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On foliations of higher dimensional symplectic manifolds and symplectic mapping class group relations
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On foliations of higher dimensional symplectic manifolds and symplectic mapping class group relations
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Content
ON FOLIATIONS OF HIGHER DIMENSIONAL SYMPLECTIC
MANIFOLDS AND SYMPLECTIC MAPPING CLASS GROUP
RELATIONS
by
Bahar Acu
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
May 2017
Copyright 2017 Bahar Acu
Ji Dayika Min Re...
Acknowledgment
First and foremost, I would like to thank my advisor Ko Honda. Without his help, guidance,
and patience, this dissertation would surely have never been possible. I am also indebted to Francis
Bonahon for his invaluable help and support throughout my Ph.D. studies. He has been a great
mentor to me from day one until now.
I have beneted greatly from many fruitful and instructive conversations with John Etnyre and
Chris Wendl. A special thanks should go to Otto van Koert for generously spending his time to
talk to me and explain concepts, that are mostly obvious to him, and suggesting many promising
future projects to me when things went south. Symplectic mapping class group relations project
is a joint work with Russell Avdek. I'd like to thank him for keeping in touch with me throughout
the completion of that project although he is not in academia anymore.
I must thank my friends and family who helped me keep my sanity, more important than the
math, and pushed me with their endless encouragements. Strangely enough, they have never lost
their faith in me. I can't thank Bulut enough for being my long gone social side and feeding my
soul with his intellectuality by sharing things that I'd nd interesting to read or listen to. Aycan
and Apo, Ay se, Erg un, Esra, and Nesli have always been by my side, despite 7000 miles away, with
their friendships and warmth. My Los Angeles crew; Baran, Elli, Alyssa, Greg, Keyvan, Hana,
and Helga who do not only feel me but also share their time and energy to be around me when I
need, sometimes even when I don't.
Last but not least, I will say a couple of words about my family though some things go without
saying. They are due most of the credit. This dissertation is dedicated to the toughest woman that
I have known, my dear mother, Dilber who has never gone to school, but is the most hardworking,
determined, and perseverant person that I have known. I can't imagine the life I'd live without
being exposed to her wonderful heart and endless energy. My dear sister and best friend Ru sen has
5
constantly lightened up the dark moments these last six years by just being herself and reaching
out to me with her rational self. And my other siblings Ay se, Baver, Berat, Leyla, and Metin have
not only been supportive but also hardworking and successful models to me which made me realize
and believe that I was allowed to dream and even better, I could achieve my dreams while I was
growing up in a city where not many people were given the chance to go to school.
Finally, my dear partner G okhan who claims to be bad at math, but can give the denition of
a contact manifold on a good day deserves more than a plain thank you. I, without a shadow of a
doubt, consider this dissertation an accomplishment we achieved together.
6
Abstract
In the present thesis, we study symplectic mapping class group relations of higher dimensional
symplectic manifolds and closed Reeb orbits of higher dimensional contact manifolds. Hence, the
thesis is structured in two main parts. In the rst part, we introduce the notions of an iterated
planar open book decomposition and an iterated planar Lefschetz bration and prove the Weinstein
conjecture for \iterated planar" contact manifolds. In the second part, we examine mapping class
group relations of some symplectic manifolds. For each n¥ 1 and k ¥ 1, we show that the 2n-
dimensional Weinstein domain W tf 0uXB
2n2
, determined by the degree k homogeneous
polynomialfPCrz
0
;:::;z
n
s, has a Boothby-Wang type boundary and a right-handed bered Dehn
twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists
along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the
case n 2 recovering the classical chain relation for the torus with two boundary components.
7
Contents
Acknowledgment 5
Abstract 7
Chapter 1. Introduction 10
1.1. Preliminaries 10
1.2. Stable Hamiltonian structures 14
1.3. Neck stretching 16
1.4. Moduli spaces of J-holomorphic curves 18
1.5. The Weinstein conjecture 21
Chapter 2. Iterated planar open book decompositions 23
2.1. Open book decompositions 23
2.2. Iterated planar Lefschetz brations 25
Chapter 3. Proof of Theorem 1.5.2 33
3.1. Setup 33
3.2. Index calculation 36
3.3. Induced complex structure on T
S
n1
38
3.4. Regularity of J-holomorphic curves 44
3.4.1. Construction of the suitable almost complex structure 44
3.4.2. Proof of the regularity 46
3.5. Compactness of the moduli space 52
3.6. Gluing 55
3.7. Neck stretching 56
3.8. Proofs of Proposition 3.1.1 and Theorem 1.5.2 59
8
Chapter 4. Symplectic mapping class group relations 61
4.1. Boothby-Wang bundles 66
4.2. Fibered Dehn twists and fractional bered Dehn twists 68
4.3. Setup 70
Chapter 5. Proofs of Theorem 4.0.1 and 4.0.4 72
5.1. Lefschetz bration construction 72
5.2. Boothby-Wang construction 74
5.3. Comparison of two views 75
5.4. Proof of the Corollary 4.0.2 77
5.5. Proof of the Theorem 4.0.4 78
5.6. Proof of the exact number of Dehn twists 82
Chapter 6. The case n 2 85
6.1. The case of arbitrary k¥ 2 85
6.2. The case k 3 87
6.3. From monodromy to mapping class group relations 89
Bibliography 93
9
CHAPTER 1
Introduction
In this chapter, we will outline some prerequisite material on symplectic eld theory which
is essential in the proof of Theorem 1.5.2. Throughout, all manifolds are smooth oriented and
dimension means real dimension unless stated otherwise.
1.1. Preliminaries
A contact manifoldpM;q is ap2n1q-dimensional manifold equipped with a smooth 2n-plane
eld TM which is locally ker, where P
1
pMq which satises ^pdq
n
0.
We say that is a contact structure onM and is a contact form which locally denes. For
any given contact form, there exists a unique vector eldR satisfyingdpR;q 0 andpRq 1.
Such a vector eld R is called the Reeb vector eld for .
A symplectic manifold pW;!q is a 2n-dimensional manifold equipped with a closed 2-form
!P
2
pWq satisfying !
n
0 called the symplectic form.
One way to relate symplectic manifolds with contact manifolds is to realize contact manifolds
as hypersurfaces of symplectic manifolds as follows:
Definition 1.1.1. A codimension 1 hypersurface M of a symplectic manifoldpW;!q is called
contact type if there is a neighborhoodN ofM such that!d for someP
1
pNq and a vector
eld Z on W determined by
Z
! positively transverse to M.
Observe that the second condition implies that |
M
is a contact form.
Definition 1.1.2. A Liouville domain is a triplepW;!;Zq, where
(1) ! is a symplectic form on W .
(2) pW;!dq is a compact exact 2n-dimensional symplectic manifold with boundary.
(3) Z is a Liouville vector eld on W , i.e.,L
Z
!! and points outward at the boundary.
10
A Liouville form is a contact form
Z
! dening the Liouville vector eldZ. In this thesis,
we are primarily interested in the following class of symplectic manifolds whose boundaries are
contact hypersurfaces:
Definition 1.1.3. LetW be a compact 2n-dimensional manifold with boundary. A Weinstein
domain structure on W is a triplep!;Z;q, where
(1) !d is a symplectic form on W .
(2) Z is a Liouville vector eld dened by
Z
!.
(3) :W ÑR is a generalized Morse function for which Z is gradient-like.
(4) hasBW t 0u as a regular level set.
The quadruplepW;!;Z;q is then called a Weinstein domain.
Notation. Throughout, we will abuse the notation and writepW;!q instead ofpW;!;Z;q.
Observe that any Weinstein domain is a Liouville domain. Also, any Weinstein domainpW;!q
has the contact type boundary pBW; kerq, where
Z
!. Moreover, a contact manifold
pM;q is called Weinstein llable if it is the boundary of a Weinstein domainpW;!q.
Definition 1.1.4. LetpM; kerq be a contact manifold. The symplectization of M is an
open symplectic manifoldpRM;dpe
s
q where s is theR-coordinate.
The contact condition on M implies that dpe
s
q is a symplectic form on the symplectization
RM. Observe thatB
s
is a Liouville vector eld transverse to the slices tsuM. Thus, every
contact manifold is a contact type hypersurface in its own symplectizationpRM;dpe
s
qq.
Definition 1.1.5. IfpM;q is the boundary of a Weinstein domainpW;!dq, then one can
smoothly attach a cylindrical end to dene a larger symplectic manifold
(1.1.1) p
x
W; ^ !qpW;!qY
BW
pr0;8qM;dpe
s
qq
called the symplectic completion ofpW;!q.
LetM be ap2n 1q-dimensional contact manifold and be a nondegenerate contact form on
M. Denote by R
the Reeb vector eld for .
11
Definition 1.1.6. Let W be a 2n-dimensional manifold. An almost complex structure is an
endomorphism J : TW Ñ TW such that J
2
1. pW;Jq is then called an almost complex
manifold.
Definition 1.1.7. An almost complex structure J on a symplectic manifoldpW;!q is said to
be compatible with ! (or !-compatible) if
(1) !pJu;Jvq!pu;vq, for any u;vPTW and
(2) !pv;Jvq¡ 0, for any nonzero vPTW .
It is known that the set of all almost complex structures on a symplectic manifold pW;!q is
nonempty and contractible. Therefore, the tangent bundle TW can be considered as a complex
vector bundle.
Definition 1.1.8. LetpF;jq be a Riemann surface andpW;Jq be an almost complex manifold.
A J-holomorphic curve (or pseudoholomorphic curve) is a smooth map u :F ÑW such that its
dierential, at every point, satises the following complex-linearity condition:
dujJdu
or, equivalently,B
J
u 0.
We will now dene a special class of compatible almost complex structures onpRM;dpe
s
qq.
Since RM is noncompact, an almost complex structure J on RM needs to satisfy certain
conditions near innity in order to obtain a well behaved space of J-holomorphic curves. Note
that the symplectization of M inherits a natural splitting of the tangent bundle TpRMq
RxB
s
y`RxR
y`, whereB
s
is the unit vector in theR-direction. We will use this information to
make the following denition:
Definition 1.1.9. An almost complex structure J onRM is called -compatible if
(1) J isR-invariant.
(2) JpB
s
qR
and JpR
qB
s
, whereB
s
is the unit vector in theR-direction.
(3) Jpq.
(4) J|
is d-compatible.
12
Notation. We will denote byJpRMq the space of all-compatible almost complex structures
on RM.
Definition 1.1.10. LetpM;q be a contact type hypersurface in a Weinstein domainpW;!
dq. An!-compatible almost complex structure J onW is called adapted toM ifJ restricts to a
|
M
-compatible almost complex structure on a collar neighborhood of M.
M
R
Z
Figure 1. Figure depicting the adapted almost complex structure in a neigh-
borhood of a contact hypersurface M. Blue lines represent the
ow lines of the
Liouville vector eld Z. Here R is the Reeb vector eld for the contact form on
M.
Definition 1.1.11. A positive end (resp., negative end) of a J-holomorphic curve u is a
neighborhood of a puncture on which u is asymptotic to some R
as sÑ8 (resp., sÑ8),
where
is a closed Reeb orbit. TheJ-holomorphic curveu is then called asymptotically cylindrical.
13
1.2. Stable Hamiltonian structures
Definition 1.2.1. A stable Hamiltonian structure (SHS)H on ap2n1q-dimensional manifold
M is a pairp
; q consisting of a closed 2-form
and 1-form dened on M with the following
properties:
(1) ker
kerd.
(2) ^
n
¡ 0.
Note that the condition p1q can equivalently be written as d g! where g : M Ñ R is a
smooth function. Observe also that the conditionp2q is equivalent to
|
is nondegenerate, where
ker is a co-oriented hyperplane distribution. In other words,
|
is a symplectic vector
bundle. Moreover, on a neighborhood p;qM of M, for ¡ 0 suciently small, dpsq
is a symplectic form where sPp;q. One can generalize this further to RM by letting the
symplectic form onRM to be !
dppsqq
where :RÑp;q is a strictly increasing
function. Observe that the symplectization ofpM;Hq is symplectomorphic topRM;!
q.
Definition 1.2.2. The Reeb vector eld R
H
of a stable Hamiltonian structure H on M is a
vector eld characterized by
(1) pR
H
q 1,
(2)
pR
H
;q 0.
Notice that the
ow of the Reeb vector eld R
H
preserves ,
, and g, i.e., L
R
H
0,
L
R
H
0, andL
R
H
g 0.
Definition 1.2.3. Let M be a hypersurface in a symplectic manifold pW;!q. A stabilizing
vector eld Z on a neighborhood of M is a vector eld satisfying the following properties:
(1) Z is transverse to M.
(2) The 1-parameter family of hypersurfacesM
s
tsuM, forsPp;q, generated by the
ow of Z preserves the Reeb vector elds. That is, the Reeb vector eld on each slice
M
s
tsuM is independent of s.
14
Example 1.2.4. LetpM; kerpqq be a contact hypersurface of a symplectic manifoldpW;!q
then H pd;q is a stable Hamiltonian structure in which R
H
is the Reeb vector eld in the
usual sense, the Liouville vector eld Z transverse to M is a stabilizing vector eld and g 1.
Recall that
|
is symplectic. One can then dene
-compatibility of an almost complex
structureJ onRM after controlling the (end) behavior of J-holomorphic curves inRM near
innity.
Definition 1.2.5. An almost complex structure J onRM is called
-compatible if
(1) J isR-invariant.
(2) JpB
s
qR
H
and JpR
H
qB
s
, whereB
s
is the unit vector in theR-direction.
(3) Jpq.
(4) gpu;vq
pu;Jvq is a Riemannian metric on RM, where u;v P , i.e., J|
is
-
compatible.
Notation. We will denote byJpHq the space of all
-compatible almost complex structures J
onr0;8qM.
15
1.3. Neck stretching
In this section, we describe neck-stretching, which is a technique originating in symplectic eld
theory (SFT), also known as the splitting of a symplectic manifoldW along a contact hypersurface
M by deforming the almost complex structure on W in some neighborhood of M. See [BEH
]
and [EGH] for more details.
LetpM;q be a contact type hypersurface of an exact symplectic manifoldpW;!dq andZ
be a Liouville vector eld positively transverse to M. Suppose that M divides W into two parts
W
andW
with two boundary componentsM
andM
, respectively, so thatZ points outwards
along M
and inwards along M
.
Let J be an !-compatible almost complex structure on pW;!q adapted to pM;q and Z be
a Liouville vector eld on W . We wish to construct a family of !-compatible almost complex
structures on W as follows:
Let I
t
rt;ts. Denote by X
s
the
ow of the Liouville vector eld Z. Dene a
dieomorphism '
t
as follows:
'
t
:I
t
MÑW;
ps;mqÞÑX
psq
pmq;
where :I
t
Ñr;s is a strictly increasing function chosen in such a way that it is close to being
the identity map near s 0.
Assume that I
t
M is equipped with the symplectic form !
dpe
psq
q'
t
p!q. Let
~
J
t
be
an almost complex structure on I
t
M such that
~
J
t
|
J|
. Assume also that
~
J
t
is invariant in
the Liouville direction.
Once can then form
pI
t
M;
~
J
t
qY
't
pW'
t
pI
t
Mq;Jq
by gluing via the dieomorphism '
t
. The resulting manifold is then symplectomorphic topW;J
t
q
whereJ
t
is the new!-compatible almost complex structure on W . The familyJ
t
of!-compatible
almost complex structures on W is then called a neck-stretching ofJ along a neighborhood of M.
16
W
T
S
n
W
M
M
t
M
8
tÑ8
Figure 2. The illustration of the stretching of a sequence of almost complex
structures along a neighborhood of a contact hypersurface M until it breaks the
manifold into 3 noncompact pieces
W ,M
8
RM andT
S
n
. HereM
t
denotes
I
t
M.
17
1.4. Moduli spaces of J-holomorphic curves
Let pW;! dq be a compact Liouville cobordism from M
to M
. Here M
denotes the
positive and negative boundarie of W , i.e., the Liouville vector eld points outward at M
and
inward at M
. Let R
be the Reeb vector eld for
|
M
.
The symplectic completion p
x
W; ^ !q of pW;!q is then the following noncompact symplectic
manifold dened by attaching cylindrical ends to some collar neighborhood of M
:
(1.4.1) p
x
W; ^ !qpp8; 0sM
;dpe
s
qqY
M
pW;!qY
M
pr0;8qM
;dpe
s
qq
Definition 1.4.1. An almost complex structure J on p
x
W; ^ !q is called ^ !-compatible if the
following conditions hold:
(1) On the endspr0;8qM
;dpe
s
qq andp8; 0sM
;dpe
s
qq,
(a) J isR-invariant.
(b) JpB
s
qR
and JpR
qB
s
, whereB
s
is the unit vector in theR-direction.
(c) Jpq.
(d) J|
is compatible with d.
(2) On W , J is !-compatible, i.e., gp;q!p;Jq is a Riemannian metric.
Notation. We will denote by Jp
x
Wq the set of all ^ !-compatible almost complex structures on
x
W .
We can now make the following denition:
Definition 1.4.2. Fix nonnegative integers k
and k
. Let
p
1
;:::;
k
q be ordered
sets of Reeb orbits of R
and let A be a relative homology class in H
2
pW;
Y
q where, by
abuse of notation,
are viewed as subsets of M
.
The moduli space of ^ !-compatible J-holomorphic curves, with m marked points, homologous
to A and asymptotic top
;
q in
x
W is a family of equivalence classes of tuples
M
x
W;m
pJ;A;
qtpu :p
9
F;jqÑp
x
W;Jq;pz
1
;:::;z
m
qq|
B
J
u 0;u
r
9
FsA;z
i
P
9
F {
with the following properties:
18
(1) F is a closed genus 0 surface.
(2)
pz
1
;:::;z
k
q and
pz
1
;:::;z
k
q are disjoint and ordered sets of positive and
negative punctures in F .
(3)
9
F
F zp
Y
q is a genus 0 surface with k punctures where kk
k
.
(4) u : p
9
F;jq Ñ p
x
W;Jq is an asymptotically cylindrical J-holomorphic curve, i.e., u is as-
ymptotic toR
i
near each z
i
P for i 1;:::;k
and dujJdu.
(5) u :p
9
F;jqÑp
x
W;Jq represents the relative homology class APH
2
pW; ~
Y ~
q.
Here the equivalence is given by
pu;pz
1
;:::;z
m
qqpu;p
1
pz
1
q;:::;
1
pz
m
qqq
where is an automorphism ofp
9
F;jq taking
to
and
to
.
We explain what it means for u : p
9
F;jq Ñ p
x
W;Jq to represent A. Let
F be the compact
surface with boundary which is obtained from
9
F by addingt8uS
1
to each cylindrical end. Let
r :
x
W Ñ W be the retraction onto W such that r|
W
is the identity map and rps;mq mP M
when ps;mqPp8; 0sM
or r0;8qM
. Then ru :
9
F Ñ W has a continuous extension
u :p
F;B
FqÑpW; ~
Y ~
q and by u representing a nonzero relative homology class ArusP
H
2
pW; ~
Y ~
q, we mean u representing A whose image under
H
2
pW; ~
Y ~
q
B
ÝÑH
1
p~
Y ~
q
is given as
BA
k
¸
i1
r
i
s
k
¸
i1
r
i
sPH
1
p~
Y ~
q:
The moduli spaceM
x
W;m
pJ;A;
q then comes with a natural map that records the evaluation
of the curves at the marked points.
Definition 1.4.3. The evaluation map ofM
x
W;m
pJ;A;
q is the map
ev :M
x
W;m
pJ;A;
qÑ
x
W
m
;
pu;pz
1
;:::;z
m
qqÞÑpupz
1
q;:::;upz
m
qq:
19
The evaluation map encodes relations between the topology of
x
W and the structure of the
moduli space of curves in
x
W . The degree of evaluation map is an algebraic count of the number
ofJ-holomorphic curves with a marked point passing through a generic point in
x
W that carry the
relative homology class A.
In dimension 4, there are particularly powerful results regarding the intersection properties of
J-holomorphic curves.
Theorem 1.4.4. [MS, Theorem 2.6.3, Positivity of intersections]pX;Jq LetX be a 4-dimensional
almost complex manifold and u
1
andu
2
be two simple J-holomorphic curves in X representing the
homology classes A
1
, A
2
P H
2
pXq. Each intersection of the images contributes a positive integer
to the homological intersection A
1
A
2
of the two curves and that number is 1 if and only if the
intersection is transverse.
In particular, the homological intersectionA
1
A
2
is always nonnegative. If it is 0, thenu
1
and
u
2
are disjoint J-holomorphic curves. We also note that the positivity of intersections is a local
result and applies to punctured J-holomorphic curves in the symplectization.
Theorem 1.4.5. [Wend, Theorem 2, Automatic transversality] Let u be an embedded punc-
tured J-holomorphic curve in the symplectization pRM;Jq of a 3-manifold M equipped with a
stable Hamiltonian structure and generic J. Then every curve inM
RM;m
pJ;A;
q is embedded.
20
1.5. The Weinstein conjecture
LetM be a smoothn-manifold. Fix a Riemannian metricg onM. Forpv;uqPT
M, consider
the canonical 1-form
can
udv on the cotangent bundle T
M. Then the restriction of
can
to
the unit cotangent bundle
ST
Mtpv;uqPT
M||u| 1u
is a contact form. The Reeb vector eld for is then the geodesic
ow under the identication
TMT
M determined by g and the associated Reeb orbits are closed geodesics.
Motivated by the discussion above, Alan Weinstein formulates the following conjecture known
as the Weinstein conjecture in 1978:
Conjecture 1.5.1. On a compact contact manifold, any Reeb vector eld carries at least one
closed orbit.
Several methods from symplectic geometry have been used to prove the conjecture in many
cases. However, it is still not known if those methods can be used to prove the conjecture in all
cases.
There are many results that move us towards understanding the Weinstein conjecture in both
3 and higher dimensions such as results of Etnyre [Etn], Floer, Hofer and Viterbo [FHV], [H],
[HV], [Vit], and also of Taubes [T]. The conjecture was proven for all closed 3-dimensional contact
manifolds by Taubes [T] by using Seiberg-Witten theory. However, it is still open in higher
dimensions. Yet there are partial results analyzing the conjecture in higher dimensions. In [Vit],
the conjecture was proven for all closed contact hypersurfaces inR
2n
,n¥ 2. In [HV] and [FHV],
the result was extended to cotangent bundles by using the presence of holomorphic spheres and
also to a large class of aspherical manifolds, respectively.
In the rst part of the thesis, we examine the Weinstein conjecture in the case of a special
class of higher dimensional contact manifolds and prove the following theorem.
Theorem 1.5.2. LetpM;q be ap2n 1q-dimensional iterated planar contact manifold. For
any on M, the associated Reeb vector eld on M admits a closed Reeb orbit.
21
Notice that when n 1, M is a planar contact manifold and in that case, the Weinstein
conjecture is known to be true [ACH].
22
CHAPTER 2
Iterated planar open book decompositions
In this section we introduce the notions of an iterated planar Lefschetz bration, an iterated
planar open book decomposition and related necessary background to introduce these two notions.
2.1. Open book decompositions
Definition 2.1.1. An abstract open book decomposition is a pairpF; q, where:
(1) F is a compact 2n-dimensional manifold with boundary, called the page and
(2) :F ÑF is a dieomorphism preservingBF , called the monodromy.
Definition 2.1.2. An open book decomposition of a compactp2n 1q-dimensional manifold
M is a pairpB;q, where:
(1) B is a codimension 2 submanifold of M with trivial normal bundle, called the binding of
the open book.
(2) :MBÑS
1
is a ber bundle of the complement ofB and the ber bundle restricted
to a neighborhood BD of B agrees with the angular coordinate on the normal disk
D.
The preimage F
:
1
pq, for P S
1
, gives a 2n-dimensional manifold with boundary
BF
B called the page of the open book. The holonomy of the ber bundle determines a
conjugacy class in the orientation-preserving dieomorphism group of a pageF
xing its boundary,
i.e., in Di
pF
;BF
q which we call the monodromy.
Since M is oriented, pages are naturally co-oriented by the canonical orientation of S
1
and
hence are naturally oriented. Also, the binding inherits an orientation from the open book decom-
position ofM. We assume that the given orientation onB coincides with the boundary orientation
induced by the pages.
23
By using this description, one can construct a closed p2n 1q-dimensional manifold M from
an abstract open book in the following way: Consider the mapping torus
F
r0; 1sF
L
p0; pzqqp1;zq:
The boundary of F
is then given by
BF
r0; 1sBF
L
p0;zqp1;zq;
since pzqz onBF . Let|BF
| denote the number of boundary components of F
. We set
M
pF;q
F
Y
§
|BF|
DBF
via the following identication:
pp;pqPBF
qpp;pqPBDBFq:
Here the boundary of each diskS
1
tptu inDBF gets glued totptuS
1
in the mapping torus.
The abstract open bookpF; q is then an open book decomposition of a closedp2n1q-dimensional
manifold M if M
pF;q
is dieomorphic to M.
The mapping torus F
carries the structure of a natural bration F
Ñ S
1
, away fromBF ,
whose ber is the interior of the page F of the open book decomposition.
Definition 2.1.3. A contact structure on a compact manifold M is said to be supported by
an open book pB;q of M if it is the kernel of a contact form satisfying the following:
(1) is a positive contact form on the binding and
(2) d is positively symplectic on each ber of .
If these two conditions hold, then the open book pB;q is called a supporting open book for the
contact manifold pM;q and the contact form is said to be adapted to the open book pB;q.
Furthermore, the contact form is called the Giroux form.
24
2.2. Iterated planar Lefschetz brations
Definition 2.2.1. A topological Lefschetz bration is a smooth map f :W ÑD, where W is
a compact manifold of dimension 2n with corners andD is a 2-disk, with the following properties:
(1) The critical points of f are isolated, nondegenerate, and are in the interior of W .
(2) If p P W is a critical point of f, then there are local complex coordinates pz
1
;:::;z
n
q
about p p0;:::; 0q on W and z about fppq on D such that, with respect to these
coordinates, f is given by the complex map zfpz
1
;:::;z
n
qz
2
1
z
2
n
.
(3) There exists a decomposition of BW into horizontal and vertical parts B
h
W and B
v
W ,
respectively, which meet at a codimension 2 corner and B
v
W f
1
pBDq and B
h
W
Y
zPD
BF
z
. HereB
h
W is the boundary of all bers including singular ones.
Denoting the critical points byp
1
;:::;p
r
PW and the corresponding critical values byc
1
;:::;c
r
P
D,r¥ 1, for eachzPDztc
1
;:::;c
r
u,F
z
f
1
pzq is ap2n2q-dimensional submanifold ofW with
nonempty boundary.
W
D
c
1 z
c
2
Figure1. The gure representing the Lefschetz brationf :W ÑD with critical
values c
1
and c
2
. Here the purple and blue curves represent vanishing cycles of
the regular ber over z. The bers with purple and blue dots represent singular
bers.
In order to talk about an open book decomposition induced by the boundary restriction of the
Lefschetz brationf, one should smooth the corners ofBW to begin with. The Lefschetz bration
f on W will then induce an open book decomposition onBW .
25
We shall now describe the compatibility conditions between a topological Lefschetz bration
f :W ÑD and a symplectic form ! on W as given in [KS].
Definition 2.2.2. A symplectic Lefschetz bration is a pairpf :W ÑD;!dq, where
(1) f is a topological Lefschetz bration.
(2) !d is an exact symplectic form on W such thatpBW;|
BW
q is a contact manifold.
(3) Each regular ber F
z
, z P Dztc
1
;:::;c
r
u, carries the structure of an exact symplectic
manifold with contact type boundary.
(4) LetF
z0
be a reference ber and denote by |
BFz
0
the pullback fromBF
z0
toDpr0; 1s
BF
z0
q. Then, on a neighborhood pr0; 1sBpB
v
WqqB
h
W of B
h
W , the contact form
|
BFz
0
Kf
pq, where is the standard Liouville form onD andK is a suciently
large constant.
Note that the condition (4) in the denition above implies that !d is an exact symplectic
form and the vector eld dened by is transverse toBW and points outwards (see McLean [Mc,
Theorem 2.15]). Therefore, one can smooth the corners ofBW to obtain a symplectic deformation
of W .
We will next examine the compatibility of an almost complex structure on with a symplectic
Lefschetz bration. Given a symplectic Lefschetz bration pf : W ÑD;! dq. Fix a complex
structure j onD. We can then make the following denition:
Definition 2.2.3. An almost complex structure J on W is compatible with a symplectic Lef-
schetz bration pf :W ÑD;!dq if
(1) f ispJ;jq-holomorphic.
(2) J J
0
and j j
0
near critical points of W , where J
0
and j
0
are the standard almost
complex and complex structures onC
2
andC, respectively.
(3) In a neighborhood ofB
h
W , J is a product almost complex structure.
(4) In a neighborhood ofB
v
W , J is invariant in the radial direction.
(5) J|
T
h
W
is !-compatible, where T
h
W is the horizontal component of the tangent bundle
TW .
26
Note that the matrix for J never achieves to be antisymmetric. Therefore, J is not !-
compatible. However, J is tamed by ! (see Seidel [Sei3, Lemma 2.1]). Denote by J
f
pWq the
space of almost complex structures compatible with a symplectic Lefschetz bration. Moreover, it
is known thatJ
f
pWq is nonempty and contractible.
Definition 2.2.4. A completion of the symplectic Lefschetz bration pf :W ÑD;!dq is
a pairp
^
f :
x
W ÑC;p !d
p
q, where
(1) The completion of a neighborhood ofB
h
W is dened by
x
W
v
pr1;8qB
h
WqYW .
(2) The contact form |
BFz
extends over the cylindrical endr1;8qB
h
W to s|
BFz
where
s denotes theR
-direction.
(3) Denote by B
v
p
x
W
v
q the vertical boundary of
x
W
v
. The completion of W is dened by
x
W
x
W
v
YpB
v
p
x
W
v
qr1;8qq.
(4) The extension of f over Y
B
v
p
x
W
v
qr0;8q is
^
f|
Y
pv;tq
^
f|
Bvp
x
Wvq
pvq.
(5) The contact form
p
|
^
F
K
^
f
pp q, where|
^
F
is the pullback fromBF toDpr1;8q
BFq, p is the standard Liouville form onC, and K is a suciently large constant.
Now we will turn our attention to the almost complex structures on the completion of a
symplectic Lefschetz bration. Let J be a compatible almost complex structure on
x
W and j be
the standard complex structure onC.
Definition 2.2.5. An almost complex structure J on
x
W is compatible with p
^
f :
x
W ÑC;p !q if
the following conditions hold:
(1) d
^
fJjd
^
f.
(2) Onpr1;8qBFqD, J is a product almost complex structure.
(3) OnB
v
Wr1;8q, J is invariant in the radial direction.
(4) J is -compatible on the completion of a neighborhood ofB
h
W .
(5) On W , J is !-compatible, i.e., gp;q!p;Jq is a Riemannian metric.
Denote byJ
h
p
x
Wq the space of all almost complex structures compatible with
^
f. It is known
thatJ
h
p
x
Wq is nonempty and contractible (c.f. [Sei3, Section 2.2]).
27
Definition 2.2.6. A Lefschetz bration f : W ÑD with dim
R
W 4 is called planar if all
bers of f have genus zero, i.e., are planar surfaces.
Note that the boundary restriction of a planar Lefschetz bration induces a planar open book
decomposition. Now we are ready to introduce a new class of Lefschetz brations for a Weinstein
domain W as follows:
Definition 2.2.7. A Weinstein domainpW
2n
;!q, n¥ 2, admits an iterated planar Lefschetz
bration if
(1) there exists a sequence of symplectic Lefschetz brations f
2
;:::;f
n
where f
i
:W
2i
ÑD
for i 2;:::;n.
(2) Each regular ber of f
i1
is the total space of f
i
, i.e., W
2i
is a regular ber of f
i1
.
(3) f
2
:W
4
ÑD is a planar Lefschetz bration.
Here the superscript 2i indicates the dimension of W
2i
. Notice that when n 2, W
4
admits a
planar Lefschetz bration.
Let us give here two important examples of manifolds admitting iterated planar Lefschetz
brations.
Example 2.2.8. For n ¥ 2, the unit disk bundle W
2n
T
S
n
admits an iterated planar
Lefschetz bration since each regular ber of a Lefschetz bration on T
S
n
is T
S
n1
and the
Lefschetz bration on T
S
2
is planar with bers T
S
1
r0; 1sS
1
.
Example 2.2.9. ConsiderA
k
-singularity which is, by denition, symplectically identied with
tpz
1
;:::;z
n
q|z
2
1
z
2
n1
z
k1
n
1upC
n
;!
std
q
for n¥ 3 and k¥ 2. It is known that A
k
-singularity can be expressed as a plumbing of k copies
of T
S
n1
. Notice that each regular ber of a Lefschetz bration on A
k
-singularity is T
S
n1
.
Hence, the discussion above implies also that A
k
-singularity admits an iterated planar Lefschetz
bration.
If f :W ÑD is an iterated planar Lefschetz bration, then, after smoothing the corners, the
boundary ofW inherits an open book decomposition whose pages are dieomorphic to the regular
28
bers of f. The following denition is motivated by looking at the open book decomposition
induced by the boundary restriction of an iterated planar Lefschetz bration.
Definition 2.2.10. An open book decomposition of a contact manifold pM
2n1
;q whose
pages admit iterated planar Lefschetz brations is called an iterated planar open book decomposition
and nally a contact manifold supporting such an open book decomposition is called an iterated
planar contact manifold.
We shall now construct an almost complex structure J on
x
W compatible with a stable Hamil-
tonian structure on
x
W to be able to study theJ-holomorphic curves on the pages of the open book
decomposition ofBW . Denote bypW;!dq a 4-dimensional Weinstein manifold with boundary
ofBW M. Suppose thatpM; kerq bounds a planar symplectic Lefschetz bration. Then
the following lemma holds:
Lemma 2.2.11. Let pf : W Ñ D;! dq be a planar symplectic Lefschetz bration. Then
there exists an almost complex structure JPJ
h
p
x
Wq compatible with a suitable stable Hamiltonian
structureH onr0;8qM such that the bers of
^
f :
x
W ÑC are holomorphic.
Proof. Recall that the space of almost complex structures compatible with the completion
of a symplectic manifold (c.f. Denition 2.2.5) is nonempty and contractible. Then, there exists
at least one J PJ
h
p
x
Wq dened as in Denition 2.2.5. It is left to show that it is also compatible
with some suitable stable Hamiltonian structure on the symplectization of M such that the bers
of
^
f are holomorphic.
We will start with the construction of a suitable stable Hamiltonian structure on M and then
examine the abstract open book decomposition of M. The advantage of working with stable
Hamiltonian structures is that if we pick a suitable stable Hamiltonian structure as in [Wenb],
then then it admits arbitrarily small perturbations that are contact structures supported by the
planar open book decomposition dened on the boundary of a planar Lefschetz bration.
Let and be the coordinates on the boundary of bers and on the boundary ofD, respectively.
Denote by s and t the vertical and horizontal collar coordinates of B
v
W and B
h
W , respectively.
Let F
z
be the ber over zP D. Consider the berwise Liouville form |
BFz
on each of the bers
such that
29
(1) d|
Fz
is a symplectic form and
(2) |
BFz
is a contact form.
Note that the berwise Liouville form |
BFz
is independent of the choice of z P D. Observe
also that|
BFz
e
s
d nearB
h
W . Let be the standard Liouville form onD such thate
t
d is
a Liouville form near the boundary ofD. By using this information, one can construct a Liouville
form near corners whose
ow smooths the corners ofBW .
f
s
t
W
M
M
a
b
B
h
W
B
v
W
D
Figure 2. The local picture representing the Lefschetz bration f after the
smoothing the corner of the total space W . Here the hypersurface M repre-
sents the smoothed boundary BW and M
denotes another hypersurface below
and suciently close to M.
We shall now construct a Liouville vector eld near the corners by turning the berwise Liou-
ville form into a Liouville form:
(2.2.1)
K
|
BFz
Kf
;
where K is a suciently large positive constant and !
K
d
K
is symplectic (see Seidel [Sei3,
Lemma 1.5]). Note that the Liouville form
K
induces a Liouville vector eldZ
K
pB
t
B
s
q near
the corner and a Reeb vector eldB
.
Now we want to dene a new vector eld Z in some neighborhood of M as follows:
30
Z :
$
'
'
'
'
'
'
&
'
'
'
'
'
'
%
Z
K
s¥a and nearB
h
W;
B
t
psqB
s
b¤s¤a;
B
t
for s¤b;
where is a suitably chosen cuto function.
In the light of this vector eld, one can construct a stable Hamiltonian structure H induced
by Z on M as follows:
:
z
!
K
|
M
;
:!
K
|
M
:
Recall that the characteristic vector eld is given by the -direction. Notice that the
ow of
Z doesn't change that line eld since there is no dependence on the s-direction. Now observe that
if we take the smooth hypersurface M and
ow it along Z, we obtain a 1-parameter family of
hypersurfaces whose characteristic line elds are all identical. In other words, Z is a stabilizing
vector eld. Hence, if we choose another hypersurface M
which is transverse to the Liouville
vector eld Z
K
, lying below M and also suciently close to it as in Figure 2, then the region in
between is a symplectic cobordism.
Observe also that on B
h
W , !|
B
h
W
Ke
t
dt^ d d|
B
h
W
d|
B
h
W
. Near the corner,
d|
BFz
e
s
ds^d. Note that this construction is true for any p2n 1q-dimensional Weinstein
llable contact manifold M.
The remainder of the proof closely follows the construction in [Wenb]. Denote bypF; q the
abstract open book decomposition of M. Let F
and B the associated mapping torus and the
binding, respectively. Here we are using the notation in Denition 2.1.1. By page, we mean the
bers of the mapping torusF
. The compatibility check ofJ onM will be done onF
andDB
separately.
OnF
, the co-oriented hyperplane distribution is tangent to the pages and is preserved under
any J we choose. Therefore, for any choice of J, the bers of the mapping torus, i.e., the pages
become holomorphic. That is to say, tsutuF in RF
is a J-holomorphic curve for any
31
choice of J. Now we need to check if that extends to a neighborhood of the binding. To do this,
we need to nd aJ-holomorphic cylinder asymptotic totr 0u near the binding that t smoothly
with the bertuF . To see this, we refer the reader to [Wenb] for a detailed analysis near the
binding.
One can then attach the cylinder to the holomorphic ber tsutuF which extends to
a holomorphic ber in the symplectization of F
to an embedded J-holomorphic curve in the
symplectization of the Weinstein llable contact manifoldM. Therefore,JPJ
h
p
x
Wq is compatible
with the symplectization of the stable Hamiltonian structureH.
32
CHAPTER 3
Proof of Theorem 1.5.2
The proof is based on an inductive argument which helps us carry several results and phenom-
ena particular to dimension 4, such as automatic transversality and positivity of intersections, to
higher dimensions.
3.1. Setup
Letf
i
:W
2i
ÑD be an iterated planar Lefschetz bration and let MM
2n1
andW W
2i
with BW
2i
M
2i1
. Let pRM
2n1
;dpe
s
qq be the symplectization of a contact manifold
pM
2n1
; kerq supporting an open book whose pages are 2n-dimensional Weinstein domains
W
2n
, wheres denotes theR-coordinate. Assume also thatW
2n
admits an iterated planar Lefschetz
bration.
Denote by
x
W the completion of a Weinstein domain W . Let JPJ
h
p
x
Wq. We will specify the
Reeb orbits
and the relative homology class A for the moduli spaceM
x
W;m
pJ;A;
q of planar
J-holomorphic curves in
x
W asymptotic to
. Notice that these curves do not have any negative
ends since W is a Weinstein domain.
When n 1, we take
to be the binding of the open book supported by M
3
. Similarly,
whenn 2, we taket0u
in the neighborhoodDBW
4
DM
3
of the binding of the open
book of M
5
.
More generally, when n¡ 1, the associated Reeb orbits for M
2n1
are
t0ut0ut0u
DDDM
3
DM
2n1
which we will simply call
throughout this thesis.
We set Art0ut0ut0uW
2
sPH
2
pW
2n
;
q where
t0ut0ut0uW
2
DW
2n2
:
33
Notation. In the rest of this thesis, we write M
x
W
pJq instead of M
x
W;m
pJ;A;
q. We will
assume m 1 unless stated otherwise.
Proposition 3.1.1. Suppose W
2n
admits an iterated planar Lefschetz bration. Then there
exists an almost complex structure J P J
h
p
x
W
2n
q compatible with a suitable stable Hamiltonian
structure H such that M
x
W
2n
pJq is regular and
x
W
2n
is \lled" by planar J-holomorphic curves,
i.e. the degree of the evaluation map ev: M
x
W
2n
pJqÑ
x
W
2n
is 1 mod 2.
The proof can be summarized as follows: Denote by f
n
: W
2n
Ñ D the associated iterated
planar Lefschetz bration. We will study the foliation of
x
W
2n
by planar J-holomorphic curves by
usingf
n
. Iff
n
has no singular points, then we can use the facts that there exists an almost complex
structure J compatible with a suitable stable Hamiltonian structure such that each regular ber
W
4
off
3
is foliated by planarJ-holomorphic curves [Wenb, Main Theorem] and each regular ber
of f
n
is W
2n2
. Hence, the lling follows.
If f
n
has singular points, we need to make sure that the holomorphic curves pass through
the singular points safely without any obstruction. Away from singular points, regular bers are
foliated by planarJ-holomorphic curves by using the inductive argument provided by the iterated
planar Lefschetz bration structure. However, around each singular point, it is not clear if we can
extend the family of J-holomorphic curves in a regular ber across the singular bers safely. The
singular point may block the holomorphic curves from passing through the singular point.
A priori, we have no control over curves when they get pinched. We want to extend the
family of curves in the regular ber across singular bers. In order to extend across the singular
bers, we will apply neck stretching to a neighborhood of a critical point since each curve in that
neighborhood will then look like limiting Reeb orbits. By neck stretching, we are able to normalize
the neighborhood of the critical point. Hence, the picture becomes clearer. This local analysis is
to make sure that there exists a planar J-holomorphic curve through each point in
x
W
2n
.
We will prove the Proposition 3.1.1 via an inductive construction examining the following
arguments inductively:
(1) All J-holomorphic curves in the associated moduli space are transversely cut out, i.e.,
the moduli space is regular.
34
(2) The compactication of the associated moduli space is nice. This will be elaborated in
Section 3.5.
(3) The associated moduli space has the expected dimension.
Once we achieve these for all n¥ 2, we can use the evaluation map to show that there exists
a unique holomorphic curve, up to algebraic count, passing through each generic point in
x
W
2n
since the evaluation map is a degree 1 map. The rest of the proof will then be reduced to applying
neck-stretching to a neighborhood of a contact hypersurface in RM
2n1
to observe which will
help us observe that all J-holomorphic curves in the moduli spaceM
RM;1
pJq are asymptotic to
closed Reeb orbits.
35
3.2. Index calculation
Let uPM
RM
pJq be a planar J-holomorphic curve inRM.
Lemma 3.2.1. The index of u (the Fredholm index of the linearized Cauchy-Riemann operator)
inM
RM
pJq is 2n.
Proof. Recall that the expected dimension of the moduli space of J-holomorphic curves in
RM is the Fredholm index of
B
J
u. This index is
(3.2.1) indpuqppn 1q 3qpq
¸
CZ
p
;q
¸
CZ
p
;q 2c
1
pu
TpRMqq;
where
and
are positive and negative ends ofup
9
Fq, respectively and is the trivialization of
along
. Note that the Conley-Zehnder index is independent of the choice of . The dimension
of the ambient manifold isp2n 2q, which explains the termppn 1q 3q.
Let k be the number of positive ends of up
9
Fq. Recall that we do not have any negative ends.
Therefore,
CZ
p
;q 0. Let us begin with the case n 2. With respect to the trivialization
for which boundary of the pages are longitudinal, the Conley-Zehnder index of each of the Reeb
orbits is
CZ
p
;q 1 for the elliptic case.
The relative rst Chern number of the bundle u
TpRMqÑ
9
F with respect to the trivial-
ization is
c
1
pu
TpRMqqc
1
pT
9
F`N
u
qc
1
pT
9
Fqc
1
pN
u
q
whereN
u
denotes the normal bundle to u. Observe also that transverse direction is trivial. Thus,
we have
c
1
pu
TpRMqqc
1
pT
9
Fq:
It remains to analyze the tangential component of c
1
. Note that the relative Chern class of a
planar surface
9
F with k positive ends is given by
c
1
pT
9
F;q;
9
F ¡ ep
9
Fq;
9
F ¡p
9
Fq 2k:
36
Having determined the relative rst Chern number, we shall compute the index for n 2:
indpuqp2 3qp2 0kqk 0 2pc
1
pT
9
Fqq
1p2kqk 2p2kq 2:
Now consider the case whenu :p
9
F;jqÝÑpRM;Jq and dim
R
pRMq 2n2. Consider the
projection map from a neighborhood nhdpM
2n3
q ofM
2n3
toD. This map gives a trivialization
of nhd(M
2n3
)M
2n3
D. We then get
CZ
p
;qpn 1q 1n:
Thus, Equation 3.2.1 implies that
indpuqppn 1q 3qp
9
Fqkn 0 2p
9
Fq
pn 2qp2kqkn 2p2kq
np2kqkn
2n:
37
3.3. Induced complex structure on T
S
n1
In this section, we consider
T
S
n1
tpu;vqPR
n
R
n
||u| 1 and uv 0uR
2n
and identify it with a ber of the completed Lefschetz bration
'
n
:C
n
ÑC;
pz
1
;:::;z
n
qÞÑz
2
1
z
2
n
:
This identication gives a complex structure J on T
S
n1
which is induced from C
n
with the
standard complex structure. We then prove:
Lemma 3.3.1. The standard complex structure onC
n
induces a complex structure on T
S
n1
which is asymptotically cylindrical at the ends of T
S
n1
.
Proof. Let us begin with examining the compatibility of T
S
n1
with the map C
n
Ñ C.
Consider the standard model
'
n
:C
n
ÑC;
pz
1
;:::;z
n
qÞÑz
2
1
z
2
n
:
Observe that the map '
n
is the completion of the map f : W
2n
Ñ D where W
2n
is the
compact total space obtained by cutting down the bers of'
n
in the following way: Regular bers
are obtained by cutting down each ber '
1
n
pwq of '
n
from T
S
n1
to the disk bundleDT
S
n1
and the base is obtained by restrictingC toD. The construction of the Lefschetz bration f from
the standard model '
n
is explained in detail in [Sei3].
Letw be a positive real number andxpx
1
;:::;x
n
q andypy
1
;:::;y
n
qPR
n
. Then consider
'
1
n
pwqtxiyPC
n
||x|
2
|y|
2
w and xy 0u:
Note that '
1
n
pwq is a symplectic submanifold of pC
n
;!q with the standard symplectic form
!
n
°
j1
dx
j
^dy
j
.
38
Consider T
S
n1
tpu;vqP R
n
R
n
||u| 1 and uv 0u R
2n
: Note that T
S
n1
is
symplectic submanifold of R
2n
with the restriction of the standard symplectic form dv^du on
R
2n
.
Consider the map
:C
n
ÑR
2n
;
xiyÞÑ
x
|x|
;y|x|
:
Denote by the restriction map|
'
1
n
p1q
. Then the ber'
1
n
p1q can be identied withT
S
n1
via the following dieomorphism:
:'
1
n
p1qÑT
S
n1
;
xiyÞÑ
x
|x|
;y|x|
:
Denote by
1
1
2
pxdyydxq|
'
1
n
p1q
and
2
vdu|
T
S
n1 the Liouville forms of '
1
n
p1q
and T
S
n1
, respectively. Note that
1
and
2
are pullbacks of the standard primitives of the
symplectic forms onC
n
andR
2n
to '
1
n
p1q and T
S
n1
, respectively.
Claim. The map :p'
1
n
p1q;
1
qÑpT
S
n1
;
2
q is an exact symplectomorphism.
Proof. In the light of the construction above, it is sucient to show that
p
2
q
1
is
exact.
pvduq
p
n
¸
j1
v
j
du
j
q;
n
¸
j1
y
j
|x|
1
|x|
dx
j
n
¸
l1
x
l
x
j
|x|
2
dx
j
;
n
¸
j1
py
j
dx
j
n
¸
l1
x
l
x
j
y
j
|x|
2
dx
j
q;
n
¸
j1
y
j
dx
j
ydx:
pvduq
1
2
pxdyydxqydx
1
2
pxdyydxq
1
2
pxdyydxq
1
2
dpxyq 0:
39
Hence, :'
1
n
p1qÑT
S
n1
is an exact symplectomorphism.
Via the dieomorphism : '
1
n
p1qÑ T
S
n1
, the complex structure on C
n
then induces a
complex structure onT
S
n1
. LetJ be a complex structure onT
S
n1
induced from the standard
complex structure onC
n
via the dieomorphism :'
1
n
p1qÑT
S
n1
.
End behavior of J: In this subsection, we consider the singular ber Q
s
given by
Q
s
$
'
&
'
%
|x|
2
|y|
2
0;
xy 0
,
/
.
/
-
and identify it with Q
s
'
1
n
p0qt0uÑT
S
n1
tzero sectionu.
Next we will analyze the behavior of the induced complex structureJ when it goes suciently
far out in the ber to see if it is cylindrical at the ends. To see this, we will examine the induced
complex structure J on the singular ber and regular ber, separately.
X
H
s
JX
S
Q
s
Figure1. Local picture representing the intersection of the singular berQ
s
with
a sphere S above a critical point. Here H
s
represents the hypersurface obtained
by the intersection of Q
s
with the sphere S given by the equation x
2
y
2
2c
2
for cPR.
Claim. The induced complex structure J is cylindrical on Q
s
.
Proof. The singular ber Q
s
is equipped with the standard symplectic form ! d where
1
2
pxdyydxq and the induced complex structure J. Denote by X the Liouville vector eld
given by X
1
2
x
B
Bx
y
B
By
. To observe the end behavior of the induced complex structure, we
40
will intersect Q
s
with some sphere depicted as in Figure 1 and dene the hypersurface, which we
will call H
s
, as follows:
H
s
$
'
'
'
'
&
'
'
'
'
%
|x|
2
|y|
2
0;
xy 0;
|x|
2
|y|
2
4:
,
/
/
/
/
.
/
/
/
/
-
(1) We rst show that the induced complex structure J sends the Liouville vector eld X to
the Reeb vector eld JX.
One can compute that the orthogonal complement to T
px;yq
H
s
at px;yq P H
s
is
spanned bypx;yq;py;xq;px;yq or equivalently,px; 0q;p0;yq;py;xq. Note that the induced
complex structure is the standard complex structure onC
n
. Therefore, one can compute
the vector eld JX as follows:
JXJ
1
2
x
B
Bx
y
B
By
1
2
y
B
Bx
x
B
By
:
Notice that JXpy;xq is tangent to the hypersurface H
s
for allpx;yqPH
s
. This
follows from the following computation.
py;xqpx; 0qyx 0
py;xqp0;yqxy 0
py;xqpy;xq|x|
2
|y|
2
0
Moreover, JX is the Reeb vector eld for . To see this,
i
JX
!|
Hs
!pJX;q|
Hs
!
1
2
y
B
Bx
x
B
By
;
|
Hs
1
2
pydyxdxq|
Hs
1
4
dp|x|
2
|y|
2
q|
Hs
0:
It also satises the rescaling condition:
pJXq
1
2
pxdyydxq
1
2
y
B
Bx
x
B
By
1
4
p|x|
2
|y|
2
q 1
41
since|x|
2
|y|
2
4 onH
s
. Thus, the induced complex structure J sendsX to the Reeb
vector eld JX.
(2) Next we show that the contact distribution is preserved by J.
Firstly, note thatpJXq
K
R px; 0q;p0;yq;py;xq;py;xq¡
K
. Forpx;yqP H
s
, we
know that the tangent bundleTH
s
KR px; 0q;p0;yq;py;xq¡ andTH
s
. Therefore,
for anypu;vqP, we have
pu;vqpx; 0qux 0; (3.3.1)
pu;vqp0;yqvy 0; (3.3.2)
pu;vqpy;xquyvx 0: (3.3.3)
Also, kerpq is perpendicular to the Reeb vector eld JX. Therefore, is spanned
by the orthogonal complement ofpx; 0q;p0;yq;py;xq; andpy;xq. That is,
R px; 0q;p0;yq;py; 0q;p0;xq¡
K
when restricted to the hypersurface H
s
. In addition to the relations above , we have
pu;vqpy; 0quy 0; (3.3.4)
pu;vqp0;xqvx 0: (3.3.5)
Next we claim that this implies that the contact distribution is preserved by J, i.e.,
Jpq. To see this, letpu;vqP. Then Jpu;vqpv;uq and
pv;uqpx; 0qvx 0 by Equation 3.3.5
pv;uqp0;yquy 0 by Equation 3.3.4
pv;uqpy; 0qvy 0 by Equation 3.3.2
pv;uqp0;xqux 0 by Equation 3.3.1
42
Therefore,pv;uqP. With that, we have veried that is preserved by the induced
complex structure.
Therefore, when H
s
ows with respect to X, the induced complex structure J becomes cylin-
drical at the ends of the singular ber Q
s
which proves the claim.
It is left to study the regular ber case. Consider the regular ber
Q
$
'
&
'
%
|x|
2
|y|
2
1;
xy 0
,
/
.
/
-
and identify it with T
S
n1
. Unlike the singular case, the induced complex structure J on Q
is not automatically cylindrical at the ends of Q. It is rather asymptotically cylindrical which
follows along the same lines as [B, Example 2] by using a cylindrical coordinate chart on the
symplectization of the unit sphere bundle ST
S
n1
.
This nishes the proof of Lemma 3.3.
43
3.4. Regularity of J-holomorphic curves
Here we recall the spaceJ
h
p
x
W
2n
q of almost complex structures compatible with the Lefschetz
bration
^
f
n
:
x
W
2n
ÑC.
Proposition 3.4.1. There exists an almost complex structure J PJ
h
p
x
W
2n
q compatible with
a suitable stable Hamiltonian structureH such that
(1) J is as desired in Lemma 3.3.1.
(2) J is an almost complex structure compatible with the symplectic connection away from
the singular bers such that the regular bers of f
n
are J-holomorphic,
(3) Each element inM
x
W
2n
pJq lies in a ber of f
n
,
(4) M
x
W
2n
pJq is regular.
Proof. The proof is structured in two parts. In Section 3.4.1, we will construct an almost
complex structure in J
h
p
x
W
2n
q that makes the bers of f
n
J-holomorphic. In Section 3.4.2, we
will show that the almost complex structure we obtained in the rst part of the proof makes the
moduli spaceM
W
pJq regular. Both parts concern analysis around a singular and a regular ber
along with an extension analysis to the rest of the total space W
2n
of f
n
.
3.4.1. Construction of the suitable almost complex structure. Consider the case
n 2. Then there exists an almost complex structure on W
4
compatible with a suitable sta-
ble Hamiltonian structure such that W
4
is foliated by nite-energy planar J-holomorphic curves
[Wenb, Main Theorem].
Next we will consider the case n¡ 2 in two parts:
Part I: Regular bers
Recall thatf
n
is an iterated planar Lefschetz bration. Then there exists a compatible almost
complex structure on each regular ber W
2n2
of f
n
such that W
2n2
is J-holomorphic and is
lled by planar J-holomorphic curves. Let pP D be a regular value away from critical values of
f
n
and consider a neighborhood N
p
of p. Above N
p
, the almost complex structure is a product
almost complex structure given by the almost complex structureJ on the berW
2n2
and the lift
of the complex structure on the base.
Part II: Singular bers
44
Recall that in a neighborhood of a critical point, the Lefschetz brationf
n
:W
2n
ÑD is given
by the complex map
C
n
ÑC
pz
1
;:::;z
n
qÞÑz
2
1
z
2
n
By Lemma 3.3.1, we know that the standard complex structure on C
n
induces an asymptot-
ically cylindrical complex structure J
0
in a neighborhood of a critical point. Therefore, f
n
is a
holomorphic map around each critical point. See Section 3.3 for a more comprehensive discussion
on the induced complex structure.
LetE
c
be a neighborhood of a critical valuecPD off
n
andE be the corresponding neighbor-
hood of the critical point aboveE
c
. We can then pick an asymptotically cylindrical product almost
complex structure J
c
on the complement f
1
n
pE
c
qE such that the the complement looks like a
product manifold. Since J
0
and J
c
have the same end behavior, there exists an almost complex
structure J such that singular bers of f
n
are J-holomorphic.
Claim. Let N
p
be a neighborhood of a regular value pPD. The almost complex structure J
extends to f
1
n
pDE
c
N
p
q.
Proof. Let N f
1
n
pDE
c
N
p
q. To study this extension, we will examine the almost
complex structure given by the transverse and ber directions of N. To extend the regularity to
the rest of the total space, we slowly interpolate between the regular almost complex structures
we obtained on the regular and singular bers. The change in almost complex structures in the
transverse direction is almost constant when zoomed in. Hence, the almost complex structure is
close to being a product structure since the variation in the transverse direction is negligible.
Notice that the transverse direction, i.e. symplectic connection direction, of N is given by the
pullback of the complex structure on D. To study the ber direction, we recall that the space
of almost complex structures compatible with the symplectic form on the bers is contractible.
Therefore, the extension in the ber direction is automatic. Hence, the almost complex structure
on a regular ber is homotopic to the almost complex structure on the singular ber.
45
c
1
c
2
c
m
D
Figure 2. Each dot represents a nondegenerate critical value c
i
P D where i
1;:::;m. Vertical line represents ber direction and horizontal line represents
transverse direction.
Denote by J
ci
the induced complex structure on the ber over each critical value c
i
, for
i 1;:::;m. Consider J
c1
and J
c2
over the critical values c
1
and c
2
, respectively. When we move
from c
1
to c
2
as shown in Figure 2, we switch from J
c1
to J
c2
by slowly interpolating between
them. Here, the change is very slow. When we zoom in the buer region, nothing changes locally.
Therefore, the induced complex structure in the buer region will almost look like a product locally.
Finally, one needs to check that no degenerations of curves, such as breaking, occur as we are
changing the almost complex structure J during this interpolation.
Consider the map
^
f
n
:
x
W
6
Ñ C. Observe that the cylindrical ends in the ber direction
containing the Reeb orbits map down to a point. Hence, by the open mapping theorem, every end
containing Reeb orbits has to live in a ber since
^
f
n
is a holomorphic map.
Letu be a curve in the cylindrical endRBW
4
containing the Reeb orbit depicted as in Figure
9. Hence, by the discussion above, u has image inside W
4
and W
4
admits a planar Lefschetz
bration whose bers are planar surfaces with no negative ends. Therefore, no suchu exists unless
it is a trivial cylinder and trivial cylinders are ignored in holomorphic buildings by convention.
Observe that one can apply this construction to W
2n
for all n¥ 2 inductively since the map
^
f
n
is a projection map from the completion of W
2n
to a point for all n, Thus, we could go one
dimension down to use the information that u is contained in a ber. Therefore, the proof of the
claim follows.
3.4.2. Proof of the regularity. We begin with examining the neighborhood of a regular
berW
2n2
lying aboveN
p
. Recall that allJ-holomorphic curves inW
2n2
are transverse due to
the inductive argument. Hence, the associated moduli space is regular away from critical points.
46
C
z
^
1
pzq
u
RBW
2n
S
1
RM
^
Figure 3. The gure on the left is the map
^
:RM ÑC. The gure on the
right is the ber
^
1
pzq over zPC and a 2-level building depicting the breaking
of curves. Here u represents the end of a broken curve containing the Reeb orbit
in
x
W
2n
.
The goal is now to do the regularity analysis in some neighborhood of a critical point. To do
this, we will use the local model where W
2n
is identied withC
n
and f
n
:W
2n
ÑD has only one
critical point. The local picture will then look like a conic bundle as in Figure 4. Note that one
can extend this model case to the whole bration by extending the base so that the base includes
all critical points.
C
n
C
tz
2
1
z
2
n1
0u
tz
2
1
z
2
n1
wu
0
Figure 4. Local picture representing the behavior of vanishing cycles above a
small neighborhood of a critical point.
47
Let us begin with the casen 3. Consider the iterated planar Lefschetz brationf
3
:W
6
ÑD,
where each regular ber W
4
of f
3
is foliated by planar J-holomorphic curves. We would like to
understand how, under ideal conditions, the foliation of W
6
degenerates as each regular ber W
4
of f
3
degenerates into a singular ber.
To see this, consider the following projection map
'
3
:C
3
ÑC;
pz
1
;z
2
;z
3
qÞÑwz
2
1
z
2
2
z
2
3
;
where'
1
3
pwqtxiyPC
3
||x|
2
|y|
2
w andxy 0u. One can similarly dene the following
projection map
:C
3
ÑC
2
;
pz
1
;z
2
;z
3
qÞÑpz
3
;wq;
wherewz
2
1
z
2
2
z
2
3
. Note that
1
pz
3
;wqtz
2
1
z
2
2
wz
2
3
u and
1
p0; 0qtz
2
1
z
2
2
0u.
Thus, around each critical point of f
3
, we have the local picture shown in Figure 5.
tz
2
1
z
2
2
z
2
3
0u
tz
2
1
z
2
2
z
2
3
wu
C
3
C C
2
0
C
3
'
3
z
3
w
wz
2
3
Figure 5. The gure on the left is the projection map '
3
onto w whose regular
bers'
1
3
pwq are dieomorphic toT
S
2
and the gure on the right is the projec-
tion map ontoz
3
andw whose regular bers aretz
2
1
z
2
2
wz
2
3
u. Here'
3
1
p0q
is the singular ber.
48
Consider a suciently small box around the origin as depicted in Figure 6. Each vertical slice
in that box will then represent the ber W
4
of f
3
. Recall that in order to show regularity, it is
sucient to prove the following local statement:
(1) Each point on the vertical slice W
4
should be regular in that slice (see Figure 6). That
is, all J-holomorphic curves in W
4
are transversely cut out.
(2) There exists a biholomorphism such that W
6
W
4
B
, where B
is a ball centered at
0 with radius inC, for ¥ 0 small.
Second condition above implies that, in a suciently small box, the almost complex structure
on W
4
which we will call J
2
P AutpTW
4
q is invariant in the horizontal direction. That is to
say, the almost complex structure on W
6
in some neighborhood of a critical point, which we will
call J
3
P AutpTW
6
q, can be expressed as a split almost complex structure pJ
2
;jq, where j is the
standard complex structure inC, i.e., in the horizontal complex direction z
3
.
In the light of the recall above, we need to write down an almost complex structure J
3
which
works with the mapC
3
ÑC such that
(1) if we take vertical slices in the box in Im() as shown in Figure 6, then the slices will
look like T
S
2
,
(2) if we move in the horizontal direction z
3
, then the almost complex structure is just a
small variation of J
3
which is regular.
Once we nd such an almost complex structure J
3
, we need to make sure that it extends to
the rest of the total space. Firstly, observe that the standard complex structure on C
2
induces
a complex structure, which we will call J
2
, on T
S
2
. See Section 3.3 for details. Consider the
projection map :C
3
ÑC
2
and a point pPC
2
close to the origin. We want to show that
1
ppq
is transversely cut out with respect to the standard complex structure in C
3
. Observe that each
vertical slice as shown in Figure 6 is transversely cut out since the dimension of each vertical slice
is 4. In order to show that it is transversely cut out entirely, we need to show J
2
varies suciently
slow in z
3
-direction.
49
wz
2
3
z
3
w
C
2
Figure 6. The graph of the singular locus of the bration , where each red
vertical line represents vertical slices in a small neighborhood of the origin.
Let :C
4
ÑC
4
be a biholomorphism dened as follows:
:pz
1
;z
2
;z
3
;wqÞÑpz
1
;z
2
;z
3
;wz
2
3
w
1
q;
tz
2
1
z
2
2
z
2
3
wuÞÑtz
2
1
z
2
2
w
1
u:
The dieomorphism is equivalent to straightening out the singular locus where each vertical
slice gets mapped to another vertical slice in the range as shown in Figure 7. Therefore, each
vertical slice will be invariant inz
3
-direction. Hence, the biholomorphism makes the total space
into a product which is sucient to guarantee regularity.
w
w
1
z
3
z
3
Figure7. Biholomorphism sending the singular locuswz
2
3
to the horizontal
line w
1
0. Each box represents a small open neighborhood of singular locus
around the origin. Red lines represent vertical slices.
The existence of the biholomorphism guarantees the fact that the induced complex structure
J
3
on the total space is split in the neighborhood of a critical point and looks like symplectization
at the ends. Therefore, all J-holomorphic curves are transversely cut out in a small neighborhood
of a critical point. To extend this to the rest of the the neighborhood of the singular ber which
is a product manifold as explained in the rst part of the proof, we assume that the generic
asymptotically cylindrical product almost complex structure we pick on the complement is regular.
50
We can then glue these two regular structures to get a regular almost complex structure on the
neighborhood of the singular ber.
The construction above applies to alln¥ 3 as the projection map onto the last two variables
z
n
and w will then correspond to the singular locus wz
2
n
as in Figure 5, and thus, the induced
almost complex structure will look like J :J
n1
pJ
n
;jq for all n¥ 2. Thus, the regularity of
the holomorphic curves follows.
51
3.5. Compactness of the moduli space
Here we study the compactness of the moduli spaceM
RM
pJq by studying the relevant com-
pactness results in the symplectic eld theory and also by analyzing the existence of broken cylin-
ders, and bubbling. For a more detailed exposition on the subject, we refer the reader to [BEH
].
Let us rst recall the celebrated compactness result in the symplectic eld theory.
Let X
8
W YpRYqYT
S
n
be a decomposition of the symplectic manifold RM
obtained by performing the neck-stretching process as in Section 1.3 and J
8
be the associated
almost complex structure. Then, the SFT compactness [BEH
] implies that these holomorphic
buildings consist of nite energy curves that are asymptotic to holomorphic cylinders on closed
Reeb orbits. Moreover, asymptotic Reeb orbits must match up in pairs. That is, each positive end
of a curve in level i is matched with the corresponding negative end of the curve in level i 1.
During the neck-stretching process, one needs to keep track of possible degenerations of J-
holomorphic curves into non-smooth objects such as bubbles or broken cylinders. Next, we will
study these degenerations and prove the following:
Lemma 3.5.1. There exist no broken cylinders and bubblings of closed curves in M
RM
pJq.
Proof. By Stokes' theorem, there exist no non-constant J-holomorphic spheres since We-
instein domains are exact manifolds. Therefore, bubbling of closed curves never occurs in all
dimensions.
As for breaking, one needs to perform a careful analysis. Let M be a (2n+1)-dimensional
contact manifold supporting an open book whose pages are 2n-dimensional Weinstein domains
W
2n
. Let h :W
2n
ÑW
2n
be the monodromy map and
M
h
r0; 1sW
2n
L
p0;hpzqqp1;zq
be the associated mapping torus which is a smoothp2n 1q-dimensional manifold with boundary.
Then there exists a natural bration
:M
h
ÑS
1
whose ber isW
2n
. When we move along the baseS
1
as depicted in Figure 8, the almost complex
structure J on each ber starts changing due to the monodromy h. By the time we come to the
52
pointqPS
1
, the almost complex structureh
pJq is not the same as the almost complex structure
J over pPS
1
.
Let
: r0; 1s Ñ S
1
be a path with
p0q q and
p1q p. Since there exists a curve thru
each point over the path
on M
h
, we can interpolate the almost complex structure on that path.
Note that we cannot explicitly construct this interpolation. However, one can use the evaluation
map over this path to show that, geometrically thru each point, there exists a curve. Hence, the
interpolation follows.
S
1
M
h
p
q
h
J
h
pJq
Figure 8. The gure representing the action of the monodromy on the bers
W
2n
and the associated almost complex structures. Herep andq represent points
on S
1
and h
:TW
2n
ÑTW
2n
.
Next we will examine possible degenerations of curves, such as breaking as we are changing
the almost complex structure J during this interpolation. To see this, consider the map
^
:RMÑC:
Notice that breaking may occur if we obtain a holomorphic building with 2 levels, none of
which is trivial, after symplectization.
Letn 2. Consider the map
^
:RMÑC where dim
R
M 5. Observe that the cylindrical
ends in the ber direction containing the Reeb orbits map down to a point. Hence, by the open
mapping theorem, every end containing Reeb orbits has to live in a ber since
^
is a holomorphic
map.
53
C
z
^
1
pzq
u
RBW
2n
S
1
RM
^
Figure 9. The gure on the left is the map
^
:RM ÑC. The gure on the
right is the ber
^
1
pzq over zPC and a 2-level building depicting the breaking
of curves. Here u represents the end of a broken curve containing the Reeb orbit
in
x
W
2n
.
Letu be a curve in the cylindrical endRBW
4
containing the Reeb orbit depicted as in Figure
9. Hence, by the discussion above, u has image inside W
4
and W
4
admits a planar Lefschetz
bration whose bers are planar surfaces with no negative ends. Therefore, no suchu exists unless
it is a trivial cylinder and trivial cylinders are ignored in holomorphic buildings by convention.
Observe that one can apply this construction to W
2n
for all n¥ 2 inductively since the map
^
is a projection map from the completion of the page W
2n
to a point for all n, Thus, we could
go one dimension down to use the information that u is contained in a ber. Therefore, the proof
follows.
At this point, it is crucial to remind the reader that the induced complex structure J is not
cylindrical on a regular ber as explained in Section 3.3. It is rather asymptotically cylindrical
and this may aect the compactness discussion above. However, thanks to [B], we know that the
compactication results in SFT generalizes to asymptotically cylindrical almost complex structures.
54
3.6. Gluing
The goal of this section is to state the gluing result that we need in Section 3.7 to examine the
moduli space of planar J-holomorphic curves near a split holomorphic map in a k-level building.
Consider the Lefschetz bration
^
f
n
:
x
W
2n
ÑC. Let E
c
be a neighborhood of a critical value
cPD of f
n
and E be the corresponding neighborhood of the critical point above E
c
.
Recall that J
0
and J
c
are asymptotically cylindrical almost complex structures dened in a
neighborhood of a critical point E and f
1
n
pE
c
qE, respectively. Let M
1
denote the moduli
space of asymptotically cylindrical planar J
0
-holomorphic curves u
1
:
9
F
1
Ñ
^
ET
S
n
. Similarly,
denote by M
2
the moduli space of asymptotically cylindrical planar J
c
-holomorphic curves u
2
:
9
F
2
Ñ
^
f
1
n
pE
c
qE.
Let
9
F
9
F
1
#
9
F
2
. Denote by G
pM
1
;M
2
q the set of J-holomorphic maps u :
9
F Ñ
^
f
1
n
pE
c
q
that are -close to breaking into u
1
and u
2
in the sense of [HT1].
Here, the almost complex structureJ onup
9
Fq is the connect sumJ
c
#J
R
#J
0
, whereJ
R
is the
almost complex structure on the neck region connecting
^
E with
^
f
1
n
pE
c
qE.
Theorem 3.6.1. For suciently small ¡ 0, there exists a suciently large R and a gluing
map
G :M
1
M
2
ÑG
pM
1
;M
2
q
pu
1
;u
2
qÞÑu
which is a dieomorphism onto its image.
Since the curves of interest are not multiply covered, the proof of this result is a direct conse-
quence of the gluing results provided in [Pa] and [BV].
55
3.7. Neck stretching
In this section, we will study the degeneration of a regular ber into a singular ber via neck-
stretching and show that the lling by planar J-holomorphic curves in the completion of W
2n
restricts to a foliation onC
n
after neck-stretching on E. Hence, one can examine the structure of
J-holomorphic curves by looking at E while passing through the critical point.
A priori, we have no control overJ-holomorphic curves in the regular berW
2i
when they get
pinched at the singular ber. Therefore, one needs to analyze the degeneration of J-holomorphic
curves across the singular ber. Most reasonable procedure to study this is to use neck-stretching
argument in SFT [EGH]. Recall that neck-stretching is a deformation of an almost complex struc-
ture in some neighborhood of a contact hypersurface in
x
W : See Section 1.3 and [EGH] for a more
detailed discussion.
In what follows, we will apply neck-stretching to a neighborhoodE of a critical pointpPW
2i
.
In this way, J-holomorphic curves in this region will then be forced to look like standard annuli
when stretching is complete.
Consider the case n 3 and the map
'
3
:C
3
ÑC;
pz
1
;z
2
;z
3
qÞÑz
2
1
z
2
2
z
2
3
;
whose bers are T
S
2
W
4
. Recall that the Liouville form
1
2
2
¸
i1
px
i
dy
i
y
i
dx
i
q
on T
S
2
induces a contact form on its boundary ST
S
2
. Recall also that
ST
S
2
Lp2; 1qS
3
{Z
2
RP
3
:
We take a small neighborhoodr;sRP
3
ofRP
3
. Following [EGH] and [HWZ2], we can
replacer;sRP
3
with
rT;TsRP
3
;
56
which we will call the neck region, since the almost complex structure onr;sRP
3
is chosen
to be invariant under the Liouville
ow. We also dene the associated almost complex structure
J
T
with the following properties:
(1) It is invariant onrT;TsRP
3
.
(2) It agrees with the induced complex structure J on T
S
2
elsewhere.
Since J
T
is adapted on the neck region, one can send T to innity. As we take the limit,
the almost complex structure J
T
will start deforming so that in the limit it breaks W
4
into the
following two noncompact manifolds with cylindrical ends:
D
S
2
Yr0;8qRP
3
T
S
2
W
4
W
4
z T
S
2
One can then use SFT compactness [BEH
] to observe that theJ-holomorphic curves in
x
W
4
converge to a holomorphic building.
One can then conclude that
W
4
is foliated by planar J-holomorphic curves away from a set
of binding orbits ofBW
4
. Hence, lling by planarJ-holomorphic curves in the completion of W
2n
restricts to the foliation on T
S
n
after neck-stretching in a neighborhood of a singular ber.
Recall that under the identication ofT
S
n
with the tangent bundleTS
n
determined by a xed
a Riemannian metric g on S
n
, one can observe that the Reeb vector eld on ST
S
n
corresponds
to the geodesic
ow on S
n
. Thus, the Reeb orbits are closed geodesics of S
n
.
Observe that this is a highly degenerate situation since these closed orbits, i.e. geodesics,
are not isolated, but come in families. That is, there exists a Morse-Bott family of closed Reeb
orbits. This degenerate situation can be converted into a nondegenerate situation by perturbing
our contact form in such a way that it induces the same contact structure on the unit sphere bundle
ST
S
n
whose Reeb
ow generates the desired nondegenerate orbits.
Let us begin our analysis with the casen 2. Recall that the closed Reeb orbits ofST
S
2
are
closed geodesics of S
2
. We wish to show that there exists a contact form on ST
S
2
RP
3
whose
associated Reeb vector eld has precisely two nondegenerate closed orbits of Conley-Zehnder index
1.
We shall now perform the perturbation of the induced contact form onRP
3
. This perturbation
is crucial for determining the following: when we stretch the neck, we have a priori no information
57
about what collection of orbits the curve in T
S
2
limits to. If we have full information about the
orbits, we can eventually say that the curve limits to the two orbits of index 1 by a Fredholm index
count.
We consider the class of J-holomorphic curves in T
S
2
that are asymptotic to 2 closed non-
degenerate Reeb orbits. Call these two orbits
1
and
2
analogously dened as
in Section 3.1.
Let u be a planar J-holomorphic curve for the Lefschetz bration T
S
2
Ñ D with bers T
S
1
.
Then
indpuqpn 3qp
9
Fq
CZ
p
1
;q
CZ
p
2
;q:
Observe that p
9
Fq 2 2gk 2 0 2 2. In order for these curves to be of index 2,
the following should hold:
CZ
p
1
;q
CZ
p
2
;q 1:
with respect to a properly chosen trivialization . Therefore, we need to perturb our contact form
so that the resulting closed Reeb orbits are of index 1. To remedy the situation, we will use the
perturbation of the contact form described in [Wenb, Section 2] which has been constructed by
using the abstract open book of the associated planar contact manifold. Therefore, T
S
2
region is
foliated by cylindrical J-holomorphic curves asymptotic to closed Reeb orbits of Conley-Zehnder
index 1.
Note that one can extend this inductively to higher dimensions by using the iterative argument
onT
S
n
. Finally, we need to glue back the cylindrical ends of each level in the holomorphic building
to study the foliation of
x
W
4
by using Theorem 3.6.1.
Note also that one can apply this construction to further generalize it to
x
W
2n
, for n ¥ 2,
inductively by using the facts that each W
2i
is a ber of the Lefschetz bration on W
2i2
for all
i¥ 2 and the evaluation map for
x
W
2i
is of degree 1 for each i¡ 2.
58
3.8. Proofs of Proposition 3.1.1 and Theorem 1.5.2
Let us now use the results of the previous sections and prove Proposition 3.1.1 and Theorem
1.5.2 by arguing that the evaluation map
ev :M
RM
pJqÑRM;
pu;zqÞÑupzq
is of degree 1, i.e., through any generic point pPRM, the mod 2 count of ev
1
ppq is 1. Observe
that when M is 3-dimensional, one can make a stronger statement, i.e., RM admits a foliation
by embedded, nite-energy planar J-holomorphic curves [Wenb].
Proof of Proposition 3.1.1: Consider W
2i
f
1
i
ppq. When i 2, we know by the Main Theorem
in [Wenb] that
x
W
4
is foliated by planar J-holomorphic curves.
Now assume that
x
W
2i
is lled by planar J-holomorphic curves and show that
x
W
2i2
is also
lled by planar J-holomorphic curves. Since f
i1
is an iterated planar Lefschetz bration, we can
conclude that each regular ber of
^
f
i1
is lled by planar J-holomorphic curves for i¥ 2.
In Sections 3.4, 3.5, and 3.2.1, we showed that near a critical point, the moduli space of
planarJ-holomorphic curves in
x
W
2i2
s
is still regular, compact and has the expected dimension 2i
whenM
x
W
2i
s
pJq is regular, compact and has the expected dimension. Then the evaluation map on
M
x
W
2i2
s
pJq is still a degree 1 map and therefore
x
W
2i2
is lled by planar J-holomorphic curves .
In light of the discussion above, we can use the evaluation map
ev :M
x
W
2n
pJqÝÑ
x
W
2n
;
pu;zqÞÝÑupzq;
where z is a marked point in W
2n
, to show that there exists a unique curve passing through each
generic point in
x
W
2n
up to algebraic count since the completion of each regular and singular ber
off
n
:W
2n
ÑD is lled by planarJ-holomorphic curves. Hence, the pagesW
2n
of the open book
supported by the contact manifold M are lled by planar J-holomorphic curves. This nishes the
proof of Proposition 3.1.1.
59
Proof of Theorem 1.5.2:
Given any nondegenerate contact form
0
compatible with the open book of an iterated planar
contact manifold M. Let be another contact form on M with nondegenerate Reeb orbits and
f¡ 1 be some positive constant function on M such that
f
0
Now pick R ¡ 0 and J as dened in Section 3.4.1. Consider the symplectization pr0;Rs
M; kerpqq inRM dened by the graph off. Notice that asRÑ8, we need to make sure that
compactness issues such as bubbling or breaking of curves do not arise. In our setting, that never
happens as explained in Section 3.5.
By using Proposition 3.1.1 and the SFT compactness, one can deduce that each planar holo-
morphic curve inr0;RsM will be asymptotic to closed Reeb orbits as RÑ8 since there exists
a curve through each generic point inRM away from the binding orbits of M. As holomorphic
curves can't have maxima, they have to have a positive end which is asymptotic to a periodic orbit.
This concludes the proof of Theorem 1.5.2.
60
CHAPTER 4
Symplectic mapping class group relations
About fteen years ago, Giroux [G] described a correspondence betweenp2n 1q-dimensional
contact manifolds and symplectomorphisms of 2n-dimensional exact symplectic manifolds via a
generalization of a construction due to Thurston-Winkelnkemper [TW]. When n 1,
0
of the
symplectomorphism group of a surface is equal to the mapping class group and hence can be studied
entirely in terms of Dehn twists along simple closed curves. Considerable progress has been made in
understanding so-called \Giroux correspondence" for contact 3-manifolds; for example, Wand [W]
recently gave a characterization of tightness in terms of the supporting open book decomposition.
Symplectomorphisms of symplectic manifolds of dimension greater than 2 are comparatively
not as well understood, although there has been substantial progress. It had been known that one
can construct symplectic Dehn twists \along" Lagrangian spheres [A]. Starting with [Sei1], Seidel
and Khovanov-Seidel systematically studied symplectic Dehn twists in [KS], [Sei], [Sei2], [Sei4],
and also Wu recently described a classication of Lagrangian spheres in A
n
-surface singularities
in dimension 4 [Wu]. However, there exist exact symplectic manifolds which do not contain
Lagrangian spheres (e.g., the 2n-dimensional disk). For such manifolds, there is no general way of
constructing symplectomorphisms that are not symplectically isotopic to the identity.
Biran and Giroux [BG] introduced another class of symplectomorphisms, namely the bered
Dehn twist which turns out to be a strong tool to study symplectic manifolds that do not con-
tain Lagrangian spheres. More precisely, let pW;!q be a symplectic manifold with contact type
boundarypM; kerpqq carrying a free circle action on Mr0; 1s. Assume also that the action
preserves the contact form . The contact boundary M is then called of Boothby-Wang type. A
bered Dehn twist along the neighborhoodMr0; 1s is a boundary-preserving symplectomorphism
61
dened as follows:
:Mr0; 1sÝÑMr0; 1s;
pz;tqÞÝÑpzrsptq mod 2s;tq;
where s : r0; 1s Ñ R is a smooth function such that sptq 0 near t 1 and sptq 2 near
t 0. This will be explained in detail in Section 4.1. Many such bered Dehn twists have been
shown not to be symplectically isotopic to the identity by a theorem of Biran and Giroux [BG]
[CDK]. However in some cases, such as in our main theorem, one can relate bered Dehn twists
with symplectic Dehn twists.
The main goal of this half of the thesis is to relate bered Dehn twists and (regular) symplectic
Dehn twists for certain classes of symplectic manifolds. We prove the following theorems which
shows that in certain cases, bered Dehn twists can be expressed as a product of right-handed
Dehn twists.
Throughout, B
2n2
denotes ap2n 2q-dimensional ball of radius ¡ 0 inC
n
centered at the
origin.
Theorem 4.0.1. LetfPCrz
0
;:::;z
n
s be a homogeneous polynomial of degreek with an isolated
singularity at 0 for n¥ 1, k¥ 1. Denote bypW;dq the 2n-dimensional Weinstein domain, where
for ¥ 0 small and pq¡ 0 small
W tfpz
0
;:::;z
n
quXB
2n2
and
1
2
n
¸
j0
px
j
dy
j
y
j
dx
j
q.
ThenW has Boothby-Wang type boundary andBW has a coherent open book decompositionOBpF;
B
q
such that a right-handed bered Dehn twist
B
P SymppF;d;BFq is boundary-relative symplecti-
cally isotopic to a product ofkpk1q
n
right-handed Dehn twists
1
kpk1q
n along Lagrangian
spheres. Here F WXtz
n
u is a degree k hypersurface in W for ¡ 0 small.
It turns out that when we restrict ourselves to a certain class of homogeneous polynomials of
degree k, we obtain a beautiful relation between the bered Dehn twist
B
and symplectic Dehn
twist . We obtain the following immediate corollary of Theorem 4.0.1.
62
Corollary 4.0.2 (Roots of bered Dehn twists). Letfpz
0
;:::;z
n
qz
k
0
z
k
n
PCrz
0
;:::;z
n
s
for n¥ 1, k¥ 1. Denote bypW;dq the 2n-dimensional Weinstein domain, where for ¥ 0 small
and pq¡ 0 small
W tfpzq
n
¸
j0
z
k
j
uXB
2n2
and
1
2
n
¸
j0
px
j
dy
j
y
j
dx
j
q.
ThenW has Boothby-Wang type boundary andBW has a coherent open book decompositionOBpF;
B
q
such that a right-handed bered Dehn twist
B
P SymppF;d;BFq is boundary-relative symplecti-
cally isotopic to
k
, where is a product ofpk 1q
n
right-handed Dehn twists along Lagrangian
spheres. That is, is the k-th root of
B
.
In Section 6.2, we prove the following corollary in the case n 2, k 3; this recovers the
classical chain relation for a torus with two boundary components by using the star relation [Ger]
which will be explained in Section 6.3, the last section of the thesis. Similar representations have
been studied in terms of Artin groups by Matsumoto [Mat].
Corollary 4.0.3. Consider the genus 1 surface with 3 boundary components S
1;3
equipped
with the embedded curves
b
;
g
;
p
;
r
;b
1
;b
2
, and b
3
as depicted in Fig. 1. Let D
i
and D
bj
represent Dehn twists along the curves
i
andb
j
, respectively, whereiPtb;g;p;ru andjPt1; 2; 3u.
Then
pD
r
D
p
D
b
D
g
q
3
D
b1
D
b2
D
b3
in the mapping class group of S
1;3
.
b
p
g
r
b
1
b
2
b
3
Figure 1. We label the blue, green, purple, and red curves as
b
,
g
,
p
, and
r
, respectively. The boundary components are labeled as b
1
, b
2
, and b
3
.
63
Finally, we will examine contact manifolds given by a weighted homogeneous polynomial. Let
f PCrz
1
;:::;z
n
s be a weighted homogeneous polynomial with an isolated singularity at 0, n¥ 1,
andLpfqtfpz
1
;:::;z
n
q 0uXS
2n1
be the corresponding link of singularity. Then the Milnor
bration theorem [M2] states that
:S
2n1
LpfqÑS
1
;
zÞÑ
fpzq
|fpzq|
is a bration, called the Milnor bration off. The closure of each ber is a compact manifold with
boundary called the Milnor ber.
The Milnor bration of f then denes an open book decomposition of S
2n1
, with binding
Lpfq and bration , called the Milnor open book for S
2n1
. Moreover, this open book supports
the standard contact structure
0
ker on S
2n1
. To see this, one needs to show that is a
contact form on the binding Lpfq and d is positively symplectic on every page. Recall that the
binding, link of singularity of f, is contactomorphic to the standard sphere. Hence, it is left to
show that d is nondegenerate on every page. Note that the
ow of the Reeb vector eld R for
is given by
e
it
pz
1
;:::;z
n
qpe
it{a1
z
1
;:::;e
it{an
z
n
q
which is also a dieomorphism from one page to another. Then the Reeb vector eld R for can
be computed as follows:
R
d
dt
pe
it{a1
z
1
;:::;e
it{an
z
n
q
n
¸
i1
1
a
i
px
i
B
By
i
y
i
B
Bx
i
q:
Observe that dpR;q 0 and pRq 1. The Reeb vector eld is transverse to the pages, hence
d is nondegenerate on pages.
Unlike Theorem 4.0.1 which examines open books of the boundary restrictions of Milnor bers,
we will consider Milnor open books of the standard contact sphere and prove the following theorem.
64
Theorem 4.0.4 (Fractional twists). The Milnor open book for the standard contact sphere
pS
2n1
;
0
kerq is the contact open book
OBp
~
F;
B
q
where the page
~
F tfpzquXB
2n
is a Weinstein domain and the monodromy
B
P Sympp
~
F;d;B
~
Fq
is a generalization of a fractional bered Dehn twist along the boundary that is symplectically iso-
topic to a product of pf; 0q right-handed Dehn twists along Lagrangian spheres.
Here
~
F tf uXB
2n
, namely the Milnor ber, is a smoothing of the (singular) Weinstein
domain F tf 0uXB
2n
, and pf; 0q is the Milnor number of f at the isolated singularity at
the origin. See Section 5 for more details.
Although this work was done completely independently, after the paper appeared on the arXiv,
it was brought to our attention that there was substantial overlap with Seidel [Sei2], in which he
more or less proved Theorem 4.0.4.
65
4.1. Boothby-Wang bundles
Definition 4.1.1. A polynomial fpz
1
;:::;z
n
q c
I
z
I
, where I pa
1
;:::;a
n
q and z
z
a1
1
:::z
an
n
, is called a weighted homogeneous polynomial or quasi-homogeneous polynomial if there
exists n integers w
1
;:::;w
n
such that dw
1
a
1
w
n
a
n
holds for each monomial of f.
Integersw
1
;:::;w
n
are called weights of the variables and the sumd is called the degree of the
polynomial f. Observe that f is a homogeneous polynomial when w
1
w
n
1.
Equivalently, f is a weighted homogeneous polynomial if and only if
fp
w1
z
1
;:::;
wn
z
n
q
d
fpz
1
;:::;z
n
q:
One can then discuss a locally free circle action on the varietytf 0u. Therefore, we obtain
aC
-action onC
n
t0u by
C
tf 0uÝÑtf 0u;
pz
1
;:::;z
n
qÞÝÑp
w1
z
1
;:::;
wn
z
n
q;
for each PS
1
.
Definition 4.1.2. Let f P Crz
0
;:::;z
n
s be a hypersurface singularity germ at the origin in
C
n
, where n¥ 0. Let S
2n1
BB
2n2
be the boundary of the closed ball with radius centered
at the origin and let Lpfqtf 0uXS
2n1
be the corresponding link of singularity. Then the
smooth locally trivial bration
:S
2n1
LpfqÝÑS
1
;
zÞÝÑ
fpzq
|fpzq|
:
is called the Milnor bration of the function germ f. This is due to [M2].
When f has an isolated singularity at the origin, then the link Lpfq is a smooth manifold.
Moreover, any ber F
t
1
ptq is a smooth open manifold. In particular, the closure of each
ber, namely the Milnor ber, is a compact manifold with boundary Lpfq [M2].
66
Note that iff has an isolated singularity at the origin, then the associated Milnor ber F has
the homotopy type of a bouquet of n-spheres [M2]. The number of spheres in this bouquet is
called the Milnor number, denoted by pf; 0q.
Iff is a weighted homogeneous polynomial with an isolated singularity at the origin and with
weightspw
1
;:::;w
n
;dq, then the Milnor number can be computed in terms of the weights [Dim]
as follows:
pf; 0q
n
¹
i1
dw
i
w
i
:
Let M
2n
be a compact integral symplectic manifold with symplectic form !, where r!s P
H
2
pM;Zq. Then there exists a complex line bundle C overM withc
1
pCqr!s as the rst Chern
class classies complex line bundles.
Consider the associated principal circle bundle of C given by :P ÑM. Then according to
Theorem 3 in [BW], there exists a connection 1-form on P such that
(1) 2
!d and
(2) is a contact 1-form on P.
The contact form is called a Boothby-Wang form carried by the circle bundle P
Ý ÑM. The
fact that the contact form is a connection 1-form implies that the vector eld,R
, generating the
circle action satisesL
R
0 andpR
q 1. This implies thatR
is the Reeb vector eld for.
The circle bundlepP
Ý ÑM;q is then called a Boothby-Wang circle bundle associated withpM;!q.
Since the curvature 2-form ! on M makes the base space M into a symplectic manifold, pP;q
is called prequantum circle bundle (or Boothby-Wang circle bundle) that provides a geometric
prequantization ofpM;!q. This is due to Boothby and Wang, 1958 [BW].
67
4.2. Fibered Dehn twists and fractional bered Dehn twists
We begin with dening bered Dehn twists: LetF be a symplectic manifold with contact type
boundaryBF carrying a freeS
1
-action on the neighborhoodBFr0; 1s that preserves the contact
form onBF . HereBFr0; 1s is a collar neighborhood ofBF BFt1u. Then we can dene a
right-handed bered Dehn twist along the boundaryBF as follows:
:BFr0; 1sÝÑBFr0; 1s;
pz;tqÞÝÑpzrsptq mod 2s;tq;
wheres :r0; 1sÑR is a smooth function such thatsptq 0 neart 1 andsptq 2 neart 0. It
is easily veried that this is a symplectomorphism ofpBFr0; 1s;dpe
t
qq which is equal to identity
near boundary whose isotopy class is independent of the choice of s. See [CDK] for more details.
This notion was introduced by Biran and Giroux to emphasize that there are examples of
symplectomorphisms such as bered Dehn twists, which are not necessarily products of Dehn
twists; see [BG].
Next we dene a fractional bered Dehn twist: Here we refer the reader to [CDK2] for a
more comprehensive discussion. The setup in this subsection is as follows: We will denote by k
the degree of the weighted homogeneous polynomial f with an isolated singularity at 0 and by
F tfpzq uXB
2n
the associated Milnor ber. Denote by P the boundary of F and let l be
any positive integer that divides k and p
~
P
Ý Ñ M;q be a Boothby-Wang bundle over pM;
k
l
!q,
where
~
P is a covering of P given by the following l-fold covering map:
pr
P
:
~
P ÝÑP;
pÞÝÑpbbp
Note that the map pr
P
is well-dened since H
2
pM;Zq is torsion-free as l divides k. Next, we
want to extend the covering from P BF to the symplectic manifold F .
Definition 4.2.1. A covering map pr
F
:
~
F ÞÑ F is called adapted to pr
P
:
~
P ÞÑ P if pr
F
restricts to pr
P
on a collar neighborhood P .
68
One can always nd such an adapted l-fold covering pr
F
:
~
F Ñ F for a 2n-dimensional
Weinstein manifold F with 2n¥ 6, because there exists a subgroup of index l in
1
pFq
1
pBFq
(this equality is true for dimensions 6 and higher) as l|k.
Given an adapted covering, one can lift a bered Dehn twist to a ~ , where
~ :B
~
FIÞÝÑB
~
FI;
pz;tqÞÝÑpzrsptq mod 2s;tq
wheres :r0; 1sÑR is a smooth function such thatsptq 0 neart 1 andsptq
2
l
neart 0 for
a positive integerl. This implies that there is an action ofZ
l
given bysptq which induces rotation
by roots of unity in some collar neighborhoodB
~
FI ofB
~
F . Hence, the map ~ can similarly be
extended to the interior of F by using deck transformation of the cover
~
F Ñ F which generates
this Z
l
action in the interior of the page. Observe that this action also rotates the margin of the
page by
2
l
.
The map ~ is symplectomorphism, see [CDK2]. Observe also that for l 1, the map ~ is a
bered Dehn twist.
Definition 4.2.2. The symplectomorphism ~ given as above is called a right-handed fractional
bered Dehn twist of power l or a fractional twist as dened in [CDK2].
69
4.3. Setup
Now we will provide a description of an adapted open book decomposition of a Boothby-Wang
circle bundle over a symplectic manifold. This construction has been introduced by Biran and
Giroux [BG] and generalized and claried by Chiang, Ding and van Koert [CDK].
Any integral symplectic manifold pM
2n
;!q has a compact symplectic hypersurface H whose
homology classrHsPH
2n2
pM;Zq is Poincar e dual to the symplectic classrk!sPH
2
pM;Zq such
that F M pHq, the complement of an open tubular neighborhood pHq of H, carries a
Weinstein domain structure, [Don] and [G]. If one can patchF ,BFr0; 1s, andBF
D
2
together,
then the hypersurfaceH is called an adapted Donaldson hypersurface. See [CDK] for the patching
argument.
Now consider the Weinstein domain pW;dq with a Boothby-Wang type boundary pP;|
P
q.
Then there are two constructions of a contact manifold using the given data. Firstly, we can
construct a Boothby-Wang circle bundle pP
Ý Ñ M;q over an integral symplectic manifold M.
We can also dene a right-handed bered Dehn twist
B
along the boundary of F , then dene a
contact open book ofBW with pageF MH and monodromy as
B
. Thus, we can talk about
the relation between the Boothby-Wang circle bundle and its associated open book. That relation
is given by the following theorem (Theorem 6:3 and Corollary 6:4 in [CDK]).
Theorem 4.3.1. [CDK] LetpW;dq be a Weinstein domain with a Boothby-Wang type bound-
arypBW P;|
P
q and ! be an integral symplectic form on M. Let H be an adapted Donaldson
hypersurface Poincar e dual tor!s, F MH, and
B
be a right-handed bered Dehn twist along
BF . Then
(1) The Boothby-Wang circle bundle pP
Ý Ñ M;q associated with pM;!q has an open book
decomposition whose monodromy is a right-handed bered Dehn twist.
(2) The open bookpF;
B
q with pageF and monodromy
B
is contactomorphic to the Boothby-
Wang circle bundlepP
Ý ÑM;q associated withpM;!q.
Definition 4.3.2. An open book decompositionpB;q is coherent with a Boothby-Wang circle
bundlepP
Ý ÑM;q if
70
(1) The contact form is adapted by the bration :P
1
pHqÑS
1
, where :P ÑM
is the Boothby-Wang circle bundle.
(2) The binding B
1
pHq for some adapted Donaldson hypersurface HM.
(3) The monodromy
B
is a right-handed bered Dehn twist.
71
CHAPTER 5
Proofs of Theorem 4.0.1 and 4.0.4
Throughout, we denote the coordinates on C
n1
by z
j
x
j
iy
j
, j 0;:::;n, and the open
book with page F and monodromy by OBpF; q.
The proof is by comparing two points of view, namely the Lefschetz bration point of view
and the Boothby-Wang bration point of view. Throughout, f is a homogeneous polynomial of
degree k with an isolated singularity at 0 and W tfpz
0
;:::;z
n
quXB
2n2
is the Weinstein
domain unless otherwise is stated.
5.1. Lefschetz bration construction
Denote by z
n
:W ÑC the projection map to the last coordinate with an isolated singularity.
Then the restriction map, denoted byh :z
n
|
t|zn|¤u
:WXt|z
n
|¤uÑt|z
n
|¤u, for small, is a
Lefschetz bration after Morsifying its critical points which will be explained in the next paragraph
and F WXtz
n
u is the generic ber of the Morsication of this bration.
Since all critical points of a Lefschetz bration are of Morse-type, we need to perturb z
n
in a
certain stable way so that the isolated degenerate singularity at 0 will split up into other isolated
singularities which are non-degenerate. This deformation is called Morsication which is a small
deformation of z
n
having r distinct Morse critical points p
1
;:::;p
r
and critical values c
1
;:::;c
r
.
Here z
n
g is a Morsication of z
n
if gpzq is a linear function in general position and is very
small. We also know that there are nitely many such Morse-type critical points, i.e., r is nite,
after Morsication of z
n
since such critical points are isolated. One can also prove that the exact
number of such critical points is rkpk 1q
n
. For the proof, we refer the reader to Section 5.6.
That is to say, we have nitely many Lagrangian spheres in the ber coming from critical
values. Moreover, it gives rise to associated monodromies. Denote the monodromy associated
to critical values by
1
kpk1q
n, where each
i
P Di
pF;BFq;i 1;:::;kpk 1q
n
, is a
72
right-handed Dehn twist. Then we have the following open book decomposition associated to the
boundary restriction of the Lefschetz bration h:
Page: F WXtz
n
u.
Binding: BBWXtz
n
0u.
Monodromy:
1
kpk1q
n.
Here observe that F is a retracted page of the open book decomposition
pB;
1
kpk1q
nq. By abuse of notation, we refer to a retracted page as a page of the open
book.
Let us consider the boundary of the Weinstein domainWXt|z
n
|¤u. We know that an open
book naturally arises as boundary restriction of a Lefschetz bration. According to the Lefschetz
bration construction above,BpWXt|z
n
|¤uq is supported by the open book decomposition with
pageF and monodromy
1
kpk1q
n. The binding of this open book is the intersection of the
link associated to the homogeneous polynomialf, i.e.,BW tf 0uXS
2n1
with the hyperplane
tz
n
0u.
The analysis above provides a description of the monodromy coming from the Lefschetz bra-
tion point of view. In order to analyze the desired isotopy between the two monodromies mentioned
in the main theorem, it suces to describe the monodromy coming from the Boothby-Wang point
of view, namely a right-handed bered Dehn twist. In order to do that, we need the data given in
Theorem 4.3.1.
73
5.2. Boothby-Wang construction
Here we closely follow the description in [CDK]. Consider the Weinstein domain W with the
boundary BW carrying a free S
1
-action which preserves the contact form. We can extend this
action to the neighborhoodBWr0; 1s so that the contact form onBW is preserved.
Let :BW Ñ M be a principal circle bundle such that the S
1
-action preserves the contact
form. Then circle bers are closed orbits of the Reeb vector eldR
, i.e., Reeb foliates circle bers.
So we can view the boundary of the Weinstein domain W lying inS
2n1
as a Boothby-Wang type
boundary.
Now consider the bration
:BWtz
n
0uÑS
1
;
pz
0
;:::;z
n
qÞÑ
z
n
|z
n
|
:
Observe that Htz
n
0u is an adapted Donaldson hypersurface in M for which
1
ptz
n
0uq BW Xtz
n
0u is the binding of the open book decomposition of BW with page F
1
p
0
q for some xed angle
0
argpz
n
q and monodromy a right-handed bered Dehn twist.
The Boothby-Wang circle bundle pBW
Ý Ñ M;q is then contactomorphic to a contact structure
supported by the open book decomposition ofBW by Theorem 4.3.1.
Also, the open book decomposition discussed above is coherent with the Boothby-Wang circle
bundlepBW
Ý ÑM;q since the contact form onBW is adapted to the open bookpB
1
ptz
n
0uq;q and the monodromy is a right-handed bered Dehn twist.
So, the coherent open book decomposition is as follows:
Page: F pBWtz
n
0uqXtargpz
n
q
0
u for some xed
0
.
Binding: BBWXtz
n
0u.
Monodromy:
B
, a right-handed bered Dehn twist.
74
5.3. Comparison of two views
Observe that the pages and the bindings of the two open books discussed above are exactly the
same up to symplectomorphism. To nish the proof, we have to check that the monodromy coming
from the Lefschetz bration point of view (product of Dehn twists along Lagrangian spheres) is
symplectically isotopic to the monodromy coming from the Boothby-Wang bration point of view
(right-handed bered Dehn twist) relative to the boundary. In order to show that, we will construct
a symplectic isotopy so that we have the desired equivalence.
Notation. Denote the page coming from the Lefschetz bration construction by
0
WXtz
n
u and the slightly retracted page coming from the Boothby-Wang construction by
1
pBWtz
n
t|tPr0; 1suqXtargpz
n
q
0
uS
2n1
.
B B
0
WXtz
n
0u
Liouville
ow
0
1
z
0
;:::;z
n1
z
n
W
2n
S
2n1
Figure 1. Pictorial description of the symplectic isotopy between the pages
0
and
1
. We label the blue dot as the binding B, the red curve as the page
0
,
and the green curve as the page
1
. Here 0 denotes the isolated singularity of f.
Claim. There exists a symplectic isotopy from
0
to
1
relative to the endpoints, i.e., there
exists a one parameter family of p2nq-dimensional symplectic submanifolds p
t
;d|
t
q, tPr0; 1s
as depicted in Fig. 1.
Proof. Let be the usual Liouville 1-form onC
n1
. Let X and Y be the radial vector eld
(the Liouville vector eld of on all ofC
n1
q and the rotational vector eld in the z
n
x
n
iy
n
75
coordinate, respectively, given as follows:
X
1
2
n
¸
j0
px
j
B
Bx
j
y
j
B
By
j
q
Y
B
B
Consider the
owgX, whereg is some suitable positive function on Intp
0
q and is the identity
along B
0
. Then one can obtain one parameter family of p2nq-dimensional submanifolds
t
by
owing from
0
to
1
along the radial vector eldgX. Furthermore,
t
is transverse to the vector
elds X and Y since it is transverse to a multiple of X. Because the boundary of each
t
is the
same, the transversality condition still holds for each hypersurface
t
at the boundary.
Let V tv
1
;:::;v
2n
u be a basis of T
t
at some point p. We want to show that d
°
n
j0
dx
j
^dy
j
¡ 0 is symplectic on each
t
to ensure
t
is symplectic for tPr0; 1s.
We computepdq
n
as follows: Recall that we already know that pdq
n1
pX;Y;Vq¡ 0 since
d is the standard symplectic formC
n1
. Then
pdq
n1
pX;Y;Vqnpdx
n
^dy
n
qpdx
0
^dy
0
dx
n1
^dy
n1
q
n
pX;Y;Vq
cpdx
0
^dy
0
dx
n1
^dy
n1
q
n
pVqq¡ 0 pq
implies thatpdq
n
pVqpdx
0
^dy
0
dx
n1
^dy
n1
q
n
pVq is positive since c¡ 0.
The last inequalitypq above is equivalent to the following condition: the projection of
t
to
the rst n coordinates (i.e., projection to the z
0
;:::;z
n1
-plane Z) is symplectic. In other words,
pr :
t
Ñ prp
t
q is a covering space over a symplectic submanifold of the plane Z. Also, notice
that
0
and
1
satisfy pq above since |
0
and |
1
are Liouville. So each
t
is symplectic.
Therefore, we have a symplectic isotopy relative to the boundary.
Thus, a right-handed bered Dehn twist on
1
is symplectically isotopic to a product of right-
handed Dehn twists on
0
. This nishes the proof of Theorem 4.0.1.
76
5.4. Proof of the Corollary 4.0.2
With Theorem 4.0.1 in hand, we are ready to complete the proof of Corollary 4.0.2. Here
W tfpzqz
k
0
:::;z
k
n
uXB
2n2
.
Proof. Follows from the proof of Theorem 4.0.1. In this case, the monodromy coming from
the Lefschetz bration is the k-th power of a product of right-handed Dehn twists. This follows
from the fact that each ber (page) F
has the following behavior:
F
WXtz
n
e
i
utz
k
0
z
k
n1
k
e
ik
u:
In other words, the monodromy on F
repeats itself k times since it goes over the same
(a product of right-handed Dehn twists) k times as varies. Observe also that the 0 case
gives us the Lefschetz ber together with the monodromy . Furthermore, when it sweeps out all
of Pr0; 2q, one gets a right-handed bered Dehn twist
B
along the boundary as monodromy
which is equal to
k
up to symplectic isotopy.
One can Morsify f by adding the linear function pz
0
z
n
q and observe that degenerate
isolated singularity at the origin splits intopk1q
n
many nondegenerate critical points in the ber.
To see this: Take partial derivatives of the Morsiedfpz
0
;:::;z
n
qz
k
0
z
k
n1
pz
0
z
n
q
and observe that each partial derivative will generatepk 1q roots. Since there are n variable, f
will havepk 1q
n
critical points. Therefore, there arepk 1q
n
many nondegenerate critical points,
i.e., is a product ofpk1q
n
right-handed Dehn twists. Hence, a right-handed bered Dehn twist
is isotopic to a product of kpk 1q
n
right-handed Dehn twists since
B
k
.
Note that, when k 1, W C
n
and a bered Dehn twist is symplectically isotopic to the
identity.
In the case when k 2, W is the cotangent bundle to the n-sphere and
B
is the same as the
square of a Dehn twist along the zero section.
77
5.5. Proof of the Theorem 4.0.4
Proof follows along the same lines of the proof of Theorem 4.0.1. Here, unlike Theorem 4.0.1,
f is a weighted homogeneous polynomial with an isolated singularity at 0 and we will study the
associated Milnor bration of the standard sphere S
2n1
.
In this subsection, we wish to prove that the Milnor open book for the standard sphere S
2n1
is the contact open book whose monodromy
B
is a generalization of a fractional bered Dehn
twist along the boundary that is symplectically isotopic to a product of pf; 0q Dehn twists along
Lagrangian spheres.
Let zpz
1
;:::;z
n
q. Consider the link of gpz;z
n1
q :fpzqz
n1
at the origin given as
L
pgqtpz;z
n1
qPC
n
C|gpz;z
n1
q 0uXS
2n1
and also its canonical lling given as
V
pgqtpz;z
n1
qPC
n
C|gpz;z
n1
q 0uXB
2n2
:
One can always smooth this lling by using a cuto function that only depends on|z|. Denote
the smoothing of V
pgq by
~
V
pgqtgpz;z
n1
quXB
2n2
:
Note that L
pgq is contactomorphic to the standard contact sphere S
2n1
. To see this, let
fpzq z
n1
in L
pgq and observe that this is just a graph of S
2n1
. Note that one can use the
bration L
pgqtz
n1
0u sending pz;z
n1
q ÞÑ
zn1
|zn1|
to study the open book for the L
pgq.
Therefore, nding an open book for the standard sphere S
2n1
is the same as nding an open
book for the boundary of
~
V
pgq tgpz
1
;:::;z
n1
q uXB
2n2
. Hence, we will use the same
techniques as in the proof of the main theorem to observe that the standard contact sphere has
the following open book:
Page:
~
F tfpzquXB
2n
.
Binding: L
pfqtfpzq 0uXS
2n1
.
Monodromy:
1
pf;0q
, where each
i
is a right-handed Dehn twist and
pf; 0q is the Milnor number of f.
78
On the other hand, one also needs to describe what the monodromy looks like on the boundary,
i.e., prove the following claim. Finally, to nish the proof, we will apply the Liouville
ow technique
as in the proof of Theorem 4.0.1 to show the symplectic isotopy between two monodromies.
Claim. The monodromy
B
is a generalization of a fractional bered Dehn twist.
Proof. Proof is by analyzing a rotation map in the interior and boundary of the page
~
F . We
will considerZ
d{wi
action on
~
F for i 1;:::;n.
Since f is a weighted homogeneous polynomial, we can construct a locally free action on
the binding. As for the pages, we need to choose a cuto function to redene the hypersurface
obtained from the weighted homogeneous polynomial f so that the slightly deformed (by cutting
o) hypersurface is Weinstein. Consider a cuto function
R
with the following properties.
R
p|z|q 1 if|z|¤R,
R
p|z|q 0 if|z|¡ 2R,
|
1
R
|
2
R
.
Given a cuto function
R
that only depends on|z|, one can smooth the lling F tfpzq
0uXB
2n
of L
pfqtfpzq 0uXS
2n1
by considering fpzq
R
pzq. Note that
~
F tfpzq
R
pzquXB
2n
could be made symplectic by choosing the cut-o function carefully. See Section
6.1.2.3 of [OvK] for more details.
Consider the following periodic unitary transformation
~
:
~
F ÝÑ
~
F;
pz
1
;:::;z
n
qÞÝÑpe
2i
w
1
d
z
1
;:::;e
2i
wn
d
z
n
q:
Since f is a weighted homogeneous polynomial, the periodic map
~
is well-dened. Notice
also that it is a symplectomorphism since it is a biholomorphism.
The map
~
is not the identity map for largez. Instead, it can be considered as a generalization
of a fractional bered Dehn twist as dened in [CDK2]. We will elaborate this idea further in this
section. It is not (isotopic to) the identity near the boundary of the page
~
F . So we need to undo
the twist at the boundary to get the identity map and show that it is the desired monodromy.
79
Firstly, consider the interior of the page
~
F , call F
1
pe
i
q. Then one can choose the
monodromy on F
as
~
, see Lemma 9.4 in [M1]. In brief, Lemma 9.4 shows that F
given by
a weighted homogeneous polynomial f is dieomorphic to a nonsingular hypersurface and one
can construct the above monodromy
~
on the dieomorphic copy of F
. This completes the
monodromy construction in the interior of the page
~
F . Note that we need to ensure that this
monodromy extends to the whole bration. In what follows, we discuss this extension.
We wish to observe that the monodromy in the interior of the pages extends to the boundary
properly. To see this, we will use the untwisting argument given in Otto van Koert's Ph.D. thesis
[OvK] to determine a symplectomorphism of the whole bration so that the monodromy looks
like the identity map near the boundary. For this, let us consider the map
t
induced by the time
t
ow of the Hamiltonian vector eld associated to a Hamiltonian function H on
~
F given by
Hpz
1
;:::;z
n
q
¸
j
w
j
2d
|z
j
|
2
:
The time t
ow of this Hamiltonian function generates a circle action given by
Fl
X
H
:pz
1
;:::;z
n
qÞÑpe
it
w
1
d
z
1
;:::;e
it
wn
d
z
n
q:
Let h be a function given by hpzq : p1
2R
p|z|qq. Denote by
~
t
, the time t-
ow of the
Hamiltonian vector eld associated to
~
H hH. Then the time 2-
ow of
~
H|
~
F
, a Hamiltonian
on
~
F , is the identity for|z| 2R, and it induces the following map:
pz
1
;:::;z
n
qÞÝÑpe
2i
w
1
d
z
1
;:::;e
2i
wn
d
z
n
q;
for|z|¡ 4R.
Thus, we can dene
:
~
F ÞÝÑ
~
F;
zÞÝÑFl
X ~
H
2
pzqpe
2i
w
1
d
z
1
;:::;e
2i
wn
d
z
n
q:
Now consider
B
~
:
~
F ÝÑ
~
F . Observe that
B
is not quite a fractional bered Dehn twist
due to the action given by the periodic unitary transformation
~
. We will call it a generalization
80
of a fractional bered Dehn twist. Therefore, the monodromy
B
is a generalization of a fractional
bered Dehn twist which is the identity at the boundary of
~
F and its neighborhood.
Note that the monodromy
B
is a fractional bered Dehn twist when f is a homogeneous
polynomial as dened in [CDK2].
81
5.6. Proof of the exact number of Dehn twists
With Corollary 4.0.2 in hand and [OvK2], we show that there exists kpk 1q
n
right-handed
Dehn twists for the bered Dehn twist when f is any homogeneous polynomial of degree k with
an isolated singularity at the origin.
Proof. Let g be a homogeneous polynomial of degree k with an isolated singularity at the
origin given by gpz
0
;:::;z
n
qz
k
0
z
k
n
and let f be a homogeneous polynomial of degree k
with an isolated singularity at the origin. We want to show that there exists a 1-parameter family
of homogeneous polynomials of degree k with an isolated singularity at the origin, denoted by p
t
connecting p
0
and p
1
with the following properties:
p
0
f.
p
1
g.
p
t
is a homogeneous polynomial of degree k with an isolated singularity at the origin.
The level set p
1
t
p0qXS
2n1
is smooth for all tPIr0; 1s.
Firstly, if we can show that the complex codimension of the singular projective variety of
homogeneous polynomials of degree k in the projective variety of the space of all homogeneous
polynomials of degree k is ¥ 1, then there is a path between any two polynomials (f and g in
our case) in the nonsingular projective variety. Because one can always travel around the singular
locus.
To see this, let V
k
denote the vector space of homogeneous polynomials of degree k and S
k
denote the subset of V
k
consisting of those homogeneous polynomials of degree k containing a
singular locus. Furthermore, we know that homogeneous polynomials of degree k with an isolated
singularity at the origin dene smooth hypersurfaces in projective space and homogeneous poly-
nomials of degree k whose zero set forms a singular variety in projective space form themselves a
singular variety in the space of all homogeneous polynomials of degree k.
Note that S
k
, the zero set of some polynomials on the vector space V
k
, is a subvariety of V
k
and it has complex codimension at least 1 (real codimension 2). Thus, its complement is path
connected. Once we get a path, one can locally travel around the singular polynomials. This
82
proves the existence of a 1-parameter family of homogeneous polynomials of degree k with an
isolated singularity at the origin as desired.
Now consider Lpp
t
q :
zPC
n1
||z|
2
1; p
t
pzq 0
(
, the link of singularity of the polyno-
mial p
t
and check that the links Lpp
t
q are all contactomorphic. If we can show that they are all
contactomorphic, then there must be open books with the same properties. In other words, we
have the same number of (kpk 1q
n
many) right-handed Dehn twists for the bered Dehn twist
in both Theorem 4.0.1 and Corollary 4.0.2.
Claim. Let C :
pt;zqPIC
n1
||z|
2
1; zPp
1
t
p0q
(
. Then the set C is a topologically
trivial cobordism between Lpp
0
q and Lpp
1
q. In particular, Lpp
0
q and Lpp
1
q are dieomorphic.
Proof. Observe that the boundary ofC consists of the linksLpp
0
q andLpp
1
q. First we show
thatC is a smooth manifold and hence a smooth cobordism betweenLpp
0
q andLpp
1
q. To see this,
dene the map
P :IC
n1
ÝÑRC;
pt;zqÞÝÑp|z|
2
1;p
t
pzqq:
Observe that C P
1
p0q. To show the cobordism, we will compute the Jacobian of P and
conclude that C is smooth. The Jacobian can be computed as follows:
z z
Bpt
Bz
0
0
B pt
B z
since it is generated by|z|
2
1;p
t
; p
t
. Then the Jacobian has rank 3 since both
Bpt
Bz
and
B pt
Bz
are
non-vanishing away from 0PC
n1
and also 0 is not an element of P
1
p0q because of the|z|
2
1
condition.
Since the Jacobian ofP has real rank 3 at all points inP
1
p0q, we conclude thatC is a smooth
submanifold of IC
n1
of codimension 3. Hence, C is a smooth cobordism between the links
Lpp
0
q and Lpp
1
q.
83
To show that this cobordism is topologically trivial, consider the function
h :CÝÑR;
pt;zqÞÝÑt:
If we can show thath has no critical points inh
1
pr0; 1sq, we then know thatC is a topologically
trivial cobordism. To see this, take any pointpt
0
;z
0
qPC and take a smooth curve cptqpt;zptqq
in C through pt
0
;z
0
q with zptq smooth. One can construct such a curve as follows: Consider a
curve inC
n1
given by
yptqz
0
sptq:
We wish to set up an ODE for s in such a way that
p
t
pz
0
sptqq 0:
Dierentiating p
t
pz
0
sptqq 0 will lead us to an ODE. Locally, we can solve the resulting ODE
to obtain yptq satisfying p
t
pyptqq 0. Now if we let zptq
yptq
|yptq|
, we get the desired smooth curve
cptqpt;zptqq:
Then hc has non-vanishing derivative at tt
0
since hcptqt and dh does not vanish in
pt
0
;z
0
q. So no pointpt
0
;z
0
qPC is a critical point of h, i.e. h has no critical points. Thus C is a
topologically trivial cobordism. Furthermore, Lpp
0
q and Lpp
1
q are dieomorphic by Theorem 3.1
of [M1]. This concludes the proof of the dieomorphism between the links Lpp
0
q and Lpp
1
q.
Since each link Lpp
t
q is dieomorphic to Lpp
0
q and has an associated contact structure
t
induced as the link of singularity, Gray's stability theorem shows that there is a contactomorphism
sending
t
to
0
. As the resulting contact manifolds are contactomorphic, there must be open
books with the same properties.
As there are kpk 1q
n
right-handed Dehn twist for the full bered Dehn twist in the case of
Corollary 4.0.2, we have an open book with the same number of right-handed Dehn twists for the
full bered Dehn twist in the case of Theorem 4.0.1.
84
CHAPTER 6
The case n 2
In this nal chapter, we combine Corollary 4.0.2 with the monodromy computations appearing
in the literature to describe mapping class group relations for some surfaces.
6.1. The case of arbitrary k¥ 2
Lemma 6.1.1. [HKP] For each integer k¥ 2, consider the Weinstein domain
W tz
k
0
z
k
1
z
k
2
uXB
6
Then the page of the open book of BW (i.e., the Milnor ber F
k;k
associated to the polynomial
z
k
0
z
k
1
0) is a surface with genus g
1
2
pk 1qpk 2q and k boundary components. It is the
`canonical' Seifert surface for the pk 1;k 1q torus link which can be drawn in R
3
as shown in
Fig. 7 in [HKP]. The monodromy for the associated open book decomposition is isotopic to the
product
k
ppD
1;k1
:::D
k1;k1
qpD
1;1
D
k1;1
qq
k
of right-handed Dehn twists D
i;j
about the curves
i;j
F
k;k
shown in Fig. 1.
Proof. The computation of the genus and number of boundary components of F
k;k
follows
immediately from the degree-genus formula for complex curves inCP
2
. The drawing of the surface
inR
3
follows from Example 6.3.10 in [GS]. The Dehn twist presentation of the monodromy follows
from the proof of Proposition 3.2 in [HKP]. Note that Fig. 1 is essentially the same as Fig. 7 in
[HKP].
Combining the above lemma with Corollary 4.0.2, provides the following:
Corollary 6.1.2. The surface F
k;k
described in Fig. 7 [HKP] is such that
(6.1.1)
pD
1;k1
:::D
k1;k1
qpD
1;1
D
k1;1
q
k
85
1;1
2;1
.
.
.
.
.
k1;1
1;k1
2;k1
.
.
.
.
.
k1;k1
Figure 1. The curves
i;j
embedded in the surface F
k;k
.
is isotopic to a product of right-handed bered Dehn twists about each of the boundary components
of F
k;k
.
Note that in dimension 2, bered Dehn twists are the same as symplectic Dehn twists so that
product of bered Dehn twists is isotopic to a product of right-handed Dehn twists about curves
parallel to the boundary components of F
k;k
.
86
6.2. The case k 3
Now we describe the casen 2,k 3 in greater detail, proving Corollary 4.0.3. For simplicity,
we will henceforth refer to F
3;3
as S.
To complete the proof, we consider arcs a
g
;a
b
;a
p
, and a
r
in S as depicted in Fig. 2. We
also label the boundary components b
1
;b
2
, and b
3
of S as shown. The orientations of a-arcs are
indicated with arrows.
b
1
b
2
b
3
a
b
a
g
a
p
a
r
Figure 2. We label the green, blue, purple, and red arcs as a
g
, a
b
, a
p
, and a
r
, respectively.
If we cut the surfaceS along the arcsa
g
;a
b
;a
p
, anda
r
, all that will remain is a 16-gon, which
is topologically just a disk as shown in Fig. 3. In Fig. 3, the segments of the boundary are colored
according to the identifying colors for the a-arcs. The small arrows indicate where the boundary
orientation of the 16-gon agrees and disagrees with the orientations of the arcs.
By regluing along the cut arcs, we obtain an easier-to-visualize depiction of S shown in Fig.
2. Call the dieomorphism from the surface in Fig. 1 to the glued up surface in Fig. 4, induced
by the cutting and gluing of the a-arcs, by .
To complete the proof of Corollary 4.0.3, one should check that maps the curves
1;1
;
1;2
;
2;1
;
2;2
87
a
b
a
b
a
g
a
g
a
r
a
r
a
p
a
p
b
3
b
1
b
2
b
3
b
1
b
3
b
2
b
1
Figure 3. The disk obtained by taking the closure of Szpa
g
Ya
b
Ya
p
Ya
r
q.
a
b a
p
a
g
a
r
b
1
b
2
b
3
Figure 4. A surface equipped with embedded arcs equivalent to Fig. 2 by the
dieomorphism .
of Fig. 1 to the curves
b
;
g
;
r
;
p
of Fig. 1, respectively. To see this, note that for each color-label iPtg;b;p;ru, the simple-closed
curve
i
intersects a
i
in a single point, and is disjoint from each a
j
for j i. These intersection
conditions completely determine the isotopy classes of the -curves. This completes the proof of
Corollary 4.0.3.
88
6.3. From monodromy to mapping class group relations
In this section, we study explicit descriptions of the symplectomorphisms in the case n 2,
which provides a visual representation of the case n 2, k 3 recovering the classical chain
relation for a torus with two boundary components.
Definition 6.3.1. Consider the arcs
1
;
2
;
3
;
4
;b
1
;b
2
; andb
3
inS
1;3
as depicted in Fig. 5.
If
1
;
2
;
3
;
4
;b
1
;b
2
; and b
3
are the isotopy classes of simple closed curves in S
1;3
then we have
the following relation:
pD
1
D
2
D
3
D
4
q
3
D
b1
D
b2
D
b3
:
b
1
b
2
b
3
1
2
3
Figure5. We label the boundary components asb
1
;b
2
;b
3
and other components
of the isotopy classes of simple closed curves in S
1;3
as
1
;
2
;
3
; and
4
.
This relation is called the star relation [Ger].
Let S be a compact connected, orientable surface of genus g ¥ 1 with nonempty boundary
and let
1
;:::;
k
be simple closed curves on S. If the intersection number ip
j
;
j1
q 1 for
all 1¤ j ¤ k 1 and ip
j
;
t
q 0 for|jt|¡ 1, then the sequence of isotopy classes of curves
1
;:::;
k
in S is called a chain.
Proposition 6.3.2. [FM] Consider a chain of circles
1
;:::;
k
inS. Denote isotopy classes
of boundary curves byb in the even case and byb
1
andb
2
in the odd case. Then each chain induces
89
one of the following relations in the mapping class group of S:
pD
1
D
m
q
2m2
D
b
if m is even;
pD
1
D
m
q
m1
D
b1
D
b2
if m is odd:
In each case, the relation is called a chain relation or a k-chain relation. There is another
version of the chain relation described in [FM] which will be useful in our case:
pD
2
1
D
2
D
m
q
2m
D
b
if m is even;
pD
2
1
D
2
D
m
q
m
D
b1
D
b2
if m is odd:
Notice how the Dehn twist along the boundary diers in each case. Now consider any two arcs,
and on S with geometric intersection number 1 (i.e., the minimum number of intersections of
curves
1
and
1
isotopic to and , respectively). Then the following relation is called the braid
relation:
D
D
D
D
D
D
:
It is known that the star relation implies the 2-chain relation
pD
1
D
2
q
6
D
b
by using the braid relation. If one can show that the relation
pD
r
D
p
D
b
D
g
q
3
D
b1
D
b2
D
b3
:
in the mapping class group of S
1;3
given in Corollary 4.0.3 is isotopic to the star relation, then
it satises the 2-chain relation. To see this, we will use the following argument explained by Dan
Margalit [Mc].
Consider the yellow curve
y
as depicted in Fig. 6 which intersects
g
once and is disjoint
from
b
and
p
. Then we have:
D
r
D
1
g
D
y
D
g
:
90
b
p
g
r
y
b
1
b
2
b
3
Figure 6. We label the blue, green, purple, red, and yellow curves as
b
,
g
,
p
,
r
, and
y
, respectively. The boundary components are labeled as b
1
, b
2
, and b
3
.
After plugging D
r
D
1
g
D
y
D
g
into the relation given in Corollary 4.0.3, we get the
following relation:
ppD
1
g
D
y
D
g
qD
p
D
b
D
g
q
3
D
b1
D
b2
D
b3
:
After multiplying outD
1
g
D
y
D
g
D
p
D
b
D
g
with itself three times and canceling
out the terms, we obtain:
D
1
g
D
y
D
g
D
p
D
b
D
y
D
g
D
p
D
b
D
y
D
g
D
p
D
b
D
g
D
b1
D
b2
D
b3
:
Using the fact that D
b1
D
b2
D
b3
is central, we can conjugate both sides by D
g
and get:
D
y
D
g
D
p
D
b
D
y
D
g
D
p
D
b
D
y
D
g
D
p
D
b
D
b1
D
b2
D
b3
:
We can also apply conjugation by D
y
. After applying conjugation, we have:
D
g
D
p
D
b
D
y
D
g
D
p
D
b
D
y
D
g
D
p
D
b
D
y
D
b1
D
b2
D
b3
;
pD
g
D
p
D
b
D
y
q
3
D
b1
D
b2
D
b3
which is the star relation.
Hence, n 2 and k 3 case of Corollary 4.0.2 recovers the star relation. The star relation
and the braid relation imply the 2-chain relation as follows.
91
If you cap o one boundary component, then two curves in the star relation become the same.
Then it is an application of the braid relation to get the 2-chain relation
pD
2
1
D
2
D
3
q
3
D
b1
D
b2
:
Therefore, n 2 and k 3 case of Corollary 4.0.2 recovers the classical chain relation for a
torus with 2 boundary components.
92
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95
Abstract (if available)
Abstract
In the present thesis, we study symplectic mapping class group relations of higher dimensional symplectic manifolds and closed Reeb orbits of higher dimensional contact manifolds. Hence, the thesis is structured in two main parts. In the first part, we introduce the notions of an iterated planar open book decomposition and an iterated planar Lefschetz fibration and prove the Weinstein conjecture for ""iterated planar"" contact manifolds. In the second part, we examine mapping class group relations of some symplectic manifolds. For each n ⩾ 1 and k ⩾ 1, we show that the 2n-dimensional Weinstein domain W = {f = 0} ∩ B²ⁿ⁺², determined by the degree k homogeneous polynomial f ∊ ℂ[z₀,...,zn], has a Boothby-Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case n = 2 recovering the classical chain relation for the torus with two boundary components.
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Creator
Acu, Bahar
(author)
Core Title
On foliations of higher dimensional symplectic manifolds and symplectic mapping class group relations
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Mathematics
Publication Date
04/28/2017
Defense Date
03/02/2017
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
contact geometry,OAI-PMH Harvest,pseudoholomorphic curves,symplectic geometry,topology
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Honda, Ko (
committee chair
), Bonahon, Francis (
committee member
), Ganatra, Sheel (
committee member
)
Creator Email
bacu@usc.edu,baharacu@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-366606
Unique identifier
UC11255761
Identifier
etd-AcuBahar-5286.pdf (filename),usctheses-c40-366606 (legacy record id)
Legacy Identifier
etd-AcuBahar-5286.pdf
Dmrecord
366606
Document Type
Dissertation
Rights
Acu, Bahar
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
contact geometry
pseudoholomorphic curves
symplectic geometry
topology