On the Giroux correspondence

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Content ON THE GIROUX CORRESPONDENCE
by
Andrew Williams
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
December 2015
Copyright 2015 Andrew Williams
Acknowledgments
First and foremost, I would like to thank my advisor, Professor Ko Honda. Without
his guidance and dedication, the completion of this project would surely not have been
possible. It has been an honor to learn from him.
Further, I would like to thank Professor Francis Bonahon for his invaluable help in
the writing of this thesis and all the encouragement he has given me.
Also, I wish to relay my sincere thanks to Professor Edmond Jonckheere for his time
and commitment.
To Amy Yung and Arnold Deal, thank you both for your understanding and patience
in these last six years.
Last but not least, I would like to thank my wonderful wife for the countless hours
she spent helping me with the edits. Without her, this thesis would also not be possible.
ii
Contents
Acknowledgments ii
List of Figures iv
Abstract v
1 Introduction 1
2 Open Books 3
3 Contact Manifolds and Convex Surfaces 7
4 Existing Results 12
5 Stabilization Equivalence 20
Reference List 38
iii
List of Figures
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iv
Abstract
Andrew Williams, December 2015 Advisors: Francis Bonahon
Ko Honda
This dissertation explores the correspondence between open books and contact struc-
tures on three-manifolds. We begin the thesis with background information necessary
to describe the correspondence. After defining both open books and contact structures,
we outline the technical results that describe the notions of compatibility, the Murasugi
sum, and stabilization equivalence. We then provide a survey of the theorems compris-
ing the correspondence, from a classical result of J. Alexander on the existence of open
books, to recent results of Emmanuel Giroux on contact structures. Our result com-
prises one of the final parts of the proof of the correspondence. It involves a certain
compact surface, called the ribbon, associated to an embedded Legendrian graph, as
well as Giroux’s construction of a cell decomposition adapted to the contact structure.
We prove a series of theorems detailing the relationship between these objects and open
book decompositions.
v
Chapter 1
Introduction
The Giroux correspondence is an important theorem in three-dimensional contact topol-
ogy. It relates two seemingly different types of structures on a given closed, oriented,
smooth three-manifoldM. The first structure is a contact structure, which is a 2-plane
field on M that is locally given as the kernel of a 1-form satisfying ^d > 0.
The other structure is purely topological; an open book decomposition ofM expresses
the manifold as the disjoint union of an embedded link B and a surface bundle over
S
1
. The theorem states that these structures are not just related but in fact are in 1-to-1
correspondence up to certain equivalences.
Theorem 1.1. There exists a 1-to-1 correspondence between contact structures onM
up to isotopy and open book decompositions ofM up to stabilization equivalence.
Isotopy is a natural equivalence relation among contact structures, while stabilization
equivalence of open books is more difficult to understand. Both are discussed in detail
in the chapters below. The theorem was first announced in 2001 by Emmanuel Giroux.
Some aspects of the theorem had been known prior to the work of Giroux (cf. work
of Thurston-Winkelnkemper and Torisu), but new ideas of Giroux allowed for the full
statement to take shape.
The proof of the theorem consists of three parts: (A) Every open book decomposition
supports a contact structure. (B) Given a contact structure onM, there exists an open
book decomposition ofM which supports the contact structure. (C) Any two open book
decompositions that support a contact structure (M;) are stabilization equivalent. (A)
was already known by the work of Thurston-Winkelnkemper and (B) was given in some
1
detail in [Etnyre ’04] and [Goodman ’03]. As for (C), only a partial sketch of a proof
was provided (cf. [Goodman ’03]) and a complete argument has yet to appear in the
literature. The goal of this thesis is to provide a survey of the theorems involved in the
Giroux correspondence and to provide a detailed argument for (C).
2
Chapter 2
Open Books
Throughout the paper we will assume our manifoldM is closed, oriented, and of dimen-
sion 3.
Definition 2.1. An open book decomposition ofM is a pair (B;) whereB is an ori-
ented link in M called the binding and :MnB!S
1
is a fibration of the compliment
ofB over the circle such that, in a tubular neighborhood of each component ofB, the
map coincides with the radial projection
0
:N(B)!S
1
defined by
0
(r;;z) =.
Here (r;;z) are cylindrical coordinates on the neighborhoodN(B) =D
2
S
1
withz
being a point onB.
For any2 S
1
we denote by

the closure of the preimage of under and we
call such a set a page of the open book. Thus, each page is a compact orientable surface
inM whose boundary isB. By pulling back the correct choice of orientation fromS
1
,
we can assume that the orientation onB is consistent with being the boundary of any
page of the open book.
A manifold can have many different open book decompositions. For example, con-
siderS
3
=f(z
1
;z
2
)2C
2
jjz
1
j
2
+jz
2
j
2
= 1g with the unknotB =f(z
1
;z
2
)2S
3
jz
1
=
0g as the binding and the projection defined by(z
1
;z
2
) =
z
2
jz
2
j
. In this case the pages
are disks which provide the standard fibration of the solid torus that is the complement
of a neighborhood ofB inS
3
.
Another important example of an open book for S
3
is given by the binding B =
f(z
1
;z
2
)2 S
3
j z
1
z
2
= 0g and the fibration (z
1
;z
2
) =
z
1
z
2
jz
1
z
2
j
. Here the binding is
actually a two-component link known as the positive Hopf link and the pages are annuli,
3
embedded with a full left-handed twist. We will refer to annuli embedded in this way as
positive Hopf bands.
The existence theorem for open books on 3-manifolds is due to Alexander and can
be considered the first and oldest part of the Giroux correspondence.
Theorem 2.2 (Alexander 1923). Every closed oriented 3-manifold has an open book
decomposition.
Proof. [Etnyre ’04, p. 5]
For the purposes of defining the stabilization equivalence of open books, it is useful
to consider a slightly different formulation known as an abstract open book.
Definition 2.3. An abstract open book is a pair (;) where is a compact, orientable
surface withk 1 boundary components and : ! is a diffeomorphism that is
the identity on@. is called the page and the monodromy.
From such a pair we can construct a 3-manifold up to diffeomorphism as follows.
First, we form the mapping cylinder

:= [0; 1]= where is the equivalence
relation ((x); 0) (x; 1). Since is the identity on the boundary, the mapping cylinder

is a 3-manifold with k boundary components, each of which is a torus. For each
boundary components of , letT
s
= sS
1
be the corresponding boundary torus of

. For each such component we glue in a solid torus via the unique (up to isotopy)
diffeomorphism :@(S
1
D
2
)!T
s
that sends the fibrationfS
1
fqgjq2@D
2
g to
the fibrationfsfq
0
gjq
0
2S
1
g. We will denote the resultant 3-manifold asM(;).
Note thatM(;
1
) is diffeomorphic toM(;
2
) if
1
and
2
are isotopic or if they
are conjugate (i.e. there exists a diffeomorphismh such thath
2
=
1
h).
In contrast with this definition, the first definition of an open book decomposition
is often referred to as an ambient open book. The primary difference between the two
4
is that ambient open books are considered up to isotopy, while abstract open books
describe the 3-manifold up to diffeomorphism. The former is the proper definition for
the statement of the Giroux correspondence but it is often convenient and instructive to
keep both perspectives in mind.
Definition 2.4. A positive stabilization of an abstract open book (;) is an open book
whose page is
0
is obtained by attaching a 1-handle to and whose monodromy is

0
=
c
, where
c
is a positive Dehn twist along a curvec that intersects the cocore
of the 1-handle exactly once. We denote the open book (
0
;
0
) by S
+
a
(;) where
a = c\ is called the arc of attachment. We will say that two open books, (
1
;
1
)
and (
1
;
2
), are stabilization equivalent if there exists a third open book that can be
obtained from either (
1
;
1
) or (
1
;
2
) by a finite sequence of positive stabilizations.
To understand this operation for ambient open books we need another definition: the
Murasugi sum.
Definition 2.5. Given two abstract open books (
i
;
i
), i = 0; 1, let c
i
be a prop-
erly embedded arc in
i
and A
i
= c
i
[1; 1] and rectangular neighborhood of
c
i
. The Murasugi sum of the open books (
0
;
0
) and (
1
;
1
) is the open book
(;) = (
0
;
0
)? (
1
;
1
) with page :=
0
[
A
0
=A
1

1
, where A
0
is identified
withA
1
such thatc
i
f1; 1g =@c
i
[1; 1], and the monodromy is =
0

1
.
The effect of this operation on the resultant manifolds was shown by David Gabai in
[Gabai ’83] to be that of connected sum.
Theorem 2.6. If (;) = (
0
;
0
)? (
1
;
1
) thenM(;)

=M(
0
;
0
)#M(
1
;
1
).
Proof. [Etnyre ’04, p. 7]
Let (H
+
;) be the open book for S
3
given above whose binding was the positive
Hopf link, and whose pages are annuli. The monodromy map can be seen to be a single
5
positive Dehn twist about the core of the annulus. As such, a positive stabilization of an
open book can be seen as the result of a Murasugi sum with (H
+
;).
In light of the above, to recognize a positive stabilization in an ambient open book,
two things need to be checked. First, the new page
0
must be isotopic to the result of
plumbing a positive Hopf band to the old page , and second, that Hopf band must be
contained in a ballB such thatB\ = A whereA is the rectangular neighborhood
of the arc of attachment. In terms of the theorem of Gabai above, this ball essentially
expressesM asM#S
3
.
6
Chapter 3
Contact Manifolds and Convex
Surfaces
In this chapter we present the results from contact geometry that are needed in the proofs
of the theorems comprising the Giroux correspondence.
Definition 3.1. A contact manifold (M;) is a smooth orientable manifoldM of odd
dimension 2n + 1 together with a co-oriented hyperplane distribution, which is locally
given as the kernel of a 1-form satisfying the contact condition: ^ (d)
n
> 0. is
called the contact structure onM.
Remark 3.2. By the Frobenius integrability theorem, the contact condition implies that
is essentially the complete opposite of a foliation and any integrable submanifolds can
have dimension at mostn.
Definition 3.3. Two contact structures,
1
and
2
, onM are said to be contactomorphic
if there is a diffeomorphism : M ! M whose tangent map sends
1
to
2
.
1
and

2
are said to be isotopic if they are connected by a smooth one-parameter family of
contact structures onM.
Theorem 3.4 (Gray’s Stability Theorem). Let
t
,t2 [0; 1], be a smooth family of con-
tact structures on a closed manifold M. Then there is an isotopy
t
of M such that
T
t
(
0
) =
t
for allt2 [0; 1].
Proof. [Geiges ’08, p. 60]
7
Gray’s Theorem says that on a closed manifold contact structures have discrete mod-
uli as opposed to other geometric structures like Riemannian metrics.
Theorem 3.5 (Darboux’s Theorem). Let be a contact form on a manifoldM of dimen-
sion 2n + 1, and let p be a point in M. Then there is a neighborhood U p with
coordinatesx
1
;:::;x
n
;y
1
;:::;y
n
;z such that
j
U
=dz +
n
X
j=1
x
j
dy
j
:
Proof. [Geiges ’08, p. 67]
This result states that all contact structures (on manifolds of the same dimension)
are locally identical so many questions about contact structures are necessarily global in
nature. We will refer to such a local chart as a Darboux chart.
Definition 3.6. A smooth submanifoldL of a contact manifold is called Legendrian if
TL and it has dimensionn.
Definition 3.7. Given a (possibly piecewise) Legendrian knotL in a contact 3-manifold
and a framingF forL, we definetb(L;F ) as the integer number of full counterclockwise
rotations made by the contact structure with respect to the framingF . Negative numbers
refer to clockwise rotations. We call this number the twisting number ofL and we will
suppress the framing notation when it is clear from context.
Definition 3.8. A Legendrian graphL = (V;E) consists of a finite set of pointsVM,
called vertices, and a finite setE of smoothly embedded, pairwise-disjoint Legendrian
arcs inM whose endpoints are contained inV , called edges. We will say a Legendrian
graph is nice if each vertexv withk 1 edges incident to it has a neighborhoodN(v)
8
that is contactomorphic to the open unit ball in (R
3
; ker(dz +r
2
d)), with respect to the
usual cylindrical coordinates (r;;z), such that:
L\N(v) =f(r;;z)jz = 0; =
1
;:::;
k
; r 0g
where the
0
i
s are distinct.
Henceforth we assume that all our Legendrian graphs are nice.
Definition 3.9. A vector fieldX is called a contact vector field if its flow preserves the
contact structure. In other words,L
X
=f where
f :M!R
+
is a non-vanishing function.
In studying contact structures it is often very useful to consider the structure in a
neighborhood of a hypersurface.
Definition 3.10. Let M be an oriented embedded surface with positive area form

. In a neighborhoodN() = R, where = 0, the contact form may be
written as =
r
+u
r
dr where
r
r2 R is a smooth family of one forms on and
u
r
is a smooth family of functions. Then consider the vector fieldX on defined by
the equation i
X

=
0
. The equivalence class of X under multiplication by positive
functions f : ! R
+
is a singular 1-dimensional foliation called the characteristic
foliation of .
Definition 3.11. An embedded surface M is said to be convex if there exists a
contact vector field that is transverse to .
Theorem 3.12 (Giroux 1991). Any closed surface in a contact 3-manifold can be per-
turbed by aC
1
-small isotopy to make it convex.
Theorem 3.13 (Honda 2000). Let M be a compact, oriented, embedded surface
with Legendrian boundary. If tb(
;F

) 0 for each component
of @, then there
9
exists a C
0
-small perturbation near the boundary (fixing @) and a subsequent C
1
-
small perturbation of the perturbed surface (fixing an annular neighborhood of @),
which makes convex.
Definition 3.14. Given a convex surface with transverse contact vector fieldX, define
the dividing set

=fp2 jX(p)(p)g
The dividing set is a nonempty embedded multicurve [Honda ’00] that encodes a lot
of information about the contact structure in a neighborhood of the convex surface. As
example, if the Legendrian boundary component has twisting numbern with respect
to the surface framing, then will have 2n endpoints on that boundary component. We
will make use of this fact in subsequent constructions.
Theorem 3.15 (Legendrian Realization Principle). Let C be a collection of closed
curves and properly embedded arcs on a convex surface with dividing set . If C
is transverse to and each connected component of nC contains a piece of , then
there exists an admissible isotopy of makingC Legendrian.
Definition 3.16. An overtwisted disk is an embedded disk D M with Legendrian
boundary such that tb(@D;D) = 0, and the characteristic foliation of D has a single
elliptic singularity in its interior. A contact structure with such a disk is called over-
twisted. If no such disk exists, the structure is called tight.
Although this dichotomy is extremely important in contact topology generally, we
will only need a few key facts for our purposes.
Theorem 3.17 (Bennequin’s Theorem). The standard contact structure onR
3
is tight.
So we know that the structure on a Darboux chart is tight and so all contact manifolds
are locally tight by Darboux’s theorem.
10
Theorem 3.18 (Giroux’s tightness criterion). If 6= S
2
is a convex surface (closed or
convex with Legendrian boundary) in a contact manifold, then has a tight neighbor-
hood if and only if

has no homotopically trivial curves. If =S
2
, then has a tight
neighborhood if and only if

is connected.
11
Chapter 4
Existing Results
Definition 4.1. We say that an open book decomposition (B;) of M supports the
contact structure if is isotopic to a contact structure
0
with a defining 1-form
0
satisfying:
1. d
0
is a symplectic form for every page

,
2.
0
> 0 on the bindingB.
Theorem 4.2 (Thurston-Winkelnkemper 1975). Given an open book decomposition
(B;) of a closed oriented 3-manifoldM, there is a contact structure supported by
(B;).
Proof. [Geiges ’08, p. 151]
Together with Alexander’s theorem on the existence of open books this shows that
every closed oriented 3-manifold admits a contact structure. It also defines a mapping
from the set of open book decompositions ofM to the set of contact structures onM.
It is this mapping that we wish to show descends to a 1-to-1 correspondence when we
identify stably equivalent open books and isotopic contact structures. To this end, there
is the following result of Giroux:
Theorem 4.3. Any two contact structures supported by the same open book decompo-
sition are isotopic.
Proof. [Geiges ’08, p. 157]
12
Furthermore, the mapping defined by the Thurston-Winkelnkemper construction is
onto:
Theorem 4.4 (Giroux 2002). Every contact structure on a closed oriented 3-manifold
M is supported by some open book decomposition ofM.
The proof of this result requires a couple of important constructions which will also
be needed for the stabilization equivalence result later.
Definition 4.5. A contact cell decomposition for (M;) is a CW-complex C for M
satisfying:
1. The 1-skeleton is a (nice) Legendrian graph.
2. Every 2-cellD is convex andtb(@D;D) =1. Moreover, for any edgee ofD,
the twisting numbertb(e;TD) is nonpositive.
3. Every 3-cell is contained in a Darboux chart (and hence is tight).
We will denote byC
(i)
thei-skeleton of the CW complexC.
Here the twisting numbertb(e;TD) is the twisting of relative toTD, whereD is
any face adjacent toe; since we may assume that =TD at the endpointsv andv
0
,
the twisting number can be taken to be an element of
1
2
Z.
We remark that, given an edge e and two 2-cells D
1
and D
2
adjacent to e, it is
possible fortb(e;TD
1
) andtb(e;TD
2
) to differ by at most1.
Proposition 4.6. Every closed contact 3-manifold (M;) admits a contact cell decom-
positionC whose 1-skeletonC
(1)
is a nice Legendrian graph and such that the boundary
of each 3-cell is an embedded 2-sphere with corners.
Proof. Since M is closed it admits a finite covering by Darboux charts. Begin with
any smooth triangulation ofM and subdivide it until each 3-cell sits inside one of the
13
charts and has the required boundary. On a small open neighborhoodN each vertexv
consider a smooth (not necessarily Darboux) coordinate chart : N ! R
3
centered
at v. By a rotational change of coordinates we may assume that the contact plane at
(v) = 0 is given byz = 0. Consider the model contact form
0
=dz +r
2
d. It agrees
with the given form at zero and so by the openness of the contact condition the form

t
= t
0
+ (1t) remains contact for allt2 [0; 1] on some neighborhood of zero.
Hence Gray’s theorem yields a contactomorphism from a neighborhood of the vertex
with the given contact form to a neighborhood of zero in the model with the form
0
,
which sends the vertex to zero. We work in this model for the remaining adjustments to
the vertices.
Lete
1
;:::;e
k
be the edges incident tov and consider their images ~ e
1
;:::; ~ e
k
under a
generic projection to the planez = 0 chosen such that the tangents of the edges at zero
are nonzero and have distinct angles
1
;:::;
k
. By aC
0
-small isotopy we can make the
e
i
’s agree with the ~ e
i
’s on some neighborhood of 0. Then, a furtherC
0
-small isotopy
will make the edges agree with the straight rays associated to the angles
i
near 0. These
rays are Legendrian for this model and so we have made the edges of the 1-skeleton
Legendrian near the vertices. Then we may use a further C
0
-small isotopy supported
away from the vertices to make the rest of the edges Legendrian. We may also stabilize
the edges via a C
0
-small isotopy supported away from the vertices such that we have
tb(@D;D)1 for each 2-cell D. Since each 2-cell D is contained in a Darboux
ball and satisfies the preceding inequality we know that: first, it can be made convex by
aC
0
-small isotopy near the boundary followed by aC
1
-small isotopy on the interior
[Honda ’00] and second, if we did so its dividing set must consist ofn =jtb(@D;D)j
arcs since any closed curves would contradict Giroux’s tightness criterion. In order to
recover condition 2 of a contact cell decomposition we find a collection A of n 1
properly embedded arcs on the convex D such that their endpoints are bounded away
14
from the vertices and such that each component of DnA has exactly one arc of the
dividing set. By the Legendrian realization principle [Honda ’00] we may perturb D
through convex surfaces so that the arcs ofA are Legendrian.
Definition 4.7. A ribbon for a Legendrian graph L = (V;E) is a compact surface
R containing L in its interior such that there exists a 1-form on R satisfying the
following:
1. d is an area form forR.
2. TL ker on the interiors of the edges ofL.
3. The unique vector fieldY

satisfyingd(Y

;) = () points transversely out of
@R.
4. There exists a contact form for such thatj
R
=.
We additionally assume that the restriction of R to a neighborhood of each vertex is
given byf(r;;z)jz = 0; r"g, where has the formdzr
2
d.
Proposition 4.8. Every nice Legendrian graphL = (V;E) admits a ribbonR.
Proof. We carry out the proof by constructingR piece by piece in various local models.
To begin we consider the given model neighborhoods around each vertexv2V . These
may be chosen sufficiently small such that they are disjoint and such that for each edge
e2E, the intersection ofe with the complement of the union of these neighborhoods is
a single Legendrian arc. Within these neighborhoods we can takeR to be the disk given
byf(r;;z)jz = 0g with the 1-form =r
2
d. is the restriction of the given contact
form =dz +r
2
d toR. The characteristic foliation ofR consists of a single elliptic
singularity atv and is radially symmetric. The vector fieldY

=r@
r
points transversely
out of the boundary ofR as required.
15
On each edgee2E choose a midpointp not contained in the neighborhoods of the
endpoints of e. For each such point we consider a neighborhood N(p) small enough
to be disjoint from the neighborhoods chosen above for the vertices. By the standard
neighborhood theorem for Legendrian submanifolds we have that in N(p), there is a
neighborhood ofe that is contactomorphic to a neighborhood ofL
0
=f(x;y;z)j x =
z = 0g in the model: (R
3
; =ydx + 2xdydz) and this contactomorphism sendsL
0
toe. In this model we wantR to be some subset of the disk given byD =f(x;y;z)jz =
0g with the 1-form = ydx + 2xdy The characteristic foliation on this disk consists
of a single hyperbolic singularity at the origin. To insure the condition regardingY

=
2x@
x
+y@
y
is satisfied, we need to chooseRD somewhat carefully. In the model
consider small transverse segments: T

x
= f(x;y;z)j x = ; < y < g and
T

y
=f(x;y;z)j y =; < x < g for some small positive and. We connect
these by integral curves of the foliation perturbed by a sufficiently small vector field
x@
x
so that the resulting boundary bounds a disk (with corners) containing the origin
such thatY

points transversely inward alongT

x
and outward elsewhere. Note that the
Legendrian edge passes through the segments T

x
. In the next steps we will work to
appropriately extendR fromT

x
to a ribbon for the entire graph.
Lete be the Legendrian arc connecting an elliptic singularityv with neighborhood
N(v) and a hyperbolic singularity p with neighborhood N(p). Using a neighborhood
of e we extend the disks defined in N(v) and N(p) by a strip S = [1; 1] [;]
with coordinates chosen such that N(v)\ S = f(x;y)j 1 x < 1 + g and
N(p)\S =f(x;y)j 1 < x 1g, and e\S =f(x;y)jy = 0g for some small
positive . We also require that T
s
S =
s
as oriented planes for all s2f(x;y)jx2
[1 + +; 1]; y = 0g for some small positive. This later condition can be
assured by integrating an appropriate section of alonge to define a strip away from the
neighborhoodsN(v) andN(p) and then smoothly connecting that strip to the disks.
16
Further, we change coordinates for the strip such that inside N(p) and N(v) the
characteristic foliation ofS is represented by the vector field@
x
. In these coordinates,
the 1-forms already constructed inN(p) andN(v) are expressed here as:

= f

dy
defined onN

:=N(v)\S and
+
=f
+
dy defined onN
+
:=N(p)\S. The contact
condition^d> 0 together with =f

dy+dz implies that (f

dy^dz)^(df^dy) =
@f

@x
dx^dy^dz > 0 and sof

is increasing inx and by an identical argument so isf
+
.
Now take a tubular neighborhood of this strip N(S)

= S [;] with vertical
coordinatez chosen so that it matches the vertical coordinate in the neighborhoodsN(v)
andN(p). We claim that with this choice the contact planes alonge are spanned by the
vector fields @
x
and f

@
z
@
y
inside N

. By the local models for N(v) and N(p),
we see thatf

andf
+
are positive functions. Therefore we may rescalef

by a small
positive constant to guarantee sup
x2N
f

(x)< inf
y2N
+f
+
(y). (Note this changes

but it still has all the properties we desire. In particular it is still the restriction of a
contact form defining the same structure since contact forms can be rescaled without
changing their kernels). This rescaling allows us to extend f

and f
+
to a smooth,
positive functionf defined on the entire strip such that
@f
@x
> 0. Fix such anf for the
construction.
Let
0
be the smooth function defined one\S =f(x;y)jy = 0g that assigns to a
pointx = (x; 0) the angle betweenT
x
S and
x
in the chosen coordinates forN(S). By
construction we have that
0
= 0 on [1 + +; 1] and
0
= tan
1
(f

) on
N

. Let (x) =
0
(x) tan
1
(f(x)). We replace the existing stripS with an isotopic
strip whose cross-section atx2 e is “tilted” by the angle (x). This does not change
the ends of the strip as the function vanishes on N
+
and N

. The result is a strip
that extends the disks constructed inN(v) andN(p) and has the contact planes alonge
spanned by@
x
andf@
z
@
y
. Now we can extend the 1-forms
+
and

to a 1-form
17
= fdy defined on the entire strip. We now must show that this is the restriction of
some contact form onM isotopic to a form defining the given contact structure.
Let be a form defining the given contact structure and consider
1
= +dz. By
choice of coordinates on the neighborhood we have that ker() = ker(
1
) on the set
H :=L[f(x;y;z)jx =1; z = 0g. Then, since the contact condition is open on the
space of 1-forms, the family
t
=t
1
+ (1t) fort2 [0; 1] remains contact for allt
on some neighborhood of ofH. This defines an isotopy through contact forms that, by
Gray’s theorem, gives a contactomorphism between a neighborhood ofH M and a
neighborhood of the same set in the model: (N(S); ker(
1
= f(x)dy +dz)) which is
the identity onH itself. By passing to a smaller strip we may get thatY

= @
x
points
transversely out the boundary of the strip. As boundary components we use integral
curves of the foliation perturbed by a small vector fieldh(x;y)@
y
, whereh is a small
positive function with
@h
@x
> 0.
After smoothing corners, we will obtain a surfaceR with boundary that containsL
and has a 1-form satisfying conditions 1, 2 and 4 above. Moreover, the vector fieldY

points transversely out of@R as desired.
Remark 4.9. The resulting ribbon has a characteristic foliation with one elliptic singu-
larity at each vertex and one hyperbolic singularity in the interior of each edge. Further-
more, the flow of the vector fieldY

will define a deformation retraction of the ribbon
onto the Legendrian graph so this surface can be assumed to lie in any small tubular
neighborhood of the graph.
Lemma 4.10. The ribbon of the 1-skeleton of a contact cell decomposition of (M;) is
the page of an open book decomposition that supports (M;).
Proof. [Etnyre ’04, p. 19]
18
Definition 4.11. A disk-decomposable contact handlebody (H;; ) consists of a han-
dlebodyH with a tight contact structure such that the boundary ofH is convex with
dividing set and there exist a set of compression disksD
1
;:::;D
g
forH whose bound-
aries each intersect exactly twice.
Claim 4.12. Thickening the ribbonR provides a tubular neighborhoodN =R [0; 1]
of the graphL that is a disk-decomposable contact handlebody.
Proof. The boundary ofN is the union ofR0 andR1 along their mutual boundary .
@N is a convex surface divided by under the contact vector field@
z
. The compression
disks are provided by the suspension of the y-axes of each hyperbolic singularity onR.
To see that N is tight, consider the double of the ribbon R
00
, which is again a convex
surface divided by . Since = @R, the curves comprising must be homotopically
essential inR
00
. So by Giroux’s tightness criterion, a neighborhood ofR
00
is tight.
19
Chapter 5
Stabilization Equivalence
In this chapter we seek to prove the final part of the Giroux Correspondence:
Theorem 5.1. Given two open books (B
1
;
1
) and (B
2
;
2
) supporting the same contact
3-manifold (M;), there exists a third open book decomposition (B
3
;
3
) supporting
(M;) that can be obtained from both (B
1
;
1
) and (B
2
;
2
) by a finite sequence of
positive stabilizations.
To do this we must first pass from open books to contact cell decompositions.
Lemma 5.2. Given an open book (;) supporting (M;), there exists a sequence of
positive stabilizations such that the resulting open book (
0
;
0
) arises from a contact
cell decompositionC for (M;). In particular, the page
0
is the ribbon of the 1-skeleton
C
(1)
.
Proof. [Etnyre ’04, p. 24]
For the proof of the main theorem below we will make a few definitions.
Definition 5.3. Two Legendrian graphsL
1
andL
2
embedded in (M;) with ribbonsR
1
andR
2
such that@R
1
= @R
2
are ribbon equivalent if there exists aC
0
-small isotopy
relative to@R
1
=@R
2
carryingR
1
toR
2
through convex surfaces.
Legendrian graphs that are ribbon equivalent may have different graph structure but
will have topologically identical ribbons. Using the Giroux flexibility theorem, one can
show that subdividing an edge or “splitting” a vertex as shown below produce ribbon
equivalent graphs.
20
Figure 5.1
Definition 5.4. LetB be a 3-cell inC
(3)
andL =C
(1)
\@B. We say thatB is flat ifL has
a smooth regular neighborhoodN with convex boundary divided by =@R, whereR is
a ribbon forL, and there exists an isotopy rel boundary of disksD
1;s
;:::;D
k;s
;s2 [0; 1],
one for each 2-cellD
i;0
of@B, such that:
1. EachD
i;s
is disjoint fromD
j;s
wherei6= j and is disjoint from all other 2-cells
ofC.
2. There exists a Legendrian graphL
0
N that is ribbon equivalent toL.
3. There exists a Legendrian graph L
00
@N Legendrian isotopic to L
0
inside N
such thatL
00
is disjoint fromC
(1)
. (We may assume thatC
(1)
\@N .)
4. EachD
i
= D
i;1
intersects@N transversely and is convex outside ofint(N); let
e
D
i
=D
i
int(N).
5. There exists an annular neighborhoodN(L
00
) @N ofL
00
such that@N(L
00
) is
Legendrian and@N(L
00
) = ([
i
D
i
)\@N.
A contact cell decompositionC is flat if each 3-cellB is flat and the isotopy of disks
satisfying (1)–(5) exists simultaneously for allB.
Theorem 5.5. Let C
1
and C
2
be contact cell decompositions of (M;). Then there
existsC
3
, a contact cell decomposition of (M;) that is a subdivision of bothC
1
and
21
C
2
, and which can be obtained from eitherC
1
orC
2
by a Legendrian isotopy and a finite
sequence of moves from the following list:
Move 1: Subdividing a 2-cell D by adding a piecewise Legendrian arc I D
such thatI intersects
D
exactly once transversely.
Move 2a: Adding a 2-cellD and a piecewise Legendrian arcI
0
with@D =I
0
[I
whereI is a piecewise Legendrian arc belonging to the existing 1-skeleton such
that
D
is a single arc with one endpoint onI and the other onI
0
.
Move 2b: Adding a 2-cellD and a piecewise Legendrian arcI
0
with@D =I
0
[I
whereI is a piecewise Legendrian arc belonging to the existing 1-skeleton such
that
D
is a single arc with both endpoints onI
0
.
Move 2c: Adding a 2-cellD and a piecewise Legendrian arcI
0
with@D =I
0
[I
whereI is a piecewise Legendrian arc belonging to the existing 1-skeleton such
that
D
is a single arc with both endpoints onI.
Move 3: Subdividing a 3-cell by adding a 2-cellD withtb(@D;D) =1 whose
boundary belongs to the existing 1-skeleton.
Proof. The proof consists of the following 4 steps.
Step 1: Let us first replaceC
1
andC
2
by ribbon equivalent Legendrian graphs (still
called C
1
and C
2
) that are trivalent. We then perturb C
1
and C
2
via a C
0
-small per-
turbation such that they intersect transversely. That is to say that after the perturbation
the 1-skeletons are disjoint from each other while intersections involving the 2-skeleton
of either complex are transverse. For the 2-skeletons this is topological but for the 1-
skeletons we need to keep them Legendrian while making them disjoint. This can be
accomplished for example by using Darboux’s theorem.
22
Step 2: Using Lemma 5.10 below we may assume thatC
(3)
1
andC
(3)
2
are flat. We
abuse notation slightly and still denote the resulting complexes byC
1
andC
2
.
Step 3A: Consider a 2-cellD ofC
(2)
1
. LetA = D\C
(2)
2
. By transversality,A is a
properly embedded 1-complex which possibly includes some closed curves arising from
interior intersections. We seek to refine the complexC
1
to includeA in its 1-skeleton.
Our first task is to describeA more carefully, since Legendrian realizingA makesA
tangent toC
(1)
2
and destroys the transversality. Take a sufficiently small tubular neigh-
borhoodN(C
(1)
2
) ofC
(1)
2
with convex boundary, which we assume intersectsD in small
disk neighborhoodsD
x
ofx2 D\C
(1)
2
. Consider
e
D
00
j;i
, which is
e
D
00
i
for the complex
C
j
,j = 1; 2, as in Lemma 5.10. We apply the Legendrian stabilization procedure from
Lemma 5.10 to stabilize the arcs of C
(1)
2
on either side of each D
x
. Hence we may
assume that, nearD
x
,@N(C
(1)
2
) intersects@
e
D
00
2;i
along arcs that intersect the same com-
ponent of
@N(C
(1)
2
)
as in Figure 5.2(a). We Legendrian realize
e
D
00
2;i
\D so that it is
a Legendrian arc which has an endpoint on
@N(C
(1)
2
)
and sufficiently negative twisting
number. Thickening the arcs and taking the union with N(C
(1)
2
) gives Figure 5.2(b).
In a similar manner, take a small tubular neighborhood N(C
(1)
1
) of C
(1)
1
with convex
boundary and consider its intersections with the faces ofC
2
in the same way. The inter-
section of
e
D
00
forD and the 2-cells ofC
2
is also assumed to be Legendrian. Consider
the union of the thickened arcs of intersection; near eachD
x
we take the union with a
neighborhood ofD
x
so that the result is as given in Figure 5.2(c). Shrinking this union
yieldsA.
We may assume after isotopy that the dividing set
D
has transverse intersection
withA and that each component of the complementDnA has nonempty intersection
with
D
. This isotopy can be supported in a neighborhood of D. The middle image
of figure 3 shows the result of this isotopy. Now whenever
D
intersects a component
ofDnA inn > 1 arcs we find additional arcs (pictured in green in the third image of
23
Figure 5.2
figure 3) onD that do not intersect
D
and that divide the component inton disks, each
containing one arc of the dividing set. We denote byA
0
the union ofA and these new
green arcs.
Figure 5.3
By constructionA
0
is a non-isolating collection of curves and arcs so we may make
A
0
Legendrian by perturbingD through convex surfaces by the Legendrian realization
principle [Honda ’00]. We still denote the resulting disk byD and the Legendrian real-
ization ofA
0
byA
0
. We also isotopeC
(2)
2
so thatD\C
(2)
2
= A A
0
. Note that each
component ofDnA
0
must topologically be a disk. To addA
0
toC
(1)
1
we follow along

D
and apply Move 1 to each component ofDnA
0
.
24
We may then repeat this process for every 2-cell inC
2
1
to obtain a refinement of the
original complex that we denote
e
C
1
, which has the property thatC
(2)
1
\C
(2)
2

e
C
(1)
1
.
Step 3B: We now perform essentially the same procedure for each 2-cellD2 C
(2)
2
by considering its intersection with C
(2)
1
, which is again an embedded 1-complex and
possibly some closed curves inD. We create the additional arcs as we did above and
this time we need only Legendrian realize these new arcs and not the entire complex as
the intersectionC
(2)
1
\D has already been made Legendrian in the preceding step. After
performing this process for each 2-cell in C
(2)
2
we obtain a refinement of the original
complexC
2
that we denote
e
C
2
, which has the property thatC
(2)
2
\C
(2)
1

e
C
(1)
2
. Note it
is not true that
e
C
1
and
e
C
2
have identical 1-skeletons but that will be achieved in the next
step.
Step 4: ConsiderB2
e
C
(3)
1
. We seek to refine
e
C
1
by adding the 1-cells and 2-cells
of
e
C
(2)
2
in the interior ofB. We induct on the number of 2-cells inint(B). If there is a
2-cellD
0
such that@D
0
@B then, since@D
0
C
(1)
1
, we may addD
0
to the complex
by applying Move 3. After adding such a 2-cell,B will be split into two 3-balls, each
with fewer 2-cells in their interiors so the induction may proceed.
On the other hand, if no such 2-cell exists inB, then there must exist an outermost
2-cellD
0
such that@D
0
=I[I
0
, whereI is a piecewise Legendrian arc in@B\
e
C
(1)
1
,
andI
0
is a piecewise Legendrian arc inint(B). We explain this in slightly more detail:
Recall the description of the neighborhood ofA from Step 3A. Lete be an edge ofC
(1)
2
passing through an intersection pointx ofC
(1)
2
\D; we writee = e
1
[e
2
, wheree
i
is
obtained by cuttinge atx. Letf
1
;:::;f
k
be the edges ofC
(2)
2
\D ending atx. Then,
by Figure 5.2(b) and the discussion from Step 3A, the unione[f
1
[[f
k
and their
thickening are as given in Figure 5.4; in particular, all thef
i
are “on one side ofe”.
IfI
0
contains the portion ofe
1
nearx, then we must also takeI to contain the portion
off
1
(notf
2
;:::;f
k
) nearx. Next we considerI
0
if it passes over an edgee
0
ofC
(1)
1
;
25
Figure 5.4
see Figure 5.5. Suppose there are two faces adjacent toe
0
which we take to beD (the
lower face in Figure 5.5) andD
0
(the upper face). By the construction from Step 3A, the
relative positions of the intersections ofD andD
0
withC
(2)
2
must be as given in Figure
5.5.
Figure 5.5
Finally observe the following key point:
(*) we may take the intersection I
00
of D
0
and @(N(A)[ N(C
(1)
1
)) so that I
00

N(L
00
), where L
00
is the Legendrian graph and N(L
00
) is the annular neighbor-
hood which appear in Definition 5.4 for B, and all the intersections of I
00
with

@(N(A)[N(C
(1)
1
))
satisfy Figure 5.6(a).
26
Figure 5.6
Such a 2-cell may be added to the complex via Move 2a, 2b, or 2c depending on the
configuration of the dividing set
D
0
. By Lemma 5.7, Move 2c is equivalent to Move
2a.
After adding such a 2-cell, we consider the result of isotoping@B acrossD
0
. The
resulting 2-sphere bounds a 3-ballB
0
B containing fewer 2-cells in its interior than
B. If the dividing set is given by Move 2a or 2b, then B
0
is close to being flat (but
not quite) when regarded as a 3-cell; see Figure 5.7 for what happens after Move 2a.
However, the induction can still proceed with the next outermost 2-cell inB
0
since (*)
is satisfied.
Figure 5.7
27
After repeating this procedure for each 3-cell B 2
e
C
(3)
1
we obtain a complex we
denote

C
1
. Similarly, we may perform the above procedure for eachB2
e
C
(3)
2
to obtain
a complex we denote

C
2
. At this stage

C
1
and

C
2
must have identical 1-skeletons
and 2-skeletons. Then, since each is a CW decomposition of the same manifold, their
3-skeletons are also identical and thus

C
1
=

C
2
. We take C
3
in the statement to be

C
1
=

C
2
and we have a refinement of both of the original complexes that can be obtained
from either one by a Legendrian isotopy and finite sequence of Moves 1, 2a, 2b, and
3.
It remains to determine the effect that each of the moves in the above theorem has
on the open books associated to the contact cell decompositions. The following lemmas
complete the proof of Theorem 5.1.
Lemma 5.6. The effects of the above Moves 1, 2a, and 2b on the open book associated
to the contact cell decomposition can be realized by positive stabilizations, while Move
3 has no effect on the associated open book.
Proof. First, since Move 3 does not affect the 1-skeleton at all, it does not change the
ribbon of the 1-skeleton and therefore it does nothing to the associated open book.
Now let I D be as in the description of Move 1. As seen in the construction
of a ribbon above, the orientation ofR is positive with respect to the coorientation of
the contact structure so ifI is added to the 1-skeleton, the condition thatkI\
D
k =
1 guarantees that the ribbon is modified by adding a strip containing I that makes a
negative half twist with respect to the framing given by D and whose endpoints lie
between the two half-twists in the ribbon for@D. This is isotopic to a Murasugi sum
with a positive Hopf band as pictured below. This Hopf band is isolated in a 3-ball
given by a small neighborhood ofD. The modifications described in Moves 2a and 2b
are similar and are pictured as well. For each move the left image shows the cell complex
28
with the blue arc being the 1-cell to be added. The other images illustrate the effect on
the ribbon and show the result is a Murasugi sum with a positive Hopf band.
Figure 5.8
Lemma 5.7. Let B 2 C
(3)
be a flat 3-cell and D a convex 2-cell with boundary
@D = I[I
0
, whereI C
(1)
\@B is a piecewise Legendrian arc andI
0
is a piece-
wise Legendrian arc in the interior of B. If the dividing set of D is a single arc with
both endpoints onI, then applying Move 2c (addingI
0
toC
(1)
) positively stabilizes the
associated open book.
29
Proof. B is flat so we may assume that @B is a convex 2-sphere containing I as part
of an equatorE dividing@B into two disks,N andS. BothD and@B are convex and
they meet at the LegendrianI so their dividing curves must interlace [Honda ’00] and
thus we may assume that
@B
intersectsI in three points, its endpoints and one point
in its interior. So in a small neighborhood ofI @B,
@B
looks like 3 arcs. Because
B is tight, the dividing set
@B
is a single closed curve by Giroux’s tightness criterion.
Isotoping @B across D is a bypass attachment as defined in [Honda ’00]. After this
isotopy the dividing curve will be locally modified as shown in the figure below but
must still be a single closed curve. This leaves only one possible way for the 3 arcs in
the neighborhood of I to be connected on @B. Since each region of @BnC
(1)
must
contain exactly one arc of
@B
, there must exist a portion of@B\C
(1)
, denotedI
00
and
pictured in blue, which subdividesN into two disks, one of which contains a single arc
of
@B
. The left image shows the 3 arcs of the dividing set in a neighborhood ofI. The
middle image shows the result of the bypass attachment to those arcs. The right image
shows the unique possible connection of the three arcs and the LegendrianI
00
.
Figure 5.9
Using the portion of the ribbon associated toI
00
(shown in blue) we can see that the
modification to the ribbon is isotopic to the result of a Murasugi sum with a positive
Hopf band.
30
Figure 5.10
For the purpose of the following lemma we first make a definition.
Definition 5.8. LetB2C
(3)
be a 3-cell in a contact cell decomposition andv a vertex
in the Legendrian graph @B\C
(1)
. Let N(v) be the model neighborhood of v with
= ker(dz +rd) described in Proposition 4.8 and letD
1
;:::;D
k
be the 2-cells in@B
which contain v in their boundaries. Then we say v is flat with respect to B if each
D
i
\N(v) maps to a distinct sector of the diskfz = 0g N(v) under orthogonal
projection in thez-direction. Ifv is not flat with respect toB, we will say it is folded
with respect toB. A vertex ofC
(1)
which is folded with respect to someB is said to be
folded.
In order to make a 3-cell flat, we would like to “unfold” each folded vertex so that
the graph L can be placed on the convex sphere isotopic to @B; see Figure 5.11. In
doing this however, the ribbon associated to L may rotate positively (clockwise) with
31
respect to@B. Figure 5.12 gives an example of a 3-cell with this obstruction. To fix this
we add in extra negative twists to the edges near a folded vertex so that the unfolding
process results in a ribbon with non-positive twisting relative to@B.
Figure 5.11
Figure 5.12
Definition 5.9. Let v be a vertex and e be an edge of C such that e is one end of v.
Let (r;;z) be cylindrical coordinates centered atv so thate is locally given byz = 0,
=
0
, r 0. Then a face D adjacent to e is to the left of (v;e) (resp. to the right
of (v;e)) if its projection to (r;) increases (resp. decreases) in degree as we go from
>
0
to<
0
.
Lemma 5.10. Every contact cell decomposition C can be made flat after a finite
sequence of modifications and these modifications positively stabilize the associated
open books.
32
Proof. For each pair consisting of an edgee ofC and a vertexv ofe, we stabilizee in
the Legendrian sense [Geiges ’08] nearv so that its twisting number becomes negative.
This stabilization is carried out by creating and then Legendrian realizing an arce
0
on a
faceD to the left of (v;e) that intersects
D
twice transversely and has both endpoints
one; see Figure 5.13. LetD
0
D be the subdisk bounded bye
0
and a subarce
00
e;
also letD
00
= DD
0
. We then effectively replacee by (ee
00
)[e
0
, modifying all
adjacent faces in the process.
Since
tb((ee
00
)[e
0
;TD
00
) =tb(e;TD) 1;
further subdivisions ofD
00
are necessary to ensure that each 2-cell has a boundary with
twisting number1. The subdivisions required onD are two applications of Move 1,
and so the effect on the ribbon associated toL is that of two positive stabilizations; see
the right-hand picture of Figure 5.13. In this figure the left image shows D and the
stabilizing arce
0
in blue, while the right image shows the result of replacing a portion of
e withe
0
and the green arc is the subdivision necessary to restore the twisting number
condition ofD. The same subdivision procedure is also performed on all theD
L
’s.
Similarly, we can replace all the other facesD
L
to the left of (v;e) by the analogously
defined disks D
00
L
such that D
L
is isotopic to the union of D
00
L
and a parallel copy of
D
0
. Such subdivisions can only create flat vertices so there are only finitely many such
modifications necessary to make all vertices flat with respect to their adjacent 3-cells.
For each of the facesD
R
to the right of (v;e), we replaceD
R
by the unionD
00
R
of
D
R
and a parallel copy ofD
0
so thatD
00
R
is adjacent to (ee
00
)[e
0
. We also add parallel
copies ofe
00
; these can be seen to affect the ribbon via single positive stabilization each
in the same manner as Move 2c. See Figure 5.14.
It remains to verify the flatness of the modified contact cell decomposition, which
we still denote by C. Let D
00
i
be the 2-cells of the new C, before the subdivisions
33
Figure 5.13
Figure 5.14
are performed. (Flatness is preserved under subdivisions of convex 2-cells, so it is not
essential to havetb(@D
00
i
) =1, justtb(@D
00
i
)1.) The goal is to draw a collection
of curves on@N(C
(1)
) representing@
e
D
00
i
as we range over all the 2-cellsD
00
i
ofC, such
that@
e
D
00
i
intersects
@N(C
(1)
)
according to Figure 5.6(a) (but not Figure 5.6(b)).
We first draw the curves@
e
D
00
i
on the portion of@N(C
(1)
) corresponding to a slight
truncation of an edgee. Refer to Figure 5.15. Assume thate ends on the vertexv which
lies slightly above the cylinders (a) and (b) drawn. The dividing set is drawn in red;
let
L
and
R
be the components of on the left-hand side and right-hand side of the
cylinder (a), respectively. If the face D
00
i
is adjacent to e and is to the left of (resp. to
the right of) (v;e), then, near the top of the cylinder,@
e
D
00
i
is an arc parallel to and close
to
L
(resp. to
R
). The slight subtlety here is that a priori we have no control over
34
deciding whether@
e
D
00
i
is to the left or to the right of
i
, since we lose control when we
move from edge to edge. However, the conditiontb(e;D
00
i
)1 (which comes from
our stabilization in the above paragraphs) allows us to always push the arcs of@
e
D
00
i
to
the left of
L
or
R
nearv in a manner consistent with Figure 5.6(a); see Figure 5.15(b).
This is essential for guaranteeing that, ifD
00
i
is adjacent toe
i
ande
j
at a vertex, the arcs
of@
e
D
00
i
on@N(C
(1)
fore
i
ande
j
can be joined either without intersecting the dividing
set
@N(C
(1)
)
or according to Figure 5.6(a).
Figure 5.15
Next we extend the @
e
D
00
i
to the portion of @N(C
(1)
) near a vertex v. Figure 5.16
depicts typical intersections of the 2-skeleton with the cylinder r = " > 0, where
(r;;z) are the usual cylindrical coordinates nearv. In the figure, the horizontal direc-
tion is the-direction and the vertical direction is thez-direction; the sides are identified.
We are assuming thatv is trivalent to reduce the number of possible cases to consider.
35
Figure 5.16
Figure 5.17 depicts the arcs of @
e
D
00
i
in both Cases (a) and (b); the modification
is done in a manner consistent with Figure 5.6(a). Although it is not exhaustive, the
procedure described is sufficient to treat all cases.
Figure 5.17
It is left to the reader to verify that each 3-cell can be made flat (in particular Condi-
tion 3 in the definition of flatness). This can be done with the help of Figure 5.18, which
shows an isotopy which unfolds a vertex.
36
Figure 5.18
37
Reference List
[Alexander ’23] J. W. Alexander. A lemma on systems of knotted curves, Proc. Nat.
Acad. Sci. USA 9 (1923), 93–95.
[Bennequin ’83] Daniel Bennequin Entrelacements et ´ equations de Pfaff, in: Third
Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), 87–
161, Ast´ erisque, 107–108, Soc. Math. France, Paris, 1983.
[Etnyre ’04] John Etnyre. Lectures on open book decompositions and contact struc-
tures, in: Floer homology, gauge theory, and low-dimensional topol-
ogy, 103141, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI,
2006.
[Gabai ’83] David Gabai. The Murasugi Sum is a Natural Geometric Operation,
in: Low-dimensional topology (San Francisco, Calif., 1981), 131–143,
Contemp. Math., 20, Amer. Math. Soc., Providence, RI, 1983.
[Giroux ’91] Emmanuel Giroux. Convexit´ e en topologie de contact, Comment.
Math. Helv. 66 (1991), 637–677.
[Giroux ’02] Emmanuel Giroux. G´ eom´ etrie de contact: de la dimension trois
vers les dimensions sup´ erieures, in: Proceedings of the International
Congress of Mathematicians, Vol. II (Beijing, 2002), 405414, Higher
Ed. Press, Beijing, 2002.
[Geiges ’08] Hansj¨ org Geiges. An Introduction to Contact Topology, Cambridge
Studies in Advanced Mathematics, 109. Cambridge University Press,
Cambridge, 2008.
[Goodman ’03] Noah Goodman Contact structures and open books, Ph. D. Thesis,
University of Texas, 2003.
[Honda ’00] Ko Honda. On the classification of tight contact structures I, Geom.
Topol. 4 (2000), 309–368.
38
[Thurston-Winkelnkemper ’75] William P. Thurston and H. Elmar Winkelnkemper. On
the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–
347.
[Torisu ’00] Ichiro Torisu. Convex contact structures and fibered links in 3-
manifolds, Internat. Math. Res. Notices 2000, 441–454.
39

Asset Metadata

Creator Williams, Andrew J. (author)

Core Title On the Giroux correspondence

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 10/01/2015

Defense Date 08/24/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag contact structures,contact topology,Giroux correspondence,OAI-PMH Harvest,open books

Format application/pdf (imt)

Language English

Advisor Bonahon, Francis (committee chair), Honda, Ko (committee member), Jonckheere, Edmond (committee member)

Creator Email lettersarethenewnumbers@gmail.com,williaaj@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-189800

Unique identifier UC11273415

Identifier etd-WilliamsAn-3964.pdf (filename),usctheses-c40-189800 (legacy record id)

Legacy Identifier etd-WilliamsAn-3964.pdf

Dmrecord 189800

Document Type Dissertation

Format application/pdf (imt)

Rights Williams, Andrew J.

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

Abstract (if available)

Abstract This dissertation explores the correspondence between open books and contact structures on three-manifolds. We begin the thesis with background information necessary to describe the correspondence. After deﬁning both open books and contact structures, we outline the technical results that describe the notions of compatibility, the Murasugi sum, and stabilization equivalence. We then provide a survey of the theorems comprising the correspondence, from a classical result of J. Alexander on the existence of open books, to recent results of Emmanuel Giroux on contact structures. Our result comprises one of the ﬁnal parts of the proof of the correspondence. It involves a certain compact surface, called the ribbon, associated to an embedded Legendrian graph, as well as Giroux’s construction of a cell decomposition adapted to the contact structure. We prove a series of theorems detailing the relationship between these objects and open book decompositions.