic to: ,131 2 (Mb).
:32 : (ritual—lbl'l : (l-2[)3(12_lb3—l, it; :t)g(121)2_1(11. t3; : bgltgtllln.

Hence 7rl(/\'1) : nl($.,) / (.31. :32. 133.13,) is isomorphic to

W1(X1) = ((11.1)1.(13.t)-_) I tub-2 : [(11.1)1] = [(12.03] = t)2(l3’);1(ll : 1).

Now take the broken fiber sum of (thl) and (X2.f2) along regular fibers F1
and [72. where ml 2 (1,. ”,2 : 1);). The gluing map 01 is unique. and we take (Dz

as the identity. Thus we get a new 4-manifold X and a near~symplectic broken

TG

Lefschetz fibration f : X —+ S2 with two even round singular circles. Note that
1n(X1\N(F1)) 15:“ 7n(X1), and 771(X2\N(F-2)) = 1, since there are spheres orthogonal
to each fiber F,- in X i. From Seifert-Van Kampen’s theorem and from the choice of

7, in the broken sum, we see that. .
771(X) = ((ll,b1,(lg,b2|b1b2 =[(11,b1]=[(12,b2]= b2a2b§1a1= a1: b2 2

Thus WAX) = 1. On the other hand, e(X) = e(X1)+e(X2)+2(g1+g2)—4 = 8, and
0(X) = 0(X1)+o(X2) = —4. Hence. X is homeomorphic to CP2#5@2 by Theorem
2.0.1. Moreover we obtain four distinct syn‘iplectic sections of self-intersection —1 in
( X , f) which arise. from the internal cormected sum of four parallel copies of the self-
intersection zero section of 82 x 52 U ll' and the four —1—sections in the Matsumoto
fibration in the broken liber sum. Symplectically bl(_)wi11g-down these sections, we
get a near—symplcctic structure with two even round circles on a homotopy 5'2 x 52,
together with a broken Lefschetz pencil supporting it.

Different choices of (3);; would simply change the self-intersection of these sections in
X, but the homeniorphism type of X would not change. (Alternatively we could take
(X2. f;) as a rational fibrat ion on a. llirzebruch surface with section of self—intersection

A: and fix the gluing 02 as the identity.)

“that makes the broken liber sum operatitm interesting is that, apriori. gluing

formulae can be given for the invariants.

Proposition 4.3.5 Let (X. f) be the broken fiber sum. of(X1.f1) and (X2. f2) along

F1 and F2 with g, — _(]-_) = h' 2 0. determined by the tuplc (71. - -- fin.) of circles on.
F1. for j = 1 ..... I". Denote the J -tih clcmcntmy cobordism corresptm.ding to 1.] by
”1° and Poin('(Ixi'c-chsclitt3 duals of 7.1 on. F, by (71-. Then. we hare

\

I‘Alx'f : LAI-‘Xll e\"i-f1l.\',g.vl 0 L1 0 ' ' i O Lk O I‘ll/«\iz‘M‘V-zdz'

x: 3 N2

«.1
«I

where Lj corresponds to wedging with cj under the Piunikhin-Salamon-Schwarz iso-
morphism (defined for a given admissible Spin” ) between Floer homologies and sin-

gular homology.

Proof: The broken Lefschetz fibration (X, f) can be decomposed as

(X,f) = (X1\N1» f1lX1\.~'1)U(l’l"1ePllU' ‘ 'U(le,Pk)U(X2\N2, f2lx-2\N2)-

where each W,; is equipped with the elementary broken fibration pi. In [65] Perutz
shows that on each (lVi, ft), the LM invariant acts as described in the statement of
the proposition. Thus the. above formula follows from the fact that LM invariants fit,
in a fibered field theory. [I]
It should be possible to formulate a similar statement. for the Seiberg-W'itten invari-
ants of X, using Seiberg—VVitten monopole F loer homology [49].

For what follows we. will be interested in a particular case where the result of a.

broken fiber sum (X. f) (resp. X) has trivial LM invariants (resp. SW invariants):

Proposition 4.3.6 Let (X f) be the result of a. broken fiber snm of (X1, f1) and
(X2. f2) along F1 and Hg with, gem-3771. {11 > gg. If any round 2-handle introduced in
the broken sum is attached to a nonseparating earre on F1 which is also a vanishing
cycle for a Lefschetz handle in f1 then LM invariants of (X, f) are all zero. If

h+(X) > 1. then the SH" invariants of .X' are also trivial.

Proof: The. fibration fl is isotopic to identity on X1 \ N, . The assumption. provides
an essential sphere 8 obtained from the 2-handle of the round 2-handle and the
Lefsclmtz handle mentioned in the statement. The 'equator‘ f." is an essential curve
on F1. so there is a dual circle a that intersects it positively at one point. Since
the monodromy is trivial. this a sweel')s out a. torus in Fl x S1 :2 0(X1 \ N1), which

has self-intersection zero. If we blmv-down S. downstairs we get an embedded torus

‘1
OO

with self-intersection +1, violating the adjunction inequality for Seiberg-Witten. It
follows from the blow—up formula that SW x E 0. For the LM invariants of (X , f),
obseve that the map on the elementary cobordism is equivalent to contractng along

7, but the continuity (from 'y to the nodal point) argument for quantum cap product

 

[63] shows that this map is trivial as the sections of the Hilbert scheme miss the
nodal points. [3
However, there are examples when the result. of a. broken fiber sum has nontrivial

LM and SW invariants:

Example 4.3.7 Let X1 = S2 x 29“ and X2 = 82 x 29 with projections f,- on
the 5'2 components. The broken fiber sum (X. f) of (X 1. f1) and (X2, f2) along
the fibers 29“ and $9 is the same as 32 x 29 #51 x S3 equipped with the step
fibration. Adapting the Example 5.1.3 from [67)]. we see that. (X.f) has nontrivial
LM invariants. It also has nontrivial SW invariants (cf. [58]). calculated in the Taubes
chamber of a compatible near-synmlectic form. (Since both S2 x pt and pt x 22 are
symplectic with respect to these near-symplectic structures. the near-symplectic forms
can be chosen so that they are limnologmis to the product synmlectic form. Therefore

SW invariants are. computed nontrivially in the same. chamber.)

Remark 4.3.8 A similar argument can. be used to calculate 5' ll" nontrivially. in. gen-
eral for the broken fibrr sum of any symplectic Lefschetz fibration (Y. f) of genus g
and b+(Y) > 1 with the. trivial fibration on 52 x 39.“. The same type. of handle cal-
culus shows that the resulting manifold is Y#Hl X 53. Since Y has nontrivial S W. so
does i'#sl x S3 [58]. Moreover in [58]. the authors shows that the dimr—rnsion of the
moduli space for such. a nontrivial solution increases to one. thus Y#Sl X 53 is not of
simple type. Hence. it is an in.trig-uing ([tlt’S'l‘ltflt to determine 'urhcther the broken fiber

sum of two simply-connected 4-nmnifolds can result in a 4-manifold with nontrivial

79

Seiberg- Witten invariants, which is likely to be of non-simple type. We currently do

not have such an example or a proof that shows this can not happen.

4.3.2 Button addition

In [64] Perutz discusses a local modification. called button addition, around a regular
fiber of a broken fibration which locally increases the genus by one, while introducing
two new Lefschetz singularities and an odd round handle singularity, and resulting in
a homology equivalent 4-manifold with the same fundamental group. We will first
show that this modification can be made indeed without changing the underlying
smooth 4-manifold X.

The construction makes use of the fibration described in terms of Kirby diagrams
in Example 4.2.6 with k = 0. Taking out. a regular neighborhood of a sphere fiber
from the lower side, we are left. with a broken Lefschetz fibration over D2. which
precisely has the diagram given in Figure 4.2 on the right. Let us (lenote this piece
by 1?. and call it the button. Now given any broken Lefschetz fibration f on X . take
a regular neiglilmrhood N of any regular fiber F" , fibered trivially over D2. Locally
there exist self-intersection zero disk sections both in N and in B. \X'e. simply take
the section sum of these two fibrations so to obtain the obvious broken Lefschetz
fibration N LI [3’ ——i D), which can be. glued back in X \ N to obtain a new broken
Lefschetz fibration f’ over S2. Furthermore. if X has a section of self-intersection s.
we can choose the local section in A7 as the restriction of this one so f’ also admits a
section with self—intersection s. Using our handlelmdy diagrams and analyzing this

operation a bit carefully. for a general X we see that:

Theorem 4.3.9 Let j : X —+ 82 be a broken Lefschetz fibration compatible with
a near-symplectic structure to, and I“ be a. chosen. fiber around which we attach a.

button. The button addition does not change the (lifi'eomorphism type of X. and the

80

resulting fibration f’ : X —> 52 supports a near-symplectic form w’ which restricts
to the original near—symplectic structure to away from F. Conversely, if there is a
button in a. simplified near-symplectic broken Lefschetz fibration, one can recover a

genuine symplectic structure on X.

Proof: In Example 4.2.6 we have shown that the total space of the button fibration
is S2 x S2, where k = 0. When we take out a regular neighborhood of an S2 fiber
and a regular neighborhood of the section, the remaining piece B can easily be seen
to be D4. The button addition amounts to taking out. the local disk section and
gluing in B. Trivializing N as l)2 x 5.39, where g is the genus of F , we express the
gluing region as the union (9])2 x 02 U D2 x 802 = S3. The horizontal gluing along
I)2 x "002 is determined uniquely by the self—intersection of the section, whereas the
vertical gluing is determined by the fact that the Inonodrmuies of both fibrations are
isotopic to the identity on ("l/)3 x 239. These certainly agree on the corners, so the
operation boils down to taking out a I)4 in the original manifold X, and putting it.
back in by a. differ)!norphism of 01.)" = S3 which we have argued to be isotopic to the
idmitity. This extends over the D" to give back X.

Alternatively. take 5'2 x E” with the projection map onto the first component. We
can then take the. sectimi sum of this fibration with the button filiiration 5'2 X 5'2 -—> 5'2
along self-intersection zero sections. The handlebody diagram of the resulting 4-
mauil’old and the broken Lefschetz tibration on it is similar to the one given in Figure
4.5 l:)efore_, except that the higher side fiber now has genus g + 1. The same calculus
as in Example 4.2.6 verifies that the total space is dillcomorphic to 32 X 239. Since.
there is a section. this fibration admits a compatible near—symplectic structure. The
button addition is equivalent to fiber summing this broken Lefschets fibration along a
regular fiber in the lower side (which has genus g) with the broken Lefschetz. fibration

f on X along F . Since the fibers are symplectic, we. can alter the near-3ympleetic

81

structure on 5'2 x 29 so that the fiber sum can be made symplectically (see Theorem
1). Hence we obtain a new near-symplectic form w’ supporting the new broken
Lefschetz fibration f’ : X ——> SQ, and restricting to w on the complement of a chosen
neighborhood of F.

The last assertion follows from the definition of a simplified near-symplectic broken
Lefschetz fibration. C]

Using consecutive button additions one can locally increase the genus of any fiber
of a given broken Lefschetz fibration without changing the ambient 4-manif01d. This
allows us to define another interesting way to generalize the symplectic fiber sum
operation as follows: Let. f,: : X,- —-> 23,-, F,- and k be as in the previous subsection.
W'e repeatedly introduce k. buttonsin a regular neighborhood N2 of F2, such that the
images of round handle singularities are arranged as a nest of ovals. Take a regular
fiber F; of genus 92 + I" with a small enough regular neighborhood \é contained in
the very center of these ovals. and take the symplectic fiber sum of X1 and X 2 along
F1 and I"; to form X 2 X1 \ N, U X2 \ N1, = X1\ N1 U W U X 2 \ Ng. Then we obtain
a broken fibration f : X —> $1#32 which restricts to the fibrations f1 : X,- \ NE ——>
E},- \ D,, but now on H" it is the trivial tibration on F} x [)1 extended by ‘button
filn'atirms‘ —- introducing A‘ new round handle singularities and ‘21: new Lefschetz
singularities. Call (X,f) the lmttonul fiber sum of (thl) and (X2,f3) along F]
and H, which is unithly determined by the choice of gluings o] : ('i)(.X'l\i'\7,) ——> (3+ W
and (02 : (7(X2 \ N2) —+ doll" preserving the fibrations as in the broken fiber sum.

Thus if (X,. f,) are symph-wtic Lefschetz fibrations with regulz—u' fiber genus g1 # 9-3,
then buttoned fiber sum allows us to still take the fiber sum, after replacing one
of the symplectic forms by a. near-symplectic form. Similar vanishing results as in
Proposition 4.3.6 works in this case as welli but we do not know if the resulting

4—111anifold would always have trivial LM or SW invariants.

8'2

4.4 Applications to near-symplectic 4—manifolds
with non-trivial invariants

We now turn our attention to near-symplectic 4—manifolds with nontrivial SW in-
variants (resp. LM invariants whenever fibrations are present). Let us refer to these
as nontrivial near—symplectic 4-manifolds for a shorthand, even though we do not
claim that the SW calculation makes use of the near-symplectic forms. However
when N = 1 we always consider the SW invariant computed in the chamber of the
near—symplectic form.

Let (X,a2) be a near-symplectic 4-manifold with zero locus Z. One of the key
observations that Taubes made in his programme is that if SW of X is nontrivial,
then there is a finite energy J -holomorphic curve (7 in X \ Z which homologically
bounds Z (more precisely, C' has the intersection number one with every linking
2-sphere of Z), where J is an almost complex structure compatible with w in the
complement. of Z [78]. We call this 'I'aubes' curve. Below we show that the converse

to this theorem is not true, together with an analogous result for LM invariants:

Theorem 4.4.1 There are infinitely many pair-arise nonhomcomorphic closed ori-
ented near-symplectic 4-man2folds (Xm, a',,,). m > 0. equipped with. broken Lefschetz

. r 1') .
fihratmns f,,, : A —> .8“ that induce w’m- such that:

(i) Each (X.m_,u2m) admits a Tauhcs' curve, but 8“} E U.

"1

(ii) For each (.X',,,, f,,,) there is an mlmissible Spine structure 5 such. that the as-
sociated moduli space of L-(lgl'fl,'Il_(]'l(7/It match ing in.variants h as non-empty moduli

with. non-"negativc (limension, hut L'i“I(Xm‘/'m) E 0.

Proof: Take Sg#Eg. g 2 1 with the step fibration. Then use the connected sum

model in Remark 4.2.8 to equip Xg = #2(S‘2#SU) with a near-symplectic broken

83

Lefschetz fibration fg : X9 -> 5'2. Let E, E“ 229 be the higher genus side of this
fibration. Take a regular neighborhood D2 x Fh of Fh, where the fibration restrics as
projection prl : D2 x F), —r D2 on the first component. Let 7,, for s 6 8D2 = 31 be
the attaching circles of the fiberwise attached 2-handles, and Z be the corresponding
round singularity. One can find a parallel disk section D of (D2 x Fh,pr1), so that
8D intersects 75 at one point for all 3. One can extend each D to a disk section
D into the round cobordism from the higher side, so that 8D 2 Z. If necessary, we
can perturb the near-symplectic form on X to make D symplectic on X \ Z, and
therefore it is J -holomorphic with respect to a compatible almost complex structure
in X \ Z. Clearly D is a finite energy curve, and the way we constructed it implies
that each C intersects with every linkng sphere of Z at one point. Setting C = D,
we obtain the desired curve. However for any g 2 1, by the connected sum theorem
for SW invariants. SV’VXQ E 0.

To show the second part. let us label the fiber components of the lower side regular
fiber [7, and the two distinct. self-intersection zero sections of f9 on X9 = #‘2(92#Eg)
by Fj and SJ ( j = 1.2), respectively. Then a straightforward calculatirm shows that

the canonical Spin" structure associated to the fibration on X \ Z has
c1(,\" \ Z) = 2171+ 217-2 - (‘29 — ‘2).5'1—(‘2g — 2)S2 — 2 D.

Then the Spin" structure assemiaterl to the class {3 : —F1—F3+(.(1—1).S'1+(g—1)S‘2+[)
has (“1(5) 2 ("1(X \ Z) + QPDW) : 0. So for every fiber component B (i.e. F1, 17-2 or
F;,), we have (c1(5).2> 2 {(23). Moreover :91 < (13,23) < —¥:) is satisfied when
g > 1. 'l‘l'ierefore 5 is an admissible Spin" structure. However Lil/KM] E O as shown
in [65]. Lastly, d(L.‘\l(Xg_fg-)(5g)) = 11(c71(5)2—30(X) —-2e(X)) = §((’)—(l—2(6—89)) 2
4g — 3 2 0.

Setting m = g + 1, we get the infinite families by varying g > 0. D

In general any Gromov type of invariant might. vanish even if the associated moduli

84

space is nonempty. Thus the above result should be regarded as an explicit demon-

stration of this phenomenon.

One might wonder if the class of nontrivial near-smplectic 4-manifolds is closed
under the symplectic fiber sum operation, as it is the case for both near-symplectic

and symplectic classes. We show that this is too much to hope:

Theorem 4.4.2 There are infinitely many topologically distinct pairs of closed near-
symplectic 4-manifolds with nontrivial SW invariants whose symplectic fiber sum T6-

sults in trivial near-symplectic 4-manifolds. The same holds for LM invariants.

Proof: As discussed in Example 4.3.7 and the succeeding paragraph, if Y has non-
trivial SW, then so does Y# S 1 X S3. Take E(n) (say with n > 1) with an elliptic
fibration, and equip it with a symplectic form making the regular'torus fiber T sym-
plectic. Also take 52 x 22 with the product symplectic form. Look at the broken
fiber sum of E(n) with n 2 2 along a. regular torus fiber T with 82 X 22 along a
genus two surface {pl} x 532. where boundary gluings an and (92 are chosen to be
identity. and ”y is chosen to be some. fixed standard generator of 2;. The result is
the nontrivial nez—rr-symplectic 4—manifold X ,, E E (n)# S" x 83.

“’e can then take the slxv'inplectic fiber sum of such X,, and Xm along the higher
side genus two fibers to get X,,.,,, . There are families of disks with their boundaries on
t)(X,, \ MEI-2)) and t)(.\',,, \ .\-’(Eg)) . coming from the broken fiber sum construction in
each piece. hltltclllllg pairs of these. disks give spheres S, with zero self—ii‘itersection,
where s is parametrizeed by the. base, 31 in the gluing region 8'1 X 22 of the fiber
sum. Denote the equator of S. sitting on the fiber sum region by 73,. and consider a
dual circle as on the same fiber. Varying 3 along 5'1 we obtain a Lagrangian torus
T, which intersects each .9", at one point. Thus 30 is an essential sphere in X mm.
Since b+(X,,‘.,,,) > 1. the existence of such a. sphere implies that SW33..." E 0. Infinite

families are obtained by varying n. m. > 1 . For the second part of the. statement, let 11s

85

use Y9 = 52 x Ea # S l x S3 equipped with the step fibration f9 over S2. Then (Y9, f”)
has nontrivial LM invarints. Taking the fiber sum of two copies of (Y , f9) along higher
genus regular fibers, we obtain a near-symplectic broken Lefschetz fibration (X in ,3).

Observe that

l , — ,' V ' .' .
LJ‘I‘\!’].fé — L“[S'2X$g\t~v(gg)-fgl O LAIH’prI O L‘\I‘Q2XEQ\JV(V9)VIQI

..4

where W is a cobordism that consists of an elementary round l-handle cobor-
dism W1 followed by an elementary round 2-handle cobordism W2. So Llllu-‘p,l =
[Al/WM” o Lilwl‘prl. However. under Piunikhin-Salamon-Schwarz isomorphism.
[ii/”22W, corresponds to wedging with the Poincare-Lefschetz dual of 7, the attach—
ing circle of the ‘2-handle of the round 2-handle. Since the round l-handle cobordism
H '1 is constructed in the same way, this 7' can be contracted along ”’1. and therefore
LAM-Hm.2 is trivial. It follows that [ally-tiff; E 0. Taking g = 0.1.2.... we obtain

the desired infinite family. [1

Remark 4.4.3 For the same eramplcs in the proof of :l’heorcm. 4 .412 if one indeed
takes the fiber sum along lou‘cr genus fibers. the result is E(n+ m)# ‘25" X 83. which
again has nontrii'iul S W. Thus the choice. of the fibers in a ncar-symplectic fiber sum.
affects the outcome drastically. .4 natural question that follows is:

Question: If X, are nontrivial near—symplectic 4-manifolds and l", are symplecti-
cally embedded surfaces in F, with minimal genus. is the (symplectic) fiber sum X of

X1 and X2 along I‘] and [7-2 nontrivml'.2

It is known that Lefschetz fibrations over 5'2 do not admit sections of nonnegative
self-intersections. and the self-intersection can be zero only when the fibration is
trivial. In general near—symplectic broken Lefschetz fibrations are not subject to this
constraint. Even when we restrict our attention to neai'-sy111ple(_'tic broken Lefschetz

fibrations on nontrivial 4-manifolds. there appears a difference:

86

Theorem 4.4.4 There are closed simply-connected 4-manifolds which admit near-
symplectic broken Lefschetz fibrations over 5'2 with sections of any self-intersection.
More precisely, for any integer k and positive integer n, there is a near-symplectic
(ka , fnf) fibercrl over 82, with a section of self-intersection k: and with b+(Xn.k) =
ii. If f : X —+ S2 is a nontrivial broken Lefschetz fibration over a nontrivial near-
symplectic 4-manifold X with b+(X) > 1, then any section S of f has negative
self-intersection. There are simply-connected examples with sections of any self-

intersection when bJr = 1.

Proof: In Example 4.2.6 we have constructed near-symplectic broken Lefschetz fi-
brations over 52 which admit sections of any self-intersection k. As the total space
of these fibrations are either 8'2 X 82 or CIP2#@2. the SW invariants are nontrivial.
(Since the near-symplectic forms can be chosen so that they determine the same cham-
ber with the usual symplectic structures. and therefore SW invariants are computed
nontrivially in the chamber of the near-syn1plectic forms.) These provide examples
for the very last part of the theorem. As described in the Example 4.2.7. we can
obtain a near-symplectic broken Lefschetz fibration on connected sums of these fibra—
tions. Using it such copies, we obtain a 4-manifold with b+ = n, which proves the
first. statement. For the remaining assertion, we simply employ the SW adjunction
inequality as in the Lefschetz fibration case (see, for instance [72]). II]

There are various examples of 1'1ons_\_'11'iplectic 4-Inanifolds which have nontrivial
SW invariants. All these examples have b+ > 0. which means that they admit. near-
symplectic broken Lefschetz pencils but not svrnplectic Lefschetz fibrations or pencils.
This can be made. explicit in F intushel-Stern's knot surgered E (12.) examples ['28, 29].
The below result gives near—s_vmplcctic broken Lefschetz filn‘ations on an infinite fam-
ily of pairwise m)nditfeomorphic closed simply-connected smooth 4-manifolds which

can not be equipped with Lefschetz lll)l'i-l.l'l()llb‘ or pencils.

8?

an]

 

 

 

Proposition 4.4.5 For any knot K, E (n) K admits a near-symplectic broken Lef-

schetz fibmtion over S2.

Proof: Think of E (n) as the branched double cover of 5'2 X S2 with branch set
composed of four disjoint parallel copies of 52 X {pt} and 2n disjoint parallel copies
of {pt} x 82, equipped with the locally holomorphic ‘horizontal fibration’ [29]. The
regular torus fiber F of the usual vertical fibration is a bisection with respect to
this fibration. We. have exactly four singular fibers each with multiplicity two. On
the other hand, if M K is obtained by a 0-surgery on a nonfibered knot K in S3,
then there is a broken fibration (no Lefschetz singularities) from S 1 x A! K to T2 as
discussed in Example 4.1.2. One can compose this map with a degree two branched
covering map from the base T2 to S2, such that the branching points are not on
the images of the round handle singularities. What we get is a broken fibration
with four multiple fibers of multiplicity two. which are obtained from collapsing two
components from all directions. An original torus section T of S1 x MK —> T2 is now
a bisection of this fibration, intersecting each fiber component at one point. Both F
and T have self-intersection zero. and thus we. can take the symplectic fiber sum of
E(n) and S" x MK along them to get. E ( n.) K. The multiplicity two singular fibers
can be matched so to have a locally holomorphie broken fibration with four singular
fibers of multiplicity two. This fibration can be perturbed to be Lefschetz as argued

in [‘29]. When K is liberal. we obtain genuine Lefschetz librations. D

88

CHAPTER 5 *

Folded-symplectic 4-manifolds

5.1 Background

5.1.1 Achiral Lefschetz fibrations and PALFs

An achiral Lefschetz fibratio-n. is defined in the same way a. Lefschetz fibration is
defined. except that the given charts around critical points are. allowed to reverse
orientation. In other words. the '2—handles can be glued with framing +1 with respect
to fiber framing, too. Also recall that a Lefschetz pencil is a map f: X\{b1, . . . , bk} —>
S"). such that around any base pol/1t bl it has a. local model _/'(z,.22) = 21/32,
preserving the orientations. and that f is a. Lefschetz fibration elsewhere. An achiral
Lefschetz pencil is then defined by allowing orientation reversing charts around the
base points as well. Critical points or base points with orientation reversing charts
are called vita/alive critical points or ncgatrrc base points. whereas the other critical
points or base points are positive. For a detailed treatment of this topic and proofs
of some facts quoted below. the reader is advised to turn to [40].

A Lefschetz fibration is said to be allowable if all its vanishing cycles are homolog-
ically nontrivial in the fiber. Particularly, we will be interested in allowable Lefschetz

fibrations over 1)2 with bounded fibers. In the literature. this type of Lefschetz fibra-

89

tion having only positive critical points is called a PALF. Similarly, when the critical
points are instead all negative, we will call the fibration a NALF. Lastly note that
the monodromy representation for an achiral Lefschetz fibration can be described in
the exact same way as in Section. 2.0.3; so we can talk about the global monodromy
and representations of any given achiral Lefschetz fibration f : X ——» D2.

N ext. is a standard fact which was first observed by Harer:

Theorem 5.1.1 (Harer [44]) Let X be a 4-mantfold with boundary. Then X ad-
mits an achiral Lefschetz fibration over D2 with bounded fibers if and only ifz't admits

a handlebody decomposition with no handle of index greater than two.

5.1.2 Open book decompositions

An open book decomposition of a 3-manifold M is a pair (B,f) where B is an
oriented link in M . called the binding. and f: M \ B —+ S] is a fibration such that
./"1(/) is the interior of a compact oriented surface Ft C M and 01'} =2 I} for all
t E .9". The surface F = E. for any t, is called the page of the open book. The
monodromy of an open book is given by the return map of a flow transverse to the
pages and meridional near the binding. which is an element [1 E I‘gm, where g is the
genus of the page F. and In. is the number of components of B 2 OF.

Suppose we have an achiral Lefschetz tibration f : X ——> D2 with bounded regular
fiber F. and let. p be a regular value in the interior of the base D2. Composing f
with the radial projectitm D2 \ {p} ——+ (9f)? we obtain an open book decomposition
on ('3.\' with binding 0f‘1(p). Identifying j'"l(p) 2’ F. we can write OX = (0F x
1);!) U f 1(6)!)2). Thus we view 0F X D2 as the tubular neighborlmod of the. binding
B = é‘)_/"l(p), and the fibers over ('11)2 as its tranmted pages. The nnmodromy of
this open book is prescribed by that of "the achiral fibration [44]. In this case. we say

the open book (8 , f laxvg) bounds or is induced by the achiral Lefschetz fibration

90

f: X —+ 0". Recalling that any closed oriented 3-manifold can be bounded by a
4-manifold with only O-. 1- and 2- handles, it is fairly easy to see that any open
book decomposition bounds such an achiral Lefschetz fibration over a disk.

We would like to describe an elementary modification of these structures: Let
f: X —» D2 be an achiral Lefschetz fibration with bounded regular fiber F. Attach
a l-handle to 0F to obtain F’, and then attach a positive (resp. negative) Lefschetz
2—handle along an embedded loop in F’ that goes over the new l-handle exactly
once. This is called a. positive stabilization (resp. negative stabilization) of f. A
positive (resp. negative) Lefschetz handle is attached with framing —1 (resp. +1)
with respect to the fiber. and thus it. introduces a positive (resp. negative) Dehn
twist on F’. If the focus is on the 3-manifold, one can totally forget. the bounding
4-mauifold and view all the handle attachments in the 3-manifold. Either way.
stabilizations correspond to adding canceling handle pairs. so ditfcomorphism types
of the underlying manifolds do not. change. whereas the achiral Lefschetz fibration
and the. open book (‘l<’*('()1'111)()sition change in the obvious way. It turns out. that

stabilizations preserve more than the underlying topology, as we will discuss shortly.

5.1.3 Contact structures and compatibility

A l-form o E Sl‘(.il) on a (‘Zn — l)-di1nensional oriented manifold M is called a.
contact form if it satisfies (1 /\ (do-)"' 1 ¢ 0. A11 ()‘I'tcntcd contact structure on .ll is
then a. hyperplane field E which can be globally written as kernel of a contact 1—form
a. In din‘iension three. this is equivalmrt to asking that do. be nondegenerate on the
plane field 5.

A contact structure { on a .‘S—inanifold .'\l is said to be suppmtcd by an open book
(B. f) if 5 is isotopic to a contact structure given by a l—form a satisfying a > 0 on

positively oriented tangents to I3 and do is a positive volume form on every page.

91

\Nhen this holds, we say that the open book (B, f) is compatible with the contact
structure 5 on M .

Improving results of Thurston and VVinkelnkemper [81], Giroux proved the fol-
lowing groundbreaking theorem regarding compatibilty of open books and contact

structures:

Theorem 5.1.2 (Giroux [36]) Let M be a closed oriented 3-7nanifold. Then there
is a one-to-one correspondence between oriented contact structures on M up to iso-

topy and open book decompositions of ill up to positive stabilizations and isotopy.

Considering contact. 3-manifolds as boundaries of certain 4—manifolds together
with some compatibility conditions is a. current focus of research in low dimensional
topology. From the contact. topology point of view, it is the study of different types of
fillings of a fixed contact manifold. In dimension four. there are essentially two con-
siderations. yet we formulate them for all dimensions: Let (.X'2".w) be a. symplectic
manifold with cooriented nonempty boundary M 2 8X. If there exists a Liouuille
rector field (aka symplectic dilation.) I/ defined on a neighborhood of ('9X pointing
out along 0X. then we obtain a positive contact structure é on (Mt, which can be
written as the kernel of contact 1—for1n o = (”u/'IOX. \Vhen this holds, we say ($1.5)
is the uJ-co'nyer boundary or strongly (.‘(NIY’P‘J‘ boundary of (X..o). For the sake of
entirety. note when l/ points inside. we obtain a negative contact. structure instead.
and in this case we say (31.5) is the tu—concasec boundary of (X, a).

Now if (.\'2”,./) is aInn)st-complex. then the complex tangencies on ill 2 0X
give a uniquely defined oriented hyperplane field. It follows that there is a
l-form a on .\I such that é : Itcro. \Ve define the Lcuifmvn on .‘\I as do|£(~. J)
If this form is positive definite then (M. 5) is said to be strictly .1-(ron.ee.r boundary of
(X. .l). and if it is J -conve.\' for an unspecified .1 (for instance when .1 is tamed by a

given symplectic form). we say (.\/.£) is strictly pseudoconHer boundary. If (Ago). .1)

9‘2

is an almost-Kahler manifold, i.e. a manifold equipped with a symplectic form to and
a compatible almost-complex structure .1 , then it can be shown that strict pseudo-
convexity of the boundary is equivalent to the condition that wlé > O in dimenson
2n = 4. We would like to remark that all these definitions can be formulated in more
generality for hypersurfaces in X 2", not necessarily for 8X only.

For detailed and comparative discussions of these concepts, as well as proofs of
some facts mentioned in the next subsection, the reader can turn to [22] and [‘24].
Also for further basic notions from contact. topology of 3-manifolds such as Legendrian
knots, Thurston-Bennequin framing. or convex surfaces, which we will occasionally

use in this paper, see for example [56].

5.1 .4 Stein manifolds

A smooth function t..'I X —+ R on a conmlex manifold X of real dimension 272. is
called strictly plarisubliarmonic if if: is strictly subharmonic on every holomorphic
curve in X. \V’e call a complex manifold X Stein. if it admits a. proper strictly
plurisubharnu)nic function 1;": X —-> [0.00) (after Grauert [41]). Thus a compact.
manifold X with boundary which is equip1_)ed with a complex structure in its interior
is called compact Stein if it admits a proper strictly plurisliliharn'mnic function which
is constant on the boundary.

Given a function 1:“2X —-> R on a Stein manifold. we can define a 2—for1n “2,. =
—(1J*dz_.":. It turns out that ii} is a strictly 1)lurisiibharmrmic functitm if and only if the
synnnetric form _(}L-.(°. -) 2 tot.(-, .l-) is positive definite. So every Stein manifold X
admits a Kahler structure a'L... for any strictly plurisul)harmonic function in: X ——+
[0. 00). It. is easy to see that the restriction of to,” to each level set L’" ‘1(/) gives a Levi
form on y;~“(l). implying that all nonsingular level sets of i are strictly pseudoconvex

hyperslu‘faces. Thus in this article, we equivalently call a. Stein manifold a. strictly

93

pseudoconvex manifold. Moreover, it was observed in [22] that the gradient. vector
field of ti) defines a (global) Liouville vector field V = Va, making all nonsingular
level sets luv-convex. Hence. Stein manifolds exhibit strongest filling properties for a
contact manifold which can be realized as their boundary.

In this article, we are mainly interested in compact Stein surfaces. Another char-
acterization of these manifolds, which might be called “the topologist’s fundamental
theorem of compact Stein surfaces”, is due to Eliashberg, and was made explicit by

Gompf in dimension four:

Theorem 5.1.3 (Eliashberg [20]; Gompf [39]) A smooth oriented compact
4-manifold with boundary is a Stein surface. up to orientation preserving difleomor-
pliais'ms. if and only if it has a handle decomposition X0 U ll} U U hm. where X0
consists of 0— and l—liandlcs and each li,. l g i _<_ m. is a. ‘2—handle attached to

X, 2 X0 U lil U U li, along a Legendrian circle 1., with framing tl)(L,») — 1.
All structures we have introduced so far meet in the following theorem:

Theorem 5.1.4 (Loi and Piergallini [50], also see [2]) An oriented compact
4-7nrmifold with bo-imdary is a Stein. surface. up to orientation preserving (liflcomor-

pliisins. if and only if it admits a. 17.4 LF.

Througlnmt the article. we give ourselves the freedom of using the prefix ‘ant i. as a
slmrthand. whenever an oriented manifold X admits a structure when the. orientation
on X is reversed: like anti-synmlectic. anti—liiihler. or anti—Stein. For Lefschetz fil')ra-
tions and open books though. we use 'positive‘ and 'negative’ adjectives to distinguish

two possible cases.

94

5.2 Simple folded-symplectic structures

The definition of symplectic (or anti-symplectic) structures can be enlarged as follows
in order to cover a larger family of manifolds, which was shown in [13] to contain entire

family of closed oriented smooth 4-manifolds:

Definition 5.2.1 A folded-symplectic form on a smooth 2n-dimensional manifold
X is a closed 2-form a} such that u)" is transverse to the O-section of A2"T"X , and

1

whenever this intersection is nonempty, can" does not vanish on the hypersurface

H = (MW-1(0), called the fold.

For an oriented X, the kernel of a; on [I integrates to a foliation called null-
foliation. Martinet's singular form .11qu /\ (lyl + ([12 /\ dyz + + (1.17,, /\ dyn on R2"
defines the standard folded-symplectic structure. as every folde<l-syrnplectic form can
be expressed in this way in an appropriate Darboux coordinate system around any
point on the fold. There is also a simple folded-structure that every even dimensional
sphere carries: \Ve think of 5'2" sitting in RZ’HI. then pull back the standard sym-
plectic form (1.171 /\(ly1 + +d.r,, /\dy,, on the unit disk bounded by the equator in R2"
to 5"" by the projection maps along the last. coordinate. and finally glue them along
the fold 5'2" " to obtain to”. This is equivalent to doubling the unit disk equipped
with its standard symplectic form (by reversing the orientation on one of the disks).
“1) call this form the standard folded-symplectic form on 5'3”.

For more on ftiltle<l-symplectic structures. the reader is referred to [14]. [13]. Here
we only consider these forms on Riemann surfaces and conmact 4—manifolds. possibly
with bom'idaries. For the former class. foltled-symplectic forms form an open and
dense set in the space. of 2-forms. whereas in dimension four openness remains but.
the nonvanishing condition implies that they are nongeneric. We say an embedded
surface 2 C X“ is a foldcd-sy'Inplei'tic submanifold of (X. a) if w‘[g is a folded-

syn'iplectic form on 2. Observe that S2 equipped with the standard form obtained

by pulling back dxl /\ dyl embeds as a folded-symplectic submanifold of S4 with the
standard folded—symplectic form defined as the pullback of drl /\ dyl + ([172 A (1312 as
above.

The following proposition provides several examples of folded-symplectic

4—1nanifolds:

Proposition 5.2.2 Let X be a closed oriented smooth 4-manifold and E be a closed
oriented surface. If f: X —-> Z2 is an achiral Lefschetz fibration such that the regular
fiber is a closed oriented surface F which is nonzero in H2(X;IR), then X admits
a folded-symplectic structure a; such that fibers are symplectic and the fold H is an
F —bu-ndle over 81. The fold H splits X into pieces X+ and X_, and f induces
symplectic Lefschet: fibrations on (1X'+,w|,\'+) and on. (—X_,wlx_). respectively. Fur-
thermore, any finite set of sections can. be made folded-symplectic for an. appropriate
choice of .4}. This form, is canonical up to deformation equivalence of folded-syrn plectic

forms.

We will call this type of folded-symplectic structures simple (after Thurston [80]).
Base spaces of the fibrations defined on X+ and —X_ are determined by an arbitrary
splitting 23 : 23+ U 2-. Here we take 2. = I)"2 for simplicity. Observe that the

fibration induces an exact. sequence
7T1(F) —* TF1(1\’)—* 7T1(:] '*—> Ti'()(F) —) 0

It. folhiws that fibers are connected if the base is si1nply-connected. Otherwise we can
define a new achiral Lefschetz fibratitm from X to the finite cover of 2 corresponding
to the finite-index subgroup f#(7rl (X)) in 7?] (E). which has ctnnurcted fibers. Finally,
one can perturb f to get a fibration which has at most one critical point on each fiber.
Hence, without loss of generality; we will assume that the fibers of f are connected

and critical values are distinct.

96

Proof: [Proof of Proposition 5.2.2] Start by connecting all negative critical points in
the base by an embedded arc in the complement of positive critical points, and cover
it by the images of orientation reversing charts so that we get a closed neighborhood
)3- ’5 I)2 of this arc away from the positive critical points. This can be done because
around the regular points we have freedom to take charts of either orientation. After
we reverse the orientation on f “1(2_), the map f : f‘1(2_) -—> 2- defines a negative
Lefschetz fibration. Set 2+ = Z\E_, C = E+flE_, X+ = f‘1(2+), X- = f’l(2_),
and H = f‘1(C). If there are no negative critical points, we can choose 2.. as a
small disk around a regular value which does not contain any critical values. Now let
(3 be a folded-symplectic form on 2 which folds over C, such that it is a positive area
form on 2+ and a negative area form on E- . These forms always exist: For example
take S2 with its standard folded form a)“, and suppose St has genus gt. Symplectic
connect sum the u])per-hemisphere of S'2 with a. closed genus 9+ surface equipped
with a. positive symplectic form, and the lower—henrisphere with a closed genus 9,-
surface e(‘luipped with a negative symplectic form. This yields a foldetl-symplectic
form on 53, folded along C.

We will construct a folded-symplectic form on X by mimicking Gompf‘s proof
which generalizes Tluirston’s result for symplectic fibrations to symplectic Lefschetz
fibrations ([80] , [10]). Let C be a closed 2-for1n on X which evaluates positively on
any closed surface contained in a fiber with the induced orientation. (we have not
made any z-issumptions on the type. of vanishing cycles, so one. might have more than
one closed surface on a fiber if there are separating vanishing cycles.) First we. wish
to define a closed 2-form 7) on all over X which is symplectic on each a, : f_l(y).
for all y E 2.

Let A be a. tubular neighlmrhood of (,7 in S which does not contain any criti-

cal values. Choose disjoint open balls I fr]. around each positive and Vs, around

each negative critical point so that these sets do not intersect f ‘1(.4) in X and
that in appropriate charts the fibration map can be written as f(zl, 22) = 2122 and

f ( :1. 22) = 24ng , respectively. Take the standard forms

“1+1: 2 (13:1 /\ dyl + (112 /\ dyg = —%(lz1/\ (131—91:2 /\ ([32
on UH: and

(4)-,1 = —(l.r1 /\ (Ii/1 + (big /\ (lg/2 = $4131 /\ (1'31 — érlzg /\ ([332

on v.1 for all 1.3!. For any y E _/'(U+,k), Fy O NH; is a J+_k-holomorphic curve,
where J”. is an almost-complex structure compatible with cork. Similarly for any
3; E f (V—J), Fy 0 V4 is J-.,-holornorphic curve, where Jr; is an almost-complex
structure compatible with (4)-]. Having expressed wiry. and w_,, in terms of Kahler
forms, we can take these almost-complex structures as (i, 2') and (—23, II). respectively.
It follows that tart] Fyflfflak is symplectic, so we can extend it to a symplectic form
n)” on the entire fiber and get a)” defined for all points in each f (U+.k) this way. Do
the same for all points in f(l""__,). for every j. Finallv. for all remaining y E 2 take
any symplectic form w,, on the fiber, and rescale every to” we have defined away from
all U+.k and V_‘1 so that they are in the same cohomtflogy class as the restriction
of C to each F,,. Next. cover 3 with finitely many balls 8,. containing at most one
critical value. and whenever they do contain a critical value, assume they are centered
at that point. Reindex [7+]. and l"r__1, and shrink them if necessary to make sure
they lie in f’l(B‘.) for some s. Define us on each f—1(Bs) as the pullback of w+.s,
“L‘s, or toy by 71., where 7‘... is the retraction of f‘1(Bs) to the fiber Fy over the
center of BS, or the union of 1*], either with closure of UH; or with closure of Vflq,
whenever B... contains a positive or negative critical value. respectively. Now we can
glue these forms to construct the ‘Z-form I] we wanted, by using a partition of unity

and that each I]... is coln'nnoh)gous to (If—NB.) as in [40].

98

we claim that can = my+ f ‘(,B) is a folded-symplectic form on X , where r. is asrnall
enough positive real number. can is clearly closed and symplectic in the fiber direction.
It follows that for any noncritical point :r 6 Pg, TTX = TIFy 69 (T xFy)”. Here
[‘03) is nondegenerate over (7} [71,)” for all :r 93 H, implying that for sufficiently
small a, a)“. is nondegenerate on X \ (H U5(U+,3 U V_,5)). On the other hand
wKIUJhS = Ker/4.3 + f*()‘3) and wfilvys = mugs + f*(/3). Therefore for any

nonzero v E TU+,S, we have
tutu-1.510 = ell/("N “)3. + .13(f.('l‘)- ’I'.I'.('v)) > 0.

where g(—, —)+,g is the metric induced from w+,s and Jrg. Likewise, for any nonzero

v 6 TV...“ we will have

an. ..1. m = we. mi. worm—214m) > o,

g(—, —)_,.c,- being the metric induced from w—s and .]_,5. (Recall that i} is negative
on 2-.) Hence (UK is symplectic everywhere on X except. [I , where it vanishes
transversely: Moreover, f ‘(d) is a ft)lded-syinplectic form on any section. so taking
5 even smaller. we can as well assume that. any finite collection of sections of f are
ft‘rlded-syniplectic. It is easy to check that the fokled-symplectic form we get satisfies
all the other declared properties. (Also see Remark 5.2.3). [I]

The homological assumptitm in the theorem is a very mild one. If S is the set of
critical points of the achiral fibration f: X —> X, then the tangencies of the fibers
define a complex line bundle I. : Ker-({1}) on X \ 5'. which extends uniquely over

X. It follows that unless we have a. torus fibration. the regular fiber F is essential,

since < c1(l.). F >2 ,\(l’). Also if the fibration is obtained from a pencil by blowing
up the base points. the exceptional spheres will become sections of the fibration,

guaranteeing that the fibers are essential in the homology.

99

Remark 5.2.3 Alternatively, the folded-symplectic form in Proposition 5.2.2 can be
constructed by using the folding operation described in [14]. Restrictions of ,{3 on 2+
and on E. gii'e well-defined area forms LL and ,13_, respectively. Gompf ’3 method
can be used to define a symplectic form n+7) + f’(/3+) on X+, where r) is a 2—
form on X that restricts to the fibers as a (positive) symplectic form and m, is a
small enough positive real number. The orientation on the base together with the
orientation on the regular fiber determines the orientation of the total space, and thus
by taking the opposite orientation on 5.3- but keeping the orientation on F _, one orients
—X_. Let 7: — X_ —> 2_ be the fibration defined by taking orientation-proseruing
charts for f: X- —> E- , then we can define a symplectic form Iii-I] +7l(—,(3_) on
—X- (as —,13_ is the area form on 23- ) by following the same construction method.
Observe that T(—d. ) = f*(./3-). Hence. setting K. 2 77ltll{h‘.+,t{-}. we obtain two
symplectic manifolds (.X'+.w+) and (~X-.w'-). u’hcrc bolt 2 H7] + f*(;'3i). Let [i
be the inclusions of bourularics into iXi. then. l;(w’+) 2 HI) = r”_(w_) and the
orientations of both null-foliations agree. Thus we can glue these pieces to obtain. a
foldcd-syrup/cctic structure on X + U X- = X. which agrees with w+ and w- in. the
complement of a tubular neighborhood of the fold 0X.- 2 H = —(')X_. (Sec [14/ for
details. ) This form. is deformation (1’qu.ie(1.l(-3n.t to the form. m] + f ‘03) in Proposition

5.2.2.

5.3 Existence of folded-symplectic structures on

closed oriented 4-manifolds

Here we show that any closed oriented smooth 4-1nanifold X can be equippml with

a folded-sym])lectic form. For the sake of conmleteness. we start by outlining Etnyre

100

and Fuller’s proof that every 4-manifold admits an achiral Lefschetz fibration after
a surgery along a framed circle [23] : Take a handlebody decomposition of X with
one 0— and one 4-handle, let X, denote the union of the O—handle, 1-handles and
2-handles, and X2 denote the union of the 3-handles and the 4-handle. By Theorem
5.1.1 there exist achiral Lefschetz fibrations fi: X,- —+ DQ, which necessarily have
bounded fibers, and stabilizing both fibrations we may as well assume the fibers
have connected boundaries. After a possible slight modification of the handlebody
decomposition, Etnyre and Fuller manipulate the contact structures on the boundaries
so that. they are both overtwisted and homotopic. as plane fields. Then it follows from
results of Eliashberg and Giroux that we have isotopic contact structures, and thus
the induced open books are the same, possibly after some stabilizations and isotopies.
Denoting the final manifolds and fibrations with X ,~ and f, again, we may therefore
assume that the open book decompositions induced by these fibrations on the common
lxtiundary H 2 8X1 : —(‘)X2 are the same, so we can glue both pieces of X back

along the truncated pages, and obtain an achiral Lefschetz fibration

f1 o f2: W = XI U X2 _. sz.

ff‘t01)9):f2“(t)02)
To recover X we need to glue 5'1 x D? to 31 x D; . where
s' x I)? = ox, \ f, ‘(or)‘~’).

Filling the boundary of ll" with an S] x D3 gives the same result. so we can View
W as X \ N where N is a. neighborhood of an embedded curve. 7 C X . Now, if we

instead add on a D2 x 5'2 so that each ODE X {pt} is identified with S" X {pt}, we

101

can extend the fibration on W by the projection on the 82 component of D2 x S2.
Hence, we obtain an achiral Lefschetz fibration over S2 on the resulting manifold Y,
where the section S of this fibration discussed in [23] can be taken as 0 x 82 coming
from the glued in I)2 x 82, implying S has trivial normal bundle in Y.

We will refer the following as the standard model : Consider S4 with the standard
folded—symplectic structure wo described before, and take S 4 0 {.r4 = 0} vertical to
the fold H0 2 S“1 0 {1:5 = 0}. Take 80 = S4 0 {1:4 = 0 = 13} C—V: S'2 which intersects
the fold along the circle C0 = {rf + .173 : 1 Ir; 2 x4 = 1:5 : 0}. It is easy to see that
too restricts to this SO as the standard folded—symplectic form on 82, folded along
(70., and symplectic on the normal disks to So. Fix a disk neighborhood MD of SO so
that an evaluates as 1 on each normal disk. That is, each normal disk projects onto
unit disk {13 +13 3 1 tr] : r2 : .rr, : 0} symplectomorpliically. By restricting to”,
we get two folded-sym})lectic manifolds .lIO E S") x D2 and .N‘}, = S4 \ .l/U E D3 x 81,
with folds 8‘ x 1)“) and D2 x .8", respectively.

The existence of the section s: S2 ——» .S' C X guarantees that the fiber of the
achiral Lefschetz fibration f: Y —> 52 is homologically essential and therefore there
exists a folrled-symplectic form to as described in Proposition 5.2.2. This restricts
to Y \ M , where ill 2 82 x [)2 is a. neighborhood of S. We may assume to is
crmstructed such that M is idcntilicd with NO in the standard model above as
follows: Let. 0: ill —> .l/O be an orientation preserving ditfeomorphism such that a)
is orientation preserving on the splmres (and on the normal directions as well), and
that it maps the imper—helnisi)here of So (where too is positive) to the p<,)sitive part
of S. Then one can start the ctmstruction in the proof of Proposition 5.2.2 with the
fr)lrled-symplectic form s‘o‘(w,,) on the base sphere, which naturally restricts to an

area form on each hemisphere. We can also modify the symplectic form m; on the

fibers so that it is symplcctomorphic to 0*(u'0) on the normal disks to 5', each of

102

which lies on a fiber.

Hence we obtain a folded-symplectic form an on X such that (ALLUIM) is folded
symplectomorphic to (ii/[0,wolMo). This allows us to trade M for N ’2‘: S1 x D3
and extend the folded-symplectic structure to (Y \ M) U N ’2- X . The effect of this
surgery on the fold of Y is to turn the surface fibration over S1 into an open book
decomposition on the resulting fold. The core curve of N sits in the 3-manifold as
the binding of this open book and therefore it carries a canonical framing. We have

proved:

Theorem 5.3.1 Every closed oriented smooth 4-manifold X admits a folded—
symplectic structure. Furthermore, there erist folded-symplectic forms on. X with
connected folds. such that a surgery along a framed curve which lies in. the fold results

in a. simple foldird-symplcctic manifold.

Remark 5.3.2 Away from. the framed curve 7 in. X, the foldcd-sympleetic model
we have e(mstructcd is the restriction of the simple model discussed in the prmrious
section. and as we will see shortly. the pieces are Stein and anti-Stein. So for any sort
of pseudo-holomo7phic curve counting with. respect to this foldcd-symplectic structure,
the focus would he understanding the limit behaviors around 7 of the curves in. the
moduli space. 'u.'licre we do have a standard model. namely (.\v"n.w0|,\v(,) above. (For a
digression. on. this topic. sec [84/ ) He would like to point out that both. the knot type
of 7' in, the fold and its framing depend on. the achiral Lefschetz filirat-ions we use in.
the construction, so does the simple model we get. The following cramp/e illustrates

this phenomenon.

Example 5.3.3 If we ccmstruct S4 following the recipe given in the proof of Theo—
rem 5.3.1. we get. W = [)2 >< [)2 up“), I)2 x [)2 z: 5'2 x If“), which can be identified

“’ith Mo. and the simple ft)ldcd—syniplectic form on Y : .92 x 5'2 = 4‘10Us'2xomill0

103

can be constructed so that its restriction to each copy of Mo is indeed the stan—
dard form (on. Note that here both open books already agree, so we do not need
to alter the contact structures and change the initial fibrations. Now if we undo the
surgery, that is if we trade N = Noand M in the proof, what we get is the standard

folded-symplectic form too on 5“.

 

0 0
—>
Q. s e .
U a 4—hru'idle CO
0

S2 x 5'2 v
Figure 5.1: On the left: O—surgerv along the binding yields a trivial S2 fibration over D2

- - 1' ') - . .
on each piece, which make up 82 X 5“. On the rlght: ()-surgery along the new bmdmg
yields a cusp neighborhood on both sides.

It is a standard fact that surgery along a. framed curve. in a siniply-connected 4-
manifold will result in connect summing with either 82 x .52 or 82;.92, depending
on the framing. which can be thought as an element of 7r1(SO(3)) = Z2. In [23] (also
see [44]) it. is (,lescribed how one can homotope the. framed knot in the 3—1nanifold
to another framed knot, which is isotopic in the ambient. 4—1nanifold to the original
one. so that their trainings differ by one and that surgering the new curve yields an
achiral Lefschetz fibration on the resulting manifold as well. Apl‘ilying this trick to our
example, we can insteeul pass to an achiral Lefschetz tibration on 8'2 $252 § (CllD2 #6332,
which is a torus fibration with two cusp fibers of opposite signs (Figures 5.1 and 5.2).

l , and the corresponding Kirbv

The monodromy of this achiral fil'ira.t..ion is to 1), [lb—1 t;
diagram is depicted in Figure 5.2 (see [37]) To verify that. this manifold is Cllflattm‘z,

104

   

 

22

 

 

 

 

 

 

J,

U two 3—handles and a 4-handle U two 3—handles and a 4-handle

 

 

CO e CO

0 +1 0 + 1
U a 4-handle

 

 

 

CP2#@2

U two 3-handles and a 4—handle

Figure 5.2: The achiral Lefschetz fibration on the second associated model. The total space
is shown to be (Cll’2#<ClP’2.

we first slide one of the vertical 2—handles over the other one, and then separate this
pair from the rest of the diagrz-un by sliding over the 0 framed 2-handle. Now the
rest of the diagram can be shown to be 5'" after obvious handle cancelations. It is
possible to see that. our construction method will give. a foltled—symplectic structure
on S4 equivz-ilent to the standard one again. Note. that the first simple model above
is obtained by surgering the unknot, whereas the second comes from surgering the

right trefoil in S3. Surgery framings on them differ by one in S".

105

5.4 Kahler decompostion theorem

While we shift our attention to Stein structures, we would like to have only non-
separating vanishing cycles in our constructions, as it is suggested by the correspon-
dence between PALFs and compact Stein surfaces established in [50] and [2]. We

start with the following lemma:

Lemma 5.4.1 Let X be a closed oriented 4-manifold. Then it can be decomposed
into two handlebodies, each of which admits an allowable achiral Lefschetz fibration
over Dz, such that the fibers hare connected boumlmies and that the induced open

books are the same.

Proof: We follow the construction of Etnyre and Fuller with more care given to
having librations allowable. Take a. handlebody dccmnposition of X with one 0—
and one 4-handle, let. X1 be the union of the U-handle, 1-handles and 2—handles.
and X2 be. the union of 3-1‘1andles and the 4-handle. As it was implicitly present in
[44]. and was also observed in [‘2] one can always build an achiral Lefschetz fibration on
a. given ‘2-handlebody so that all vanishing cycles are non-separating. Therefore. we.
can start. with allowable fibratitms and then proceed with stabilizations as described
in [23] to match the induced open books. A stabilization is given by gluing a. positive
or a negative Lefschetz ‘2-handlc along a new l-handle added to a. regular fiber.
and in order to keep the binding connected. we always introduce another adjacent.
stabilization. T herefore. all tarnishing cycles introduced during stabilizations are. also

nonseparating. Induction concludes the proof. D

Theorem 5.4.2 Let X be a closed oriented smooth 4-manijold. Then X admits a
decompositioi'i into two codimension zero submanifolds X+ and X_, such that X+
and —X_ are both (.'()7np(ict Stein manifolds with strictly p.S(?‘ll.(l0("()'H-‘l’CLI? boundaries.

These Stein structures can be chosen so that the induced contact structures 6+ on

106

8X+ and 5_ on —(‘3X_ are isotopic. Furthermore, there are PALFs on each piece
such that the open book decompositions they induce on 8X+ and —8X_ are compatible
with 6+ and 6-, respectively, and they coincide. In short, all data match on the

hypersurface H = UX+ = —(')X_.

Proof: The lemma above gives us a decomposition of X into two pieces X1 and
X2 with allowable achiral Lefschetz fibrations f1 and f2 on them, such that induced
open books on the boundaries match. As in the proof of Theorem 5.3.1, we glue these

two pieces along the truncated pages to get:

w = XI U X2.
f{"(302)=f2'1(802)

Next. we glue in S2 x D2 to cap off the fibers and establish an achiral Lefschetz
fibration f: Y —+ 52 with closed fibers.

We wish to split the base of this libration into two disks D+ and D- so that all
the. positive critical values lie in the interior of D+ and all the negative ones he in
the interior of D_. As discussed earlier. restrictions of f give. a positive Lefschetz
fibration on X+ = f‘1(D+) and a negative Lefschetz fibration on X- = f“1(D_),
respectively. It also prescribes a surface bundle over S] 2 (912+ = ——(')D__ on the
hypt-‘rsurface separating Y3, and Y. . This time we would like to describe the splitting
more carefully by taking into consideration how the global monodromies of the new
fibrations are. related to the original ones.

Let. m be the monodromy of the achiral Lefschetz fibration f; on X1 and [12 be
the monodromy of the fibrat ion [2 on X2. Fix a representation of m by using arcs
s], - - - ,sll and a representation of he by using arcs 3.1, - - . .sz . Corresponding criti—
cal values are denoted by 11,1 and yf. l\lonodromies of the open book decompositions
bounding each fibration are given by m and [.12 as well. and they coincide under
an orientation reversing diffeonn)rphism, so [11 = (#2)". Let V be a small neigh-

. r) . . . . .
borhood of a. regular vahw 1n the base S". “e obtain an achiral Lefschetz fil‘n‘atirm

107

 

Figure 5.3: New monodromies from old ones. On the left: #1 is given by solid arcs, and
[12 by dotted ones. On the right: Solid arcs are the positive arcs representing at, whereas
conjugated dotted arcs are negative, providing a representation of it. after closing the base
back to 32.

f: W \ f"(V) —t D = S2 \ V, which closes up to a fibration over S2. If g is the
genus of the page, then this fibration is determined by the relation 111/12 2 1 in F9.)
and is mapped under the maps F,“ ——> F; —> F!) to the relation that describes the
achiral Lefschetz fibration f: Y \ [1(V) ——> D. Since this map factors through 1‘51],
the achiral Lefschetz fibration f comes naturally with a section S of self—intersection
zero. We denote images of the elements in F 9,1 under this map with the same el—
ements. so ply-2 = 1 is the global monodromy of the achiral Lefschetz fibration on
Y \ f “(V). Note that we can use the same arcs s} and .312. to represent the global
monodromy of this fibration. Now. if we move counterclockwise and choose only the
arcs that run through positive Dehn twists, we establish a monodromy p+. Call
these arcs positive. Next, we choose a nearby base point. and move counterclockwise
by running through the negative Dehn twists only, while avoiding intersecting any

positive are. This way. we obtain a monodromy ,u- . The new set of ares involved in

108

this monodromy will be referred as negative arcs. Observe that each negative are is

obtained by traveling around some old arcs 3,1 and .92-

J in order to avoid intersecting

positive arcs, then going around the aimed negative critical point once, and finally
going all the way back on the same detour (Figure 3). That is, each negative are
corresponds to a conjugate of a negative Dehn twist in Pg, which defines a negative
Dehn twist, too. By taking regular neighborhoods of these arcs such that positive and
negative arcs stay apart, we get a disk enclosing only positive critical points, and an
annulus containing only negative critical points. Closing the fibration to a fibration
over 82, the latter becomes a disk as well. Now we can enlarge any one of these disks
so both disks share the same boundary, and call the one containing positive values
D+ , and the other one 1)_ . So we have a new factorization of the global monodromy
of f. given by the relation [Lt/1.- = 1. The section S prescribes how to lift the new
elements ,in of [‘9 to 1‘; uniquely.

we proceed with taking out. the tubule—1r neighborhood '2 X D2 of the section
from Y = )3, U l".. and we get an inherited splitting X+ U X _. The discussion
above shows that p+ defines a positive Lefschetz fibration on X+ and it._ defines a
negative libration on , "_ . To recover the original 4-manifold X we need to put back
in 81 x D“, which has the same effect as gluing each other the tubular neighlmrhotids
51 x Di and S] x l)“: of the bindings of open books on 0X+ and ('3X_ , respectively.
”l‘l‘icrefore we can think of X as decomposing into X + and X_. we claim that. this
decompositimi 1.)ossesses the desired properties.

When we take out a tubular neighborlmod of S from Y. we turn the positive
and negative Lefschetz iibrations on Y+ and ll into a. PALF on X+ and a NALF
on X_. respectively. In the meantime the surgery converts the surface fibratiou
that separates l; and Y. to an open book decomposition on the common boundary

II = UX+ = —(‘l.—\'__ . The binding of this open book is the identified bindings of ()X+

109

and —8X-, the page F is the bounded closed surface obtained by cutting off a disk
from the regular fiber of f, and the monodromy is induced from the fibration on
either side. Noting that. the NALF on X - becomes a PALF on —X_, we see that
both PALFS induce the same open book decomposition on their boundaries.

By Theorem 5.1.4, both X+ and —X- admit Stein structures. We will construct
these Stein structures using Eliashberg’s characterization so that they match on the
common boundary. The technique we are going to use is the same as the one which
was presented in [2]: The PALF on X + is obtained by attaching positive Lefschetz
handles hl - . - 11m to X 0 = F X D2, which has the obvious PALF defined by projection
onto [)2 component. The same is true for the PALF on —X- . F x 1)2 has a natural
Stein structure by Theorem 5.1.3. We can assume all vanishing cycles (coming from
either side) sit in various pages of the open book on H . Read backwards. we can think
of the. fibrations as being constructed by attaching positive and negative Lefschetz
handles to H on either side in a sequence following the monodromy of the open
book. Thus we can induct on the number of handles. Assume that the PALF on
.\',_1 = X0 U hl U U 12,-] (i g m) induces an open book decomposition on its
boundary. and it carries a Stein structure such that the. contact structure induced on
the boundary is cmnpatible with this open book. Let. C be. the. vanishing cycle of
the positive Lefschetz handle [1, lying on a page I” of 0X,-1. \N'e open up the open
book decomposition and choose a page against F. and glue them together along the
binding 8 to form a. smooth closed convex surface 2 in the. 3.3-manifold (‘L\',-1. As
(7' is mm-separating. Z \ C' is connected and it contains the dividing set. namely B.
So we can use the Legendrian realizatimz. principle ([33], [45]) to isotope 2 through
convex surfaces to make C' Legendrian. Note that this is done by a small (73° isotopy
of the contact structure supported in a. neighborhood of S, which fixes the binding

pointwise. Hence the framing of (1' relative to the fiber [7 is the same as its contact.

110

framing, implying that the Lefschetz handle h,- is attached along a Legendrian curve
with framing tb — 1. By Theorem 5.1.3 the Stein structure extends over this handle,
and as shown by Gay in [33], the new open book on BX, will be compatible with the
new induced contact structure on t)X,-. This completes the induction. Repeating the
same argument dually for —X- , we see that the compatible open books on 8X+ and
8(—X-) are isotopic, and therefore the induced contact structures Q on 8X+ and
6- on 8(—X-) = —8X- are isotopic as well. So we fulfill all the matching conditions

listed in the statement. of the tlie(_)rem. C1

Remark 5.4.3 In [I] it was asked if one could decompose a given closed oriented
smooth. 4 -m.o.ni fold into Stein pieces so that the induced contact structures on the sepa-
rating 3-mo'nifold coincide. Our theorem gives an. ojfirrnative answer to this question.
In the some article authors remark that it is possible to alter their Stein. decomposition
to make the induccd contact plane distributions homotopic. but the tightness of the
contact structure precludes the use of Eliashbcrq's (clcbratcd theorem. on. omrrtuiisted
contact .S'lT'IH'l'IH't’S to conclude more. Considering the underlying PALFS and isotopies

of open hooks gives a may around this difficulty. thanks to Girours Theorem.

Remark 5.4.4 In. [08]. Quinn studied so-callcd dual decompositions of
4-munifolds: descriptions of 4-munifolds as a union of two 2-h.u.ndlebodics.
The author fornuilutcs the some question. as in [1 / in. terms of i‘icccssaiy se-
quence of Kirby moves to relate a. possibly nonmotching Stein dccomposition.
Theorem. 5.4.2 provides an implicit answer to this question. and are would like.
to take this as an. opportunity to suHit/nonrizc the handle calculus behind our con-
struction: An arbitrary Stein dcco/nposition X 2 X1 U X2 comes with. some
PALF poir. Using the stabilization. moves of Etnyre and Fuller. we first change

111

‘.

 

 

this PALF pair with a matching pair. This corresponds to adding canceling
1— and 2- handles to each X,, or in other words, we add canceling handle pairs
of index 1—, 2— and 3— in the original handlebody decomposition of X. In the
next step, we pass to a cobordant 4-manifold Y so that we can split the positive and
negative Lefschetz handles. Then we ‘undo’ the surgery and get the decomposition
X = X+ U X- with Stein structures on each piece that coincide on the common
boundary. Having the simply-connected case in mind, this intermediate step can
be seen as a stabilization. Let W E“ X \ S1 x D3 be the. complement of a regular

neighborhood of the framed knot 7 in X, then the first surgery dcfincs a cobordism

[0. 2] x H' U ([0,1]x s1 x D3)U1x51x52 ([1. 2] x 1)2 x 82)).
[(1.2]x51x52

which is identity on the first component. We trade ‘2-handles of X1 and X2 in Y
by making use of the two ertra handles of inder 2. Finally, the composition of two
cobordisms that gives back. X can. be seen as the double of the cobordism above, and

thus it deformation retracts to

n' U (stoned-est).

S‘ x S3

This cobordism is built: by attaching cells to Ull' : 8‘ X 5'2. where I)2 X 82 is attached
uniquely and the framing of 7' indicates in. 'u,..'hi(.'h. one of the two ways we shall glue
the other two 5'1 x I)3 pieces. Although here we started with. a {nonmatching) Stein.
decomposition. it is clear that the some discussion can. be carried out in our main
construction. as (cell. 'I'hcrcforc (l.’(,' have a. uu'll—dcfincd process. during which we first
inflate the number of handles in a given dt.composition of X. and then trade some. of

the 2-harnlles through a cobordism to achieve the desired dccomposition at the end.

112

5.5 Folded-Kahler structures and folded Lefschetz
fibrations

Unlike symplectic structures, random folded-symplectic structures do not need to bear
any information about the geometry or topology of the manifold they are defined on.
In order to specify more meaningful members of this family, one first of all needs
to impose some boundary conditions on the folding hypersurface. We would like to
acknowledge a result of Kronheimer and Mrowka: In [48]. the authors prove that
a compact symplectic 4—manifold (Kai) with strictly pseudoconvex boundary has
S\*Vy(K) = 1, where K is the canonical class of w. This motivates us to see such
manifolds as building blocks of 4-manifolds. and yields a good boundary constraint
for folded-symplectic structures. at least in this dimension. Henceforth. we assume
that the fold II = (a'")‘1((l) of a given folded-symplectic manifold (X2". to) is always
connected and nonempty. \Ve will generalize the notion of a. Kiililer structure on a.
smooth 27'1-11'1anifold by considering a distinguished subset of the family of folded-

sympleetic structures. and we then present some }_)roperties of these. structures:

Definition 5.5.1 A folded-symplcc-tic form a: on. an oriented ‘2n -dimensional man-
ifold X is called a folded—Kahler structure. if there is a. tubular neighborhood N of

H s uch. that:

1. The closure of each. component of X \ N is a compact Kcihler manifold

(:t/X'Jru'ItV-g ) with strictly pseudoconrer boundary.

2. (.\ tolV) is folded sympleitomorpbic to ( [—-1, 1] X H. d((t2+ 1) 7r*(o) ). where a

is a contact l-form on the fold H. and ft is the projection. 7r: [—1. l] x H —-—> H.

In addition. if each, (ixia‘w'lani is strictly pseudoconeer. we say to is a nicely

folded—l’x'iihler structure on. X.

113

In the definition above, nice folding can be reformulated as folding Stein mani-
folds along matching strictly pseudoconvex boundaries. Recall that if 1,!) is a proper
strictly plurisubharmonic function on a complex manifold S ._ then the associated 2-
form my. = —d.l‘di,b is Kahler. and importantly, the symplectic class of (X .ww) is
independent of the choice of w [22]. Therefore, to complete our alternative formula-
tion, we ask that each piece :1:Xi should admit some proper strictly plurisubharmonic
function It} . so that wlix:t = a)“ . In short, it is built in the definition that a nicely
folded-Kéihler manifold is folded-Kahler. Finally note that, due to a theorem of Bo-
gomolov [12]. any compact folded-Kahler manifold X can be made nicely folded after
deforming the complex structure and blowing down any exceptional curves. Even
though these definitions narrow the family of folded-symplectic structures quite a lot,
it is important to note. at least in dimension four. that we still have an adequately

large. family in the light of the following result:

Theorem 5.5.2 Any closed oriented 4-manifold X admits a nicely folded-thler

structure.

Proof: By Theorem 5.4.2. X can be decomposed into two compact Stein manifolds
X+ and -—X- with strictly psemdoconvex boundaries such that both induce the same
contact structure on the common boundary H = 0X+ = —(‘)X_. \\-'e begin with
adding collars illI to (iii/Yi.w'i). and extending the symplectic structures to to;
on :lzX'f = :l:(.\'i U (Q) so that new boundaries 0(iX'f) are still convex and con-
tactomorphic. Let. £3; be the induced contact. structures on 0(iX’i) and 15: be a.
contactomorphism between them. Using the symplectic cut-and-paste argument of
Etnyre [24]. we can add a symplectic collar to (i).-\"+.a"+) so that the new bound-
ary is not only contactomorphic to ——0X"_ but also the induced contact forms agree
up to a multiple Ir 6 1133+. For the sake of brevity. let us assume that U+ above

cmitains this collar part as well. So after rescaling u}: (and to-) by k if I'ietf'cssary.

114

 

we see that restrictions of symplectic forms w’+ ax; and kw: l-ax: agree via ill, and
orientations of null-foliations (which correspond to Reeb directions) are the same.
Therefore, once again we can apply the folding technique of [14] to obtain a folded-
symplectic structure w on X L U X ’_ such that w agrees with w’i on the complement

of a small tubular neighborhood of the fold H. We enlarge this neighborhood to
include U+ and U- and call it N. It follows that X = X+ U N U X- ’5 X+ U X_,

 

and tube. = UJ+, whereas w x_ = kw- . Also note that, the folding operation provides
us with the desired local model on N , that is. (N ,wIN) is folded symplectomorphic
to ([—1. +1] x H, (l((t'2 + 1) 7r*(a)) by construction [14].

Lastly. suppose ”oi: :t Xi ——> [0.00) are proper strictly plurisbuharmonic
functions such that iBXi correspond to the maximum points of mi. and wt =
~-—d.]'*dii'f. respectively. If X: 3A 1, we can replace 1.; with k-‘u; and obtain kw-

21.l)(_)\'(_‘. as a Kahler form of a strictly [)seudoconvex manifold. Equipped with these

properties. «I is a nicely folded-Kahler form on X . D

Remark 5.5.3 I t is clear that Theorem 5.5.2 is a re finement of Theorem. 5.3.]. Since
the folded forms we have constr-ucted in both proofs are obtained through similiar steps,
one e;I.'pects that these striu'tures are actually eguiealent. Neartz. we i.’e7“'ify this fact.
and this any we get an. insight of how folded-Kahler forms are ‘supported' (precise
definition. is given below) by Lefschetzfib-rations as was illustrated in Proposition. 5.2.2.

Take the PALF on .\'+ in the. proof of Theorem. 5. 3.1. and attach a symplectic
2-li(uidle along the binding of the induced open book on (9X; as described by Eliash-
berg in [21]. This yields a symplectic Lefschetz fibralion over Di. Dually the same
argument for the NA LF an X_ gives an anti—sgmplectic Lefschetz fibration ore/r DE,
and these handle attachments can be done so that two fibrat-ions agree on the com-
mon bomulaxijy. illorcorcr. we can ass-(1.7m: that these fibrations have genus at least

two, so the fibrations can be matched as symplectic surface jibrations over a circle.

115

as it was pointed out in [21]. At the end we get a simple folded-symplectic mani-
fold Y obtained from X after a surgery along a framed curve 7. However, any two
simple folded-symplectic forms compatible with a fixed achiral Lefschetz fibration are
deformation equivalent by Proposition 5.2.2. Moreover, we can normalize both forms
on the disks which are parallel copies of cocores of new 2-handles that were used
to cap 017 the fibers. Hence. these two folded forms are deformation equivalent on
Y \ $2 $< D2. As the folded-symplectic structure on D3 x S1 which is glued back in to
recover X is standard, the folded-symplectic form. constructed in the proof of Theorem
5. 3.1 and the folded-Kahler form obtained in Theorem 5.5.2 are indeed equivalent as

fol(led-symplectic structures. El

l\=lotivated by symplectic and near—symplectic cases ([17], [9]), we can conclude our

discussion above by defining the Lefschetz fibration analogue for our structures:

Definition 5.5.4 Let X be a closed oriented 4-manifold. and decompose 82 as the
union of the upper-hemisphcre 0+ and the lower—hemisphere D- which are glued
along the equator C = if)1)+ = —(')D_. Then a smooth map f: X —> 82 is said to
be a folded Lefschetz fibration on X. if it restricts to a PA LF over D+, to a NALF

over D_, and to an open book over (7 bounding both fibrations.

Definition 5.5.5 Let X be a. closed oriented 4-manifold equipped with. a nicely
folded-[y’dhler form. or}. Then a folded Lefschetz fibmtion f: X —> 82 is said to be
compatible. with u} if each Stein piece Xi corresponds to f "1(Di), and if the contact
structure they induce on [I : _/"‘1((1') is compatible with the open book decomposi-
tion coming from. the fibration f. In this case. we also say that nicely folded-Kahler

manifold (Xnv) is supported by the folded Lefschetz fibration f .

The. cmnpatibility in the above definition is completely on the symplectic level.

This l_)ecomes more visible if once again we recall that surgering the binding '7 of

116

the open book f I H\,: H \ 'y —> SI, we pass to a simple model where the folded
Lefschetz fibration can be extended to a folded symplectic achiral Lefschetz fibration
f with closed fibers. Also note that, since Stein manifolds harbor less topological
obstructions in complex dimensions > 2, it is very likely that they admit higher
dimensional analogues of PALFs with similar topological correspondences. If that
is established, last two definitions, as well as several results in this paper, can be
generalized to all 2n-dimensions.

The complete statement of Theorem 5.4.2 combined with Theorem 5.5.2 shows
that. given a closed oriented 4-manifold X , one can always find a nicely folded-Kahler
structure w on X together with a compatible folded Lefschetz fibration. Next, we

prove that this property in fact holds for any nicely folded-Kahler structure:

Proposition 5.5.6 Any nicely folded-Krihler structure to on X, up to orientation

preser‘m'ng (ltfft()771()'I"])l1.t37lt, admits a compatible folded Lefschetz fibration.

Proof: Each Stein piece X+ and —X_ admits a. PALF by Theorem 5.1.4. If we
crmstruct these fibrations following the algorithm of [2] and keep track of the associ-
ated open books. the work of Plamenevskaya [67] shows that we can establish PALFs
fi : :t Xi —> [)2 with the property that the open book decomposition on the bound-
aries are con’ipatible with the contact structures induced from the Stein structures
on iXi, respectively. As the contact structures are assumed to be the same, The—
oreln 5.1.2 "tells us that we can match these open books after p(;)sit.i\-'e staljiilizaitions.

Consequent 1y. we get a eonmatible folded Lefschetz tibration. [:1

Remark 5.5.7 A folded Lefschetz fibration that supports a given fola’ed-Kahler struc-
ture fails to be unique. In fact. one can find infinitely many pairwise inequivalcnt such
fibrations. This can, be seen for egrample from the construction. of [2. by using dif—
ferent (p. q) torus knots in their algorithm which we adopt for building our achiral

Lefschetz fibrations.

117

Example 5.5.8 The easiest examples are doubles. If Y4 is a compact Kahler man-
ifold with strictly pseudoconvex boundary, then X = Y U ——Y is equipped with a
folded-Kahler structure. When Y is indeed Stein, we get a nicely folded structure.
The first folded structure constructed in Example 5.3.3 is the double of standard
D4 C C2, whereas the latter is a ‘monodromy double’ of a cusp neighborhood minus
a section. Here by ‘monodromy double’ we mean that the pieces are first glued along
the pages of the open books. and if the monodromy of the folded Lefschetz fibration

on one piece is u, then the monodromy on the other one is n‘l.

Example 5.5.9 There is a construction which also allows us to see the nicely folded-
Kahler structure together with a compatible folded Lefschetz fibration. Take a contact
3-manifold (11,6), and fix a positive open book decomposition (B. f) compatible with
{. Different PALFs bounding this open book describe (possibly) different Stein fillings
of (Hf). Indeed there. are examples of infinitely many pairwise non-diffeomorphic
contact 3-manifolds each of which admit infinitely many pairwise non-diffeomorphic
Stein fillings constructed this way [57]. Thus for every pair of PALFs X1 and X2
that fill the same open book, we can construct. a nicely folclt‘Cl-Kiilllt‘l‘ form on X 2

X1 U —X2. as (_lesignated in the proof of 'l‘heorem 5.5.2.

Example 5.5.10 The main steps of our construction are depicted in the follow—
ing simple. albeit instrluftive example: Start with classical handlebody decom-
position of X = #352 X 5'2 with one (l—handle. sixteen 2—handles. and a. 4-

handle. Let X; be the. union of 0—. '2 handles. and X2 be the 4-handle.

Each piece admits a. [)2 fibration over D"). However we wish to ccmstruct al-
lowable librations. so we. introduce two 1— and 2— canceling handle pairs and
two 2- and 3- (Lranceling handle pairs in the. original lumdlebody decomposi-
tion of X. “"0 start. building the fibrations from the scratch: Add the 1-

lnuulles to the ()—handle and 3-handles to the. 4-handle. Attach the two canceling

118

2-handles with framing —1 to the union of the 0- handle and 1- handles. Attach
the other two the same way to the handlebody X 2, which is the union of 3- handles
and the 4- handle. To simplify our description, we will label the l-handles of the
first handlebody as a and b, which generate 7r, of the torus fiber with one boundary
component, and we do the same for the 1-handles of X 2 under the obvious identifi—
cation. So we obtain two achiral Lefschetz fibrations over disks with bounded torus
fibers; one with monodromy t; 1 t; 1 , and one with tbta. One can verify by Kirby
calculus that each time we insert a pair of Lefschetz handles prescribed by ta t; 1 or
tb 1,71, we introduce an 5'2 X 82. (See Figure 5.2, and observe that. here we slide off
the 2-handle pair over a +1 framed '2-handle instead.) Doing this consecutively,
we attach all the remaining 2-handles to the first handlebody, and obtain an achiral

Lefschetz fibration on X1 with monodromy

m = glib—1n, t,, n, 1,, math!” tgltgltg‘tg‘tu‘ltg‘t." ltg‘.
whereas X2 still has the monodromy

,12 = 1,1,, = (1171171)“.

Both open books that bound these fibrations contain negative Dehn twists (recall
that. on —(")X2. the monodromy is [1.31), and therefore the contact structures they
support are overtwisted. As we have already manipulated the monodromy that way.
contact structures and open books are isotopic. so we can glue X1 and X2 along
the truncated pages. Putting in 52 x D2 we pass to a. torus fibratimr f: Y ——> $2
with global numodroiny [11 -/r-_). ( pplying the handle slides given in Example 5.3.3
repeatedly. and proceeding with the same handle cancelations. one can indeed check
that Y = #882 x S2#CIP’2#@2 .) Now the monodromy splits easily as explained in
the proof of Theorem 5.4.2. and we get l1+ = (1,, tn)“ and 11-. = (I;1 tb—1)5. It is not

hard to see. that when we take out the section now. we. get pieces X+ and X- , which

119

are diffeomorphic to —-E3 and E8, respectively. So X decomposes into a Stein piece
—E8 and an anti-Stein piece E3. This defines a nicely folded-Kahler structure a) on
X , folded along the Poincaré homology sphere 2(2, 3,5), and it is supported by a
folded Lefschetz fibration which is the monodromy double of a torus fibration over

D2 minus a section.

5.6 Addendum: Interactions between the two gen-
eralizations

From the very definitions of the two different symplectic generalizations we know that
folded-symplectic structures and near-symplectic structures on smooth 4-manifolds
can not be generalized from each other. It is also apparent that achiral Lefschetz
fibrations and broken Lefschetz fibrations are not generalizations of each other either.
However, one can consider a simultaneous generalization of symplectic structures so

to deal with both of them as follows:

Definition 5.6.1 Let w be a closed ‘Z-form, on a smooth. 4-7namfold X such. that
there exists a smooth emhcddcd l-mamifold Z m X, and such that .22 intersects the
O-sectton of A“T*X transversally at every point on. X \ Z and w‘ = U at every point
on Z. We then. call a) a genera-r1 symplectic structure on X provided that at each
point .r E Z , if we use local coordinates on. a. neighborhood U of 1* to identify the
map u) : U ——> .‘\2(T*U) as a smooth map a) : R‘ ——> R“, then. its tincartzatton at 1',
D...)I : IR‘ -—> R“, has rank 3. ”"6? call H = w2(()) \ Z the, fold singularity and Z the
round singular loci. When I] = (0, (X,w) is a near-symplectic manifold if w2 2 0
and an ant.i-rrear-symplectic manifold if w2 3 0. If Z = (t) and H yé (D, then. we
obtain. a foldcd-.S'y-n‘iplcrrttc numtfold. Lastly, when. H U Z = (Z) "we have a symplectic

. . I) . . . . . a
4-manzfotd If at“ > 0 and an. an.tz-symplectic 4-manzfold if w2 < 0.

120

Note that around every component of Z, we either have w2 2 O or w2 s 0, so for
an appropriate choice of metric g, the map to : U -—) A2(T‘U) above is either onto
the subspace of self-dual 2-forms, or onto the subspace of anti-self—dual 2-forms with
respect to g ——both of which have rank 3. (Compare [9])

In a similar manner, we define a general Lefschetz fibratz'on to be a fibration where
we allow both achiral Lefschetz singularities and round singularities. In fact, what
motivates us to introduce the above notion of general symplectic structures is the
study of these fibrations by David Gay and Rob Kirby in [34] (under the name

“broken achiral Lefschetz fibration”), where the authors proved:

Theorem 5.6.2 (Gay-Kirby [34]) Let X be an arbitrary closed oriented 4-
manifold and let F be a closed surface in X with F - F = 0. Then. there arts-ts

a general Lefschetz fibralion from X to S2 with. F as a fiber.

This theorem suggests an alternative way to study general 4-1nanifolds through

generalize(tons of Lefschetz fibrattons.

5.6.1 General symplectic structures on broken achiral

Lefchetz fibrations

An open question stated in [34] was to give a meaningful formulation of a cohomolog-
ical condition that would allow one to obtain a ‘2—for1n a: on a given general Lefschetz
fibration. as in Theorem 2.0.4 and Theorem 4.1.3. The fibrations constructed in [34]
can always be arranged so that the round handle singularities all project to the tropics
of Cancer and Capricorn. with their high genus sides towards the equator and with
all Lefschetz and negative Lefschetz singularities over the equator. For what. follows.
it would be convenient to introduce yet another name to refer to general fibrations ar-
ranged in this peculiar way. Let us call these "simplified general Lefschetz fibrat ions"

for this purpose.

121

Proposition 5.6.3 Let X be a closed oriented smooth 4-manifold and f: X -—r S2
be a simplified general Lefschetz fibration such that the regular fiber is a closed ori-
ented surface F which is nonzero in H2(X; IR), then X admits a general symplectic
structure a) such. that fibers are symplectic away from the critical points, the fold II
is an F -bundle over S l, and the round singular locus Z“, coincides with the round
singular locus of f. The fold H splits X into pieces X+ and X_, and f induces
near-symplectic Lefschetz fibrations on (X+,wlx+) and on (—X_,w[x_ ), respectively.
Furthermore, any finite set of sections can be made folded-symplectic for an appro-
priate choice of w. This form is crmonical up to deformation equivalence of general

symplectic forms.

Proof: We can perturb the equator circle of the base 82 to an are which passes
from the south of the image of each Lefschetz singularity. and from the north of each
negative Lefschetz singularity. Let this are (7 be the new equator of the base 5'2. It
is easy to see that C splits off a disk D+ from 8'2 that contains only the images of
positive Lefschetz singularities. the complement of which in $2 is another disk I)_
that contains only the images of negative Lefschetz singularities. Both might contain
images of round handle singularities.

The rest of the proof is very much the same as the proof of Proposition 5.2.2 except.
on the. pieces X+ = f‘l(D+) and X_ = f‘1([)_) one now also needs to deal with
the round singularities. However this can be done as in the proof of Theorem 4.1.3,
where the authors generalize Gompf’s construction to the case of broken Lefschetz
librations [9]. [3
It is easy to generalize the above proposition to any (not necessm‘illy simplificd)
general Lefschetz fibration f: X —> E. For that though, one will need to prefer
one splitting of the base over another. since in general there is no canonical way of

separating the images of negative and positive. Lefschetz singularities in S.

122

5.6.2 FI‘orn achiral to broken Lefschetz fibrations

In [34], another question that the authors ask is whether one can avoid achirality in
their construction. In this subsection we will show that this can be achieved after
blowing-up the ambient manifold sufficiently many times. Note that blow-ups do not
kill standard smooth 4-manifold invariants and in fact. their effect on the invariants
is well-understood. Therefore, as far as the smooth invariants are concerned, this
modification is very welcome.

To get. rid of the negative Lefschetz singularities. we will consider a local mod-
ification around the image of an isolated negative singularity to obtain a new gen-
eral Lefschetz fibration where this negative Lefschetz singularity is traded with an
additional round singularity. We then show that. this amounts to blowing-up the
Il-manifold at that critical point. Our modification can be seen to be equivalent to
performing the local operation described in the third example of [9]. page 113, but
”in orientation reversing charts. For the convenience of the reader, we briefly describe
this mmlification below.

Consider an isolated negative Lefschetz critical point of a general libration f on X ,
with vanishing cycle a loop '7- in the nearb_v generic fiber. \Ve remove a neighbourhood
of this singular fiber and insert in its place a configuration where '7." is now taken as
the vanishing cycle of a round 2-handle e(_)bor(lisin. The critical values form a. simple
closed loop (5. The inner most part of this round handle cobordism is a trivial fibration
with a fiber of one. less genus. or otherwise with two fiber components. depending on
whether 7 is nonseparating or separating. This way we add a. new component to the
round locus. See the Figure 5.6.2 which is taken from [9] after a slight modification
for our case.

This mmlification yields a. new general libration j". Let (5 be the new round

singular circle. The fibers outside (5 are. obtained from those inside by attaching a.

123

 

 

—1

9/

6 6
Figure 5.4: Replacing a negative Lefschetz singularity by a round singularity: f (left) and
f’ (right)

 

 

 

 

 

 

handle joining two points q. q’ as shown in the Figure 5.6.2. Along 6 the points q. q’
describe a trivial braid, but the relative framing differs from the trivial one by ——l,
so that on the outer side the monodromy around 6 consists of a single negative Dehn
twist along y. which compensates for the loss of the isolated singular fiber.

Next we would like to understand the total space X’ that f ' is defined on. The
local model for f is simply a Ali—ball. On the other hand, the total space of the new
local model for j" contains a smoothly embedded sphere S, obtained by considering
the two points q and q' in each of the fibers inside 6 together with the equator 6.
Since the monodromy around 6 is a negative Dehn twist along 7. we deduce that
S has self-intersection —1. Also observe that the preimage of the interior region
V is the disjoint union of two 02 x Dz’s. giving a disk bundle over S 0 f”‘(V).
The preimage of the outer region is again a disk bundle over a neighbourhood of the
equator in S . and it is diffeomorphic to S 1 X 03. Hence. the total space of f’ is
a disk bundle over the sphere S with self-intersection —1. Therefore our operation
locally (and thus globally) amounts to blowing—up X. That is, X’ = X #@2_

We can also depict this operation in terms of handle diagrams. For simplicity.
assume that 'y is a nonseparating curve. Clearly the vanishing cycle 7 can be very
complicated in general. However. there exists a self-diffeomorphism of the fiber which

takes 7 to any nonseparating curve. This self-diffeomorphism can be extended to an

124

orientation preserving self-difieomorphism o of the piece 8f‘1(V). So it suffices
to study our modification in the local model in Figure 5.5, and glue the new piece
back via the same diffeomorphism (f) on the boundary, which matches the boundary

monodromies as indicated by the negative Dehn twist along the original '7.

Jmhié—VQ

 

Figure 5.5: Neighborhood of a negative nodal fiber which has a simple nonseparating
vanishing cycle.

After blowing—up in this piece, one can obtain a new diagram with no Lefschetz
singularity but with a new round handle as shown in Figures 5.6 and 5.7. In Figure
5.6. we first slide the +1—framed 2-handle over the —1-framed 2-handle so that its
framing becomes 0. Then the two strands of the O—framed 2-handle can be slid off the
1—handle using the new O-franied 2-handle. and now they go through the —1 framed
‘2-handle as shown in the third diagram. The new O-framed 2-handle and the 1-
handle becomes a canceling pair. which we remove. from the diagram to get to Figure
5.7. The last step is just an isotopy which puts the diagram in the standard form of
a trivial fibration with a fiber of one less genus, and a round l—handle attached to it.

Observe that. the framing of the 2-handle of the round l—handle is — 1. compensating

125

l

(Dig ...flflf—ifijr—fi
i ‘ i
t C , 354

Figure 5.7: After an isotopy, we obtain a Kirby diagram of a round l-handle attachment
to a product neighborhood of a fiber with one less genus.

22

 

 

 

 

for the loss of the singular fiber on the boundary monodromy.

Since the modification is made locally around a critical point, it works for any
general fibration containing negative Lefschetz singularities. In particular our folded
Lefschetz fibrations in Section 5.5 can be replaced with folded fibrations with only
broken and positive Lefschetz singularites on blow-ups of the given manifold. The
same argument applies to any achiral Lefschetz fibration as well; for instance to those

that Etnyre and Fuller obtained in [23].

127

5.6.3 Comments on describing invariants on general 4-

manifolds

A question that remains unanswered is if one can define smooth invariants in the
most general setups discussed in this chapter. We finish with a few rather speculative
comments regarding this issue.

One might. hope a Kahler decomposition for a given 4—manifold to be what a Hee—
gaard decomposition is for a given 3-manifold. For this to work. one needs to relate
any given two Kahler deconmositions by a finite set of ‘moves’, i.e. some relative ‘al-
culus which would take us from one decomposition to another. To reveal the difficulty
in this task we shall point out that. our ‘construction’ of a Kahler decomposition on
a given 4—manifold is far from being explicit. This is due to the. two results we have
utilized: Eliashbergs theorem on the existence of some isotopy between homotopic
overtwisted contact structures. as well as Giroux's theorem on the existence of com-
mon stalfilizations of two open books supporting isotopic contact structures. Neither
one of these theorems provide us with explicit algoritlnns.

On the other hand. it is a curious question to determine whether one can define an
invariant for a firm] Kahler decomposition. The difficulty lies in the fact that gauge
theoretic invariants are very sensitive to the orientation change. Even though the
invariants (Seil>erg-\\'itte11 or Heegaard—Floer) of compact Stein manifolds are well-
known. it. is unclear how one can make. use of this information on the piece with the
rtwei'secl orientation. A possible approach of course. is to go beyond the gauge theory
setting. as one can not. avoid for example while dealing with S“. The work of Jens
von Bergmann in [84] runs in this vein. but various technical details pit-went us from
adapting the same arguments for our case in any straightforward way. Nevertheless.
if this task together with the previos one can be accomplished in a compatible way.

then one can derive a (hopefully nontrivial) invariant of general J-manifolds.

128

In a different direction, and motivated by the Theorem 5.6.2 of Gay and Kirby,
we might start with a generalized fibration compatible with a generalized symplectic
structure on a given closed smooth oriented 4-manifold. By homological reasons,
these exist precisely on 4-manifolds with nontrivial second homology. One can then
try to generalize Perutz’s invariant to this setting. The crucial step is to describe a
meaningful Lagrangian matching condition along the fold. Since the fold is away from
the round singularities, it would suffice to define a Lagrangian matching condition for
achiral fibrations, which then can be used to define an invariant of general symplectic

fibrations through Perutz’s work. i

BIBLIOGRAPHY

1.30

[ll

BIBLIOGRAPHY

S. Akbulut and R. lylatveyev, A convert: decomposition theorem for 4-7nanz'folds,
International Math Research Notices 7 (1998), 371—381.

S. Akbulut and B. Ozbagci. Lefschetz fibrafions on compact Stein surfaces, Geom.
Topol. 5 (2001), 319—334 (electronic).

A. Akhmedov, Small erotic 4-manzfolds, preprint, math.GT/ 0612130.

A. Akhmedov. S. Baldridge, RI. Baykur. B. D. Park and P. Kirk, Simply con.-
nected minimal symplectic 4-manzifolds with sigmzfare less than —1, preprint.

arXiv:0705.0778v1.

A. Akhmedov. R. I. Baykur and B. D. Park. Construct-nu] infinitely many smooth.
structzues on small 4-man2folds, preprint, 1natltGT/0703480.

A. Akhmedov and B.D. Park. Erotic smooth structures on. small 4-mamfolrls.
preprint, materT/0701064.

D. Asimov. Round hand/rs and non-snag-alar Morse-Smalc flows, Ann. of Math.
(2) 102 (1975), no. 1, 41-54.

D. Auroux. SK. Donaldson and L. Katzarkov. L'IrltITin..qer surgery along La.—
grangian tori and non-isotopy for singular symplectic plane curves. Math. Ann.
326 (2003), 185 203.

D. .Auroux. S. K. Donaldson and L. Katzarkov. Singular Lefschetz pencils. Geom.
Topol. 9 (2005), 1043—114.

S. Baldridge and P. Kirk. A symplectic manifold homcomorph(Tc but not. cliflco-
morphic t0 CP2#3@I$2. preprint. mathGT/07022l1.

R.I. Baykur, thler Decomposition of (manifolds. Alg. Geom and Topol. 0
(2000).1239_1267.

131

[12]

[13]

[14]

[15]

[16]

[17]

Ml

[191

[‘20]

[21]

[221

[‘26]

F. Bogomolov, Fillability of contact pseudoconvex manifolds, Gottingen Univ.
preprint, Heft 13 (1993), 1—13.

A. Cannas da Silva, Fold-forms on four-folds, preprint, 2002.

A. Cannes da Silva, V. Guillemin and C. Woodward, 0n the unfolding of folded
symplectic structures, Mathematical Research Letters 7 (2000), 35—53.

S. K. Donaldson, An application of gauge theory to four-dimensional topology, J.
Differential Geom. 18 (2) (1983), 279—315.

S.K. Donaldson, Irrationality and the h-cobordism conjecture, J. Differential
Geom. 26 (1987), 141—168.

S. K. Donaldson. Lefschetz pencils on. symplectic manifolds, .1. Differential Geom.,
53 (2) (1999), 205—236.

S.K. Donaldson, Topological field theories and formulae of Casson and Meng-
Taubes, Geom. Topol. Monogr. 2 (1999). 87—102.

S. K. Donaldson and I. Smith. Lefschetz pencils and the canonical class for sym-
plectic 4-manifolds, Topology 42 (2003), no. 4, 74-3 785.

Y. Eliashl‘mrg. Topological characterization. of Stein. manifolds in dimension. > 2,
Int. Journ. of Math. 1 (1990), 29—46.

Y. Eliashberg. A few 7"em.a.rhs about symplectic filling. Geom. Topol. 8 (2004).
271—293 (electronic).

Y. Eliashl')erg and M. Gromov Convex symplectic manifolds, Proc. of Syrup. in
Pure Math. 57 (1991), 135—162.

J. B. Etnyre and T. Fuller. Realizing 4 -rll[(lllfilf01d8 as Achiral Lefschetz F ib-I'ations.
Int. Math. Res. Not. 2000, Art. ID 70272, 21 pp.

J. B. Etnyre. Symplectic: conrmuiti/ in lowdinzensional tolmlogy. 'Ibpology and its

App]. 88 (1998). 3—25.

R. Fintushel, B. D. Park and R. J. Stern, Reverse. enginecring small 4-mmzifolds,
preprint, InathCT/0701840.

R. Fintushel and R. J. Stern. Innnr—msed spheres in 4- -m(.zsn.ifolds and the immersed
Thom conjecture. Turkish J. Math. 19 (1995), 145—157.

132

[271

[28]

[29]

[30]

[311

[321

[33]

[341

136]

[371

[38]

[39]

[.40]

[41]

R. Fintushel and R. J. Stern, Rational blowdowns of smooth 4-manifolds, J.
Differential Geom. 46 (1997), 181-235.

R. F intushel and R. J. Stern, Knots, links and 4-manifolds, Invent. Math. 134
(1998),363—400.

R. F intushel and R. J. Stern, Families of simply connected 4-manifolds with the
same Seiberg-Witten invariants, Topology 43 (2004), 1449—1467.

R. Fintushel and R. J. Stern, Double node neighborhoods and families of simply
connected 4-manifolds with b+ = 1, J. Amer. Math. Soc. 19 (2006), 171—180.

M. H. Freedman, The topology of four-dimensional manifolds, J. Differential
Geom. 17 (1982), 357—453.

R. Friedman, Vector bundles and 80(3) invariants for elliptic surfaces. J. Amer.
Math. Soc. 8 (1995), 29—139.

D.T. Gay, Egrplicit concave fillings of con/art 3-manifolds, l\v'la.th. Proc. Carn-
bridge. Philos. Soc. 133 (2002), 431—441.

D. T. Gay and R. Kirby, Constructing Lefschetz- type fibrations on four-manifolds,
preprint, 1nath.GT/0701084.

E. Giroux. Structures dc contact an dimension trois et bifurcations des fen-il-
letages de surfaces, Invent. Math. 141 (2000) no. 3, 615—689.

E. Giroux. Geometric dc contact: de la dimension. trois ners lcs dimensions
supérieures. Proceedings of the ICM, Beijing 2002, vol. 2, 405—414.

RE. Gornpf. Sums of elliptic surfaces, J. Differential Geom. 34 (1991) 110.1,
93—114.

R. E. Gonipf, A new construction of symplectic manifolds. Ann. of Math. 142
(1995),527—595.

RE. Gornpf. Handlebody construction of Stein. surfaces, Ann. of Math. 148
(1998).619~693.

R. E. Gonlpf and A. I. Stipsicz, 4-nmnifolds and Kirby calculus, Graduate Studies
in Math. 20, Amer. Math. Soc... Providence, RI, 1999.

i

H. Grauert. On Levi's problem. Ann. of Math. 68 (1958), 460—472.

133

[42] M. Gromov, Partial diflerential relations, Springer-Verlag, Berlin and New York
(1986).

[43] M. J. D. Hamilton and D. Kotschick, Minimality and irreducibility of symplectic
four-manifolds, Int. Math. Res. Not. 2006, Art. ID 35032, 13 pp.

[44] J. L. Harer, Pencils of curves of 4-manifolds. PhD thesis, Univeristy of California,
Berkeley, 1979.

[45] K. Honda, 0n. the classification of tight contact structures -1, Geom. Topol. 4
(2000), 309—368 (electronic).

[46] K. Honda, Transnersality theorems for harmonic forms, Rocky lV'Iountain J.
Math. 34 (2004) 629—664.

[47] K. Honda, Local properties of self-dual harmonic 2-forms on a 4—manifold, J.
Reine Angew. Math. 577 (2004) 105—116.

[48] P. B. Kronheimer and T.S. Mrowka, Monopoles and contact structures, Invent.
Math. 130 (1997) no.2, 209 255.

[49] P. B. Kronheimer and T. S. l\~lrowka. ll-lonopoles and T hree-M anifolds, Cambridge
University Press; 1 edition (2007).

[50] A. Loi and R. Piergallini, Compact Stein surfaces with boundary as branched
covers of B4 , Invent. Math. 143 (2001), 325—348.

[51] K. M. Luttinger, Lagrangian. tori in. 1R4, J. Differential Geom. 52 (1999), 203—
222.

[52] J.\V. Morgan, T. S. Mrowka and Z. Szabo, Product formulas along T3 for
Seiberg-li’itten inrariants, Math. Res. Lett. 4 (1997), 915—929.

[53] D. McDu'ff and D. Salannfm. J-holomorphic curves and symplectic topology,
American l\r-Ia.thematical Society Colloquium Publications, 52. American Math-
ematical Society, Providence, RI (2004).

[54] J.VV. Morgan, Z. Szaho’ and C. H. Taubes. A product formula. for the Seiberg-
l/Vitten invariants and the generalist—d Thom conjecture, J. Differential Geom. 44
(1996),706—788.

[55] J. McCarthy and J. “lolfson, Symplectic gluing along hypersuifaccs and resolu-
tion of isolated orbifold singularities, Invent. l\rla.th. 119 (1995). 129- 154.

134

155]

[57]

[58]

[59}

[60]

[61]

[(33]

[in]

[(36]

[(57]

[(38]

[(39]

[m]

B. Ozbagci, A.I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces,
Bolyai Society Mathematical Studies, 13 (2004), Springer-Verlag, Berlin.

B. Ozbagci, A. I. Stipsicz, Contact 3—manifolds with. infinitely many Stein fillings,
Proc. Amer. Math. Soc. 132 (2004) no. 5, 1549—1558 (electronic).

P. S. Ozsvath and Z. Szabé, Higher type adjunction inequalities in Seiberg- Witten
theory, J. Differential Geom. 55 (2000), 385—440.

P. S. Ozsvath and Z. Szahé, On Park '3 erotic smooth four-manifolds, in Geometry
and Topology of Manifolds, 253—260, Fields Inst. Commun., 47, Amer. Math.
Soc., Providence, RI (2005).

J. Park. Simply connected symplectic 4-manifolds with b: = 1 and (if = 2,
Invent. Math. 159 (2005), 657—667.

J. Park. Erotic smooth structures on 3CP2#8@2, Bull. London Math. Soc. 39
(2007), 95» 102.

J. Park, A. I. Stipsicz and Z. Szalx’), Erotic smooth structures on CP2#5@2,
Math. Res. Lett. 12 (2005). 701—712.

T. Perutz, Surface-fibrations. four-manifolds, and symplectic Flocr homology.
PhD. thesis. University of London (2005).

T. Perutz. Lagrangian matching invariants for fibrtd four-maniolds: I, Geom.
Topol. 11 (2007). 759 828.

T. Perutz. Lagrangia'n matching in'i'ariants for fibred four-manifolds: ll. preprint,
Inatli.SG/0606(")62.

T. Perutz, Zero-sets of near-symplectic forms. preprint, niath.SG/0601320.

O. Planrenevskaya, Contact structures with. distinct Heegaard F loer invariants.
Math. Res. Lett. 11 (2004) no. 4, 547—561.

F. Quinn. Dual dccornpositions of 4-manifolds, Trans. Amer. Math. Soc. 354 4
(2002), 1373-1392 (electronic).

I. Smith, Scrre-Taubes duality for pseudoholomorphic curries. Topology 42 (2003),
no. 5, 931- 979.

R. J. Stern. Lecture at the 'I‘opology of 4-l\lanifolds Conference in honor of R.
Fintushel's 60th birthday. Tulane University, November 10 12. 2006.

135

[71]

[78]

[79]

[84]

[85]

AI. Stipsicz and Z. Szabo, An exotic smooth structure on (31132946477152, Geom.
Topol. 9 (2005), 813—832.

A.I. Stipsicz, Sections of Lefschetz fibrations, Turkish Math. J. 25 (2001), 97--
101.

Z. Szabé, Erotic 4-manifolds with bfr = 1, Math. Res. Lett. 3 (1996), 731—741.

C.H. Taubes, The Seiberg- Witten invariants and symplectic forms, Math. Res.
Lett. 1 (1994), 809—822.

C.H. Taubes, The Seiberg- Witten and Gromov invariants, Math. Res. Lett. 2
(1995),221»238.

C. H. Taubes, The geometry of the Seiberg- Witten invariants. Docurnenta Math—
ematica, Extra Volume ICM 1998 II, 493- 504.

C. H. Taubes, Seiberg- I'Vitten invariants and pseudo-holomorphic subvarieties for
self-dual, harmonic 2-forms, Geom. Topol. 3 (1999), 167 210.

C. H. Taubes. A compeni‘lium of pseudoholomorphic beasts in IR X (S1 x SQ).
Geom. Topol. 6 (2002), 657—814.

C. H. Taubes, A proof of a theorem. of Luttinger and Simpson, about the number of
vanishing circles of a near-symplectic form on a 4-dimensional manifold, l\lath.
Res. Lett. 13 (2006), no. 4. 557—570.

“KP. Thurston, Some. simple eramples of symplectiij' manifolds, Proc. Amer.
Math. Soc. 55 (1976), 467-468.

W. P. Thurston. H. E. \\"inkelnkemper. On. the cristenec of contact forms. Proc.
Amer. Math. Soc. 52 (1975). 345 347.

M. Usher, The Gromov 'l‘tt't’tl'l‘itl'nt and the Donaldson-Smith standard surface
count. Geom. Topol. 8 (2004). 565—610 (electronic).

M. Usher. illinimality and symplectic sums. Internat. Math. Res. Not. 2006, Art.
ID 49857, 17 pp.

.l.von Bergmann. Pseudo—halo-morphic maps into folded symplectic four-
manifolds, Geom. Top. 11 (2007), 1-—45.

E. \Vitten, Monopoles and four-manifolds. Math. Res. Lett. 1 (1994), 769- 796.

136

   

liiiiiiigii[iii]