Nonorientable contact structures on 3-manifolds

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Content NONORIENTABLE CONTACT STRUCTURES ON 3-MANIFOLDS
by
David Crombecque
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
December 2006
Copyright 2006 David Crombecque
Dedication
I would like to dedicate this work to my family, friends and the whole city of Los Ange-
les that I miss terribly.
ii
Acknowledgements
I would like to thank Ko Honda for his unconditional help and patience through this
journey. Thanks also go to Francis Bonahon for sharing his great advice with me for five
years. Finally, I would like to address a special thanks to Camille Conley, Christopher
Hiatt, Valerie Pirktl and Justin Verduyn for their support and technical help.
iii
Table of Contents
Dedication ii
Acknowledgements iii
List of Figures v
Abstract vi
1 Introduction 1
2 Convex surfaces 5
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Giroux’s Flexibility Theorem . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Genericity of convex surfaces . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Tight versus overtwisted . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Classification of orientable tight contact structures . . . . . . . . . . . . 17
3 Nonorientable contact structures on T
2
×I 20
3.1 Convex torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Analysis of a basic slice . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Nonorientable contact structures on the solid torus 27
4.1 First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Analysis of the solid torus with boundary slope−
p
q
. . . . . . . . . . . 30
5 Example on the torus bundle over S
1
39
Reference List 41
iv
List of Figures
3.1 Bypass attachment for #Γ > 1. . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Bypass attachment for #Γ = 1 and−
1
2
≤ r < 0. . . . . . . . . . . . . 22
3.3 Bypass attachment for #Γ = 1 and−1 < r <−
1
2
. . . . . . . . . . . . 22
3.4 Interpretation of Case (3) on the Farey tessellation. . . . . . . . . . . . 24
4.1 Potentially allowable configurations. . . . . . . . . . . . . . . . . . . . 28
4.2 Disallowed Configuration. . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 First state transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Bypass Attachment for curve of slope−
p
q
along a curve of slope 0. . . . 31
4.5 Configuration for curve of slope−
4
3
. . . . . . . . . . . . . . . . . . . . 33
4.6 Graph leading to a disallowed configuration. . . . . . . . . . . . . . . . 35
4.7 Graph of a bypass attachment. . . . . . . . . . . . . . . . . . . . . . . 36
4.8 State transition for slope−
10
9
obtained from state for slope−
6
5
. . . . . . 37
v
Abstract
Since Bennequin’s work [1], it has been well-known that in 3-dimensional topology,
there is a dichotomy between tight and overtwisted contact structures. Overtwisted con-
tact structures are well-understood through their classification due to Y. Eliashberg [2]
but the study of tight contact structures from a 3-dimensional perspective is still at its
early stage. In 1991 though, E. Giroux [7] introduced convex surfaces which enjoy two
essential proprerties: flexibility and genericity. In 2000, K. Honda [10] introduced the
notion of bypasses, which he combined with convex surfaces to classify tight contact
structures on such manifolds as the solid torus, lens spaces and torus bundles which
fiber over the circle [10], [11], [8].
In most studies, contact structures are always considered oriented. (Recall that a
contact 3-manifold is always orientable but its contact structure does not have to be).
It is often thought that if one has to deal with nonorientable contact structures, one
may work with its orientation double cover. Although it is true that the tightness of
the double cover implies the tightness of the corresponding nonorientable contact struc-
ture, our motivation is to realize that one cannot merely switch to the orientation double
vi
cover without loss of information when studying tightness. In this thesis, we systemat-
ically study the tightness of nonorientable contact structures and produce examples of
3-manifolds equipped with nonorientable tight contact structures for which the orienta-
tion double cover is overtwisted.
vii
Chapter 1
Introduction
The birth of contact topology can be attributed to the work of Huygens, Hamilton and
Jacobi on geometric optics more than two centuries ago but the past twenty years wit-
nessed the main developments in the field. A contact 3-dimensional manifold is a man-
ifold which carries a completely nonintegrable field ξ of planes. Such a field is locally
determined as the kernel of a one formα which satisfiesα∧dα%= 0.
Theorem 1.0.1 (Martinet [14]). There is a contact structure on every 3-manifold.
Consider on R
3
the contact structure ξ
0
given as the kernel of α = dz− ydx. ξ
0
is called the standard contact structure on R
3
. In contact geometry, there are no local
invariants:
Theorem 1.0.2. Every contact 3-manifold (M,ξ) is locally diffeomorphic to (R
3
,ξ
0
).
A contact structure on a 3-manifold is said to be overtwisted if there exists an embed-
ded disk which is tangent to the contact structure everywhere along its boundary. A con-
tact structure is tight if it is not overtwisted. Overtwisted contact structures happen to be
fairly easy to construct and work with, whereas tight contact structures are more difficult
to understand but contain much more information about the manifold supporting it.
1
Theorem 1.0.3. Let ξ
t
, t ∈ [0,1], be a 1-parameter family of contact structures on a
closed manifold M. Then there exists a 1-parameter family of diffeomorphismsφ
t
such
thatφ
0
= id andφ
∗
t
ξ
t
= ξ
0
.
This means that a homotopy of contact structures leads to a contact isotopy. This is
important for classification results.
Theorem 1.0.4 (Bennequin [1]). (R
3
,ξ
0
) is tight.
Theorem 1.0.5 (Eliashberg [2]). Given a closed 3-manifoldM, letH be the set of homo-
topy classes of 2-plane fields on M andC
O
be the set of isotopy classes of overtwisted
contact structures on M. The natural inclusion map C
O
into H induces a homotopy
equivalence.
In other words, the classification of overtwisted contact structures is the same as
the homotopy classification of 2-plane fields. As a consequence, any closed 3-manifold
supports an overtwisted contact structure. On the other hand:
Theorem 1.0.6 (Etnyre-Honda [5]). There exists a closed manifold that does not support
any tight contact structure.
Makar-Limanov [13] discovered that some (oriented) tight contact structures
become overtwisted when pulled back to the universal cover
˜
M via the covering map
π :
˜
M → M. Therefore, we defined a universally tight contact structure to be one
which remains tight when pulled back to its universal cover and we call a contact struc-
ture virtually overtwisted if it becomes overtwisted when pulled back to a finite cover.
2
Every tight contact structure falls into one of the two categories, thanks to the resolution
of the Geometrization Conjecture, due to Perelman.
Contact topology lies at the interface between 3-dimensional and 4-dimensional
topology since contact manifolds are closely related to symplectic manifolds. In par-
ticular a contact manifold can be embedded in its symplectization as a hypersurface of
contact type. Hence much of the early progress in 3-dimensional contact topology came
from holomorphic techniques. In particular, here is a useful technique to show tightness:
Definition 1.0.7. A contact structure (M,ξ) is (weakly) symplectically fillable if there
exists a compact symplectic 4-manifold (X,ω) such that∂X = M andω|
ξ
> 0.
Theorem 1.0.8 (Gromov-Eliashberg [9], [4]). A symplectically fillable contact structure
is tight.
A first 3-dimensional approach to contact geometry is one using foliation theory:
Definition 1.0.9. a rank 2 foliation on a 3-manifold is an integrable 2-plane field distri-
bution. The surfaces obtained by integrating the plane field are called leaves.
Definition 1.0.10. A foliation is called taut if there is a closed transversal curve through
each leaf.
Theorem 1.0.11 (Eliashberg-Thurston [6]). Let M be a closed, oriented 3-manifold
%= S
1
×S
2
. Then every taut foliation admits aC
0
-small perturbation into a tight contact
structure.
3
In our study of nonorientable contact structures, we use a 3-dimensional approach
which consists of a cut-and-paste theory where we cut contact manifolds along convex
surfaces and check for tightness using dividing curves and bypasses. We define all of
the above in the next chapter where we also recall some classification results on oriented
tight contact structures. We then study nonorientable contact structures on T
2
× I,
S
1
×D
2
and T
2
-bundle over S
1
.
4
Chapter 2
Convex surfaces
2.1 Definitions
Definition 2.1.1. A contact structure on a 3-dimensional differentiable manifold M is a
completely nonintegrable distributionξ of planes in the tangent space TM. Locally, the
planes can be described as the kernel of a one-formα, namely,
ξ ={X∈ TM|α(X) = 0}
and the nonintegrability condition can be stated as
α∧dα%= 0.
The above conditions are independent of the choice of α and the sign of α∧ dα is
independent of the sign ofα: if in local coordinatesϕ
j
: U
j
→ R
3
,ξ is given byα
j
, then
5
there exist non-vanishing functions f
ij
: U
i
∩U
j
→ R such thatα
j
= f
ij
α
i
on U
i
∩U
j
.
The volume formsα
j
∧dα
j
satisfy
α
j
∧dα
j
= f
2
ij
α
i
∧dα
i
.
It follows thatξ induces an orientation on M. Hence a contact 3-manifold is always
orientable.
Although note that dα%= 0 onξ, dα does depend on the sign ofα, soξ may or may
not be oriented. Nowξ is said to be transversely oriented if the local 1-formsα
j
can be
chosen such that all transitions functions f
ij
> 0 . Then, using a partition of unity, one
can construct a globally defined 1-formα.
Definition 2.1.2. A contact vector field on (M,ξ) is a vector field X ∈ X(M) whose
flow preservesξ.
Definition 2.1.3. Consider a properly embedded surface S in (M,ξ). Except at the
singular points p∈ S where the tangent plane T
p
S coincides with the contact planeξ
p
,
TS∩ξ defines a line field. This line field determines a singular foliationF
S
called the
characteristic foliation of S.
If S is oriented andξ is transversely oriented, thenF
S
will also be oriented.
Definition 2.1.4. A properly embedded surface S in (M,ξ) is called convex if there is
a contact vector field X transverse to S.
6
Since the contact vector field X induces a transverse orientation of S, a convex
surface is always oriented. Also X allows us to find an I-invariant neighborhood S×
I⊂ M of S where S = S×{0}.
Definition 2.1.5. A contact structure ξ is vertically invariant on S× I if it is invariant
by the flow of
∂
∂t
where t is the coordinate in the I-direction. It is locally defined by
equations of the typeβ
j
+u
j
dt whereβ
j
andu
j
are respectively a 1-form and a function
defined on an open subset U
j
of S such that u
j
dβ
j
+β
j
∧du
j
%= 0.
Remark 2.1.6. On a convex surface S, if ξ is transversally oriented, given a volume
form ω on S, the characteristic foliation F
S
is directed by a vector field Y which will
induce an equation forξ, namely i(Y )ω +dt = 0.
Definition 2.1.7. Consider S a closed convex surface, X a contact vector field trans-
verse to S. DefineΓ
S
= {p∈ S where X(p)∈ ξ(p)}. Γ
S
is called the dividing set of
S. Note that the isotopy type ofΓ
S
does not depend on the choice of the contact vector
field X.
Remark 2.1.8. Suppose the contact structureξ is oriented in a neighborhood of a con-
vex surface S, the dividing setΓ
S
separates S into subsurfaces R
+
* R
−
where R
+
is
the subsurface where the orientations of X and the normal orientation of ξ coincide,
and R
−
the subsurface where they are opposite. In the case of nonorientable contact
structure,Γ
S
does not divide S into positive and negative subsurfaces anymore.
7
Definition 2.1.9. Let S be a convex surface in (M,ξ). If F is a singular foliation on
S, a properly embedded multi-curveΓ is said to divideF if there exists some vertically
invariant contact structureξ on S×I such thatF is the characteristic foliationF
S×{0}
on S×{0} andΓ is the dividing set for S×{0}.
2.2 Giroux’s Flexibility Theorem
Theorem 2.2.1 (Giroux’s Flexibility Theorem). Let S be a convex surface in (M,ξ), X
a contact vector field transverse to S andΓ the dividing set of S. LetF be a singular
foliation on S adapted toΓ. Then there exists an isotopyΦ
s
, s∈ [0,1] of S such that
1. Φ
0
(S) = S;
2. Φ
1
(F) is the characteristic foliation ofΦ
1
(S);
3. Φ
s
(S) is transverse to X for any s∈ [0,1];
4. the dividing set associated to X onΦ
s
(S) isΦ
s
(Γ).
Proof. The key to this proof is to realize that if S is a convex surface, then on each
connected component of S\Γ
S
, the contact structure will be transversely oriented since
the contact vector field is transverse to it. (The orientation will be reversed when going
through the dividing set.)
Let us denote by ξ
0
the vertically invariant structure on S× I induced by the flow
of X and F
S
the characteristic foliation of S. Consider a neighborhood N(Γ) of Γ
8
on S. On (S\N(Γ))× I, ξ
0
is transversely oriented and thus is given by an equation
β
0
+ dt = 0 where β = i(Y
0
)ω, ω being a volume form on S and Y
o
a vector field
directing F
S
. On the other hand, since F is adapted toΓ, there exists another contact
structure ξ
1
vertically invariant on S× I given by β
1
+ dt = 0, where β = i(Y
0
)ω.
Let us then consider the convex family of 1-forms β
s
= (1− s)β
0
+ sβ
1
for s∈ [0,1].
We get a familyξ
s
of vertically invariant and transversally oriented contact structures on
(S\N(Γ))×I given byα
s
= β
s
+dt.
Now on a neighborhood U of N(Γ), take two vector fields Y
$
0
and Y
$
1
directing F and
F
S
such that Y
$
i
= ±Y
i
on U∩ (S\N(Γ)). ξ
0
is given on N(Γ)× I by i(Y
0
)ω + g
0
dt
where g
0
vanishes onΓ and equals±1 where Y
$
0
= Y
0
. We can operate similarly with
Y
$
1
. Denote Y
$
s
= (1− s)Y
$
0
+ sY
$
1
so that you can then obtain a family of functions g
s
such that g
s
=± where Y
$
s
=±Y
s
. We now get a one- parameter familyξ
s
of vertically
invariant contact structures on S×I.
We now use Moser’s method to construct an isotopy{Φ
s
} satisfying
Φ
∗
s
(α
s
) = f
s
α
0
(2.1)
for some function f. Differentiate and get
Φ
∗
s
(L
Xs
+
dα
s
ds
) =
df
s
ds
α
0
, (2.2)
9
where X
s
is a vector field generatingΦ
s
(i.e., X
s
=
dΦs
ds
). These two equations lead to
L
Xs
α
s
=−
dα
s
ds
+g
s
α
s
(2.3)
for some function g
s
. Set g
s
= 0 and solve
i
Xs
(dα
s
) =−
dβ
s
ds
. (2.4)
and
i
Xs
(β
s
+dt) = 0 (2.5)
The non-degeneracy of dα
s
leads to the existence and uniqueness of X
s
which we inte-
grate to getΦ
s
. Note that X
s
= 0 on N(Γ).
Φ
s
(Γ) is by construction the dividing set onΦ
s
(S),Φ
1
(F) is the characteristic foli-
ation ofΦ
1
(S) andΦ
s
(S) is transverse to X for any s∈ [0,1].
2.3 Genericity of convex surfaces
We now want to show that any closed surface in a manifold equipped with a nonori-
entable contact structure can become convex after a C
∞
-small pertubation.
Definition 2.3.1. A singular foliationF on a closed surfaceS is said to be Morse-Smale
if it satisfies the following conditions:
10
1. the singularities and closed leaves ofF are hyperbolic;
2. the limit set of every trajectory is either a singularity or a closed leaf;
3. there are no saddle-saddle connections.
Proposition 2.3.2 (Giroux). Let S be a closed oriented surface in (M,ξ). If the char-
acteristic foliationF is Morse-Smale, then S is convex.
Theorem 2.3.3. On a closed oriented surface, the characteristic foliation is generically
Morse-Smale.
Proof. We want to apply Peixoto’s theorem [15] which asserts that the set of Morse-
Smale vector fields on a closed orientable surface is open and dense in the space of all
vector fields on S with the C
1
-topology. Recall that the characteristic foliation is given
locally by vector fields X
i
such that i(X
i
)θ = β
i
where β
i
is a local contact form and
θ a local volume form on S. Now F is not orientable since ξ is not. So we cannot
find a global vector field X directing F. Let us then define the space of line fields on
S, denotedL(S), byL(S) = C
∞
(S)\{−1,1} equipped with the induced C
1
-topology.
The characteristic foliationF of S is then given a line field L ofL(S).
Lemma 2.3.4. A line field L can be approximated by a line field L
1
satisfying (1).
Proof. This becomes a local problem and can be treated locally where we can choose
an orientation for the line field, namely, a local vector field.
Lemma 2.3.5. L
1
can be approximated by a line field L
$
1
satisfying (1),(2) and (3).
11
Proof. The main ingredient is a coordinate “square“. To any ordinary point p of L
$
1
, we
associate a small curvilinear closed quadrilateral containingp in its interior and bounded
by two arcs of trajectories of L
1
and two arcs of the line field orthogonal to L
1
. In local
coordinates, this can be described as a square of length one centered at p. In order to
satisfy 2), it can be shown that we can always get out of a non-trivial minimal set either
a new closed orbit or a new connection between saddle points, and the process has to be
finite. We can then follow Peixoto’s proof and by mean of successive C
1
-perturbations,
approximate L
1
by a line field satisfying (2) and (3).
Corollary 2.3.6. Let S be a closed surface embedded in a 3-manifold (M,ξ) where ξ
is nonorientable. Then there exists a C
∞
-small isotopy of S so that it becomes a convex
surface.
Proof. This follows directly from Proposition 1 and Theorem 2.
Definition 2.3.7. Let (M,ξ) be a contact manifold. A curveγ∈ M is called Legendrian
if it is everywhere tangent toξ.
Definition 2.3.8. We define the twisting number t(γ,Fr) of a closed Legendrian curve
γ with respect to a fixed framing Fr to be the number of counterclockwise 2π twists of
ξ along γ, relative to Fr. If γ is a connected component of the boundary of a compact
surface S, we denote by Fr
S
the normal framing to TS along γ. We write t(γ) for
t(γ,Fr
S
).
12
Corollary 2.3.9 (Honda [10]). LetS be a compact, oriented, properly embedded surface
with Legendrian boundary and t(∂S)≤ 0. Then there exists a C
∞
-small isotopy of S
so that it becomes a convex surface.
2.4 Tight versus overtwisted
Definition 2.4.1. In a contact manifold (M,ξ), the contact structureξ is said to be over-
twisted if there exists an embedded disk D in M such that its boundary is everywhere
tangent to the contact planes, i.e.,ξ
p
= T
p
D for any p∈ ∂D. If (M,ξ) does not contain
such a disk,ξ is said to be tight.
The Flexibility Theorem of Giroux allows us in particular to determine which convex
surfaces have tight neighborhoods:
Theorem 2.4.2. (Giroux’s criterion) IfΣ%= S
2
is a convex surface (closed or compact
with Legendrian boundary) in a contact manifold (M,ξ), thenΣ has a tight neighbor-
hood if and only ifΓ
Σ
has no homotopically trivial curves. IfΣ = S
2
, Σ has a tight
neighborhood if and only if #Γ
Σ
= 1.
The following proposition allows one to determine whether a collection of curves,
and arcs on convex surfaceΣ can be made Legendrian.
Proposition 2.4.3. A closed curve C on a convex surfaceΣ can be realized as a Legen-
drian curve if C is transverse toΣ and C∩Σ%= 0.
13
Definition 2.4.4. LetΣ∈ M be a compact, oriented, properly embedded surface with
Legendrian boundary. Assume t(γ) < 0 for all boundary componentγ. We sayγ has a
standard annular collar A if A = S
1
× [0,1] with coordinates (x,y) andγ = S
1
×{0}.
Its neighborhood A = S
1
× [−1,1] has coordinates (x,y,t), and the contact 1-form on
A = S
1
× [−1,1] isα = sin(2πnx)dy +cos(2πnx)dt.
SupposeΣ⊂ M is a properly embedded oriented surface with ∂Σ⊂ ∂M. Using
the Legendrian realization principle, we may arrange ∂Σ to be Legendrian and perturb
Σ so that it becomes convex with Legendrian boundary. Now cut M longΣ to obtain
M\int(Σ×I), which is a manifold with corners. We round the edges using the follow-
ing lemma:
Lemma 2.4.5. (Edge-rounding) [Honda [10]] LetΣ
1
andΣ
2
be convex surfaces with
collared Legendrian boundary in (M,ξ). (Namely, there exist convex annuliA
i
= γ
i
×I,
i = 1,2 which are in standard form and so that γ
i
× {1} is a boundary component
of Σ
i
.) Assume A
1
and A
2
intersect transversely along γ
1
× {1} = γ
2
× {1}. The
neighborhood of the common Legendrian boundary is locally isomorphic to the neigh-
borhood {x
2
+ y
2
≤ +} of M = R
2
× R/Z with coordinates (x,y,z) and contact
1-form α = sin(2πnz)dx + cos(2πnz)dy, for some n ∈ Z
+
. After possible per-
turbation relative to the boundary, we may take A
1
= {x = 0,0 ≤ y ≤ +} and
A
2
= {y = 0,0 ≤ x ≤ +}. If we join Σ
1
and Σ
2
along x = y = 0 and round the
common edge, the resulting surface is convex and the dividing curve z =
k
2n
onΣ
1
will
connect to the dividing curve z =
k
2n
−
1
4n
onΣ
2
, where k = 0,...,2n− 1.
14
We now introduce another essential tool to our study of tightness. This is the notion
of bypass and is due to K. Honda [10].
Definition 2.4.6. LetΣ be a convex surface. A bypass is a half-disk D with∂D = α∪β
Legendrian for which the following hold:
1. α =Σ∩D;
2. Γ
Σ
∩{p
1
,p
2
,p
3
}, where p
1
,p
2
,p
3
are distinct points;
3. α∩β ={p
1
,p
3
};
4. for one orientation of D, p
1
and p
3
are both elliptic singular points of D, p
2
is
negative elliptic, and all the singular points along β are positive and alternate
between elliptic and hyperbolic.
Suppose we cut along M along convex surface with Legendrian boundary. The
following are ways in which a bypass can occur.
Lemma 2.4.7. LetΣ = D
2
be a convex surface with Legendrian boundary inside a tight
contact manifold. If t(∂Σ) <−1, then there exists a bypass along∂Σ.
Proposition 2.4.8. (Imbalance Principle) LetΣ = S
1
×[0,1] be convex with Legendrian
boundary inside a tight contact manifold. If t(S
1
×{0}) < t(S
1
×{1})≤ 0, then there
exists a bypass along S
1
×{0}.
Definition 2.4.9. We define an abstract bypass move as follows
15
1. Start with a multicurveΓ on a closed or compact surface S, and an arc δ which
transversely intersects exactly three points ofΓ, two of them at∂δ.
2. ModifyΓ toΓ
$
, obtained as though there were an actual bypass and the dividing
set were modified under an isotopy of S.
For an abstract bypass move, the physical presence of a bypass is not necessary.
Definition 2.4.10. A bypass attachment is called trivial if it does not change the config-
uration of the dividing set.
We next describe the State Transition method due to K. Honda [12]:
Let M be a handlebody of genus g so thatΣ = ∂M is convex and D
1
,...,D
g
be
compressible disks so thatM\(D
1
∪...∪D
g
) = B
3
. FixΓ
Σ
. Suppose that|∂D
i
∩Γ
Σ
|%=
0 and #(∂D
i
∩Γ
Σ
) = |∂D
i
∩Γ
Σ
|. We make ∂D
i
Legendrian by applying Proposition
2.4.3 toΣ and then perturb D
i
so that it becomes convex.
LetC be the configuration space, namely the set of all possible configurations C =
(Γ
D
1
,...,Γ
Dg
) and introduce a directed graphG = (C,T ), where the configuration space
C is the set of vertices andT ⊂C×C is the set of directed edges, called allowable state
transitions. We will write C→ C
$
for (C,C
$
)∈T .
A configuration C gives rise toΓ
B
3 after edge-rounding.C is said to be potentially
allowable ifΓ
∂B
3 = S
1
. We say a state transition is allowable and write C→ C
$
if
1. C is potentially allowable;
16
2. C
$
can be obtained fromC via a single nontrivial abstract bypass attachment along
some D
i
.
3. Performing an abstract bypass move along a Legendrian arc on ∂B
3
from the
interior of B does not change #Γ
∂B
3.
C → C
$
implies C
$
→ C, unless C
$
is already not potentially allowable. A con-
figuration C is allowable if every C
$
∈ C in the same connected component of G is
potentially allowable. DenoteC
0
the set of allowable C∈ C. OnC
0
the graph is reflex-
ive and we writeπ
0
(C
0
) to mean the connected components ofC
0
.
Theorem 2.4.11. LetT (M,Γ
Σ
) be the set of isotopy classes of tight contact structures
on M, relative to the boundaryΣ. Then the map ψ: C
0
→ T (M,Γ
Σ
) is surjective and
factors through
π
0
(C
0
) ˜ →T (M,Γ
Σ
).
Corollary 2.4.12. Let [C]∈ π
0
(C
0
) be the connected component containing C. If [C]
contains only one configuration, then the corresponding contact structure is universally
tight.
2.5 Classification of orientable tight contact structures
In this section we recall some classification results for oriented tight contact structures.
In what follows, contact structures will be positive.
17
Theorem 2.5.1 (Eliashberg [3]). Assume there exists a contact structure ξ on a neigh-
borhood of∂B
3
which makes∂B
3
convex with dividing set having one component. Then
there exists a unique extension ofξ to a tight contact structure onB
3
up to isotopy which
fixes the boundary.
Consider a tight contact structure on T
2
× [0,1]. Fix an oriented identification of
T
2
withΣ with R
2
/Z
2
. Given a convex torus T in T
2
× I, its set of dividing curves is
determined, up to isotopy, by:
1. the number #Γ of dividing curves, and
2. their slope s(T), defined by the fact that each curve is isotopic to a linear curve of
slope s(T) inΣ with R
2
/Z
2
.
Definition 2.5.2. Consider a tight contact structureξ on T
2
× I with convex boundary
and boundary slopes s
i
= s(T
i
), i = 0,1, where T
i
= T
2
× {i}. ξ is said to be
minimally twisting in the I-direction if every convex torus parallel to the boundary has
slope s between s
1
and s
0
. (If s
1
≤ s
0
, then we want s ∈ [s
1
,s
0
]. If s
1
> s
0
, then
s∈ (s
1
,∞]∪ [−∞,s
0
).
Theorem 2.5.3 (Honda [10]). Consider T
2
× I with convex boundary and #Γ
T
0
=
#Γ
T
1
= 2. Assume, after normalizing via SL(2,Z), thatΓ
T
0
has slope−1 andΓ
T
1
has
slope−
p
q
, where p > q > 0, (p,q) = 1. Assume also we have fixed a characteristic
foliation which is adapted toT
0
andT
1
. ThenT
2
×I has exactly|(r
0
+1)(r
1
+1)...(r
k−1
+
18
1)r
k
| tight contact structures with minimalI-twisting wherer
0
,...,r
k
are the coefficients
of the continued fraction expansion of−
p
q
, namely
−
p
q
= r
0
−
1
r
1
−
1
r
2
−...
1
r
k
,
with r
i
<−1 for all i.
Theorem 2.5.4 (Honda [10]). Consider the tight contact structures on S
1
× D
2
with
convex boundary T
2
, for which #Γ
T
2 = 2 and s(T
2
) = −
p
q
, p ≥ q > 0, (p,q) = 1.
Fix a characteristic foliation which is adapted toΓ
T
2. There exist exactly|(r
0
+1)(r
1
+
1)...(r
k−1
+ 1)r
k
| tight contact structures on S
1
× D
2
, up to isotopy fixing T
2
. Here
again, r
0
,...,r
k
are the coefficients of the continued fraction expansion of−
p
q
.
19
Chapter 3
Nonorientable contact structures on
T
2
×I
Now, equipped with Giroux’s flexibility theorem for convex surfaces, we are able to
analyze nonorientable contact structures on T
2
× [0,1].
3.1 Convex torus
In this section, we will assume thatΣ is a convex torus in a manifold M equipped with
a tight nonorientable contact structureξ. We choose an oriented identification ofΣ with
R
2
/Z
2
, where the dividing setΓ
Σ
consists of 2n + 1 parallel, homotopically essential
curves of slope∞.
Theorem 3.1.1. LetΣ = T
2
be a convex torus with dividing setΓ
Σ
consisting of 2n+1
parallel, homotopically essential curves of slope s(T
2
) =∞, and assume a bypass D
is attached from the back along a Legendrian ruling of slope r with−1 < r < 0. Then
there exists a neighborhood T
2
×I ofΣ∪D with∂(T
2
×I)=T
1
−T
0
, so thatΓ
T
0
=Γ
Σ
andΓ
T
1
will be as follows, depending on whether #Γ
T
0
= 1 or #Γ
T
0
> 1:
20
1. If #Γ
T
0
> 1 and−1 < r < 0 then s(T
0
) = s(T
1
) =∞ but #Γ
T
1
= #Γ
T
0
− 2.
2. If #Γ
T
0
= 1 and−
1
2
≤ r < 0, then #Γ
T
1
= 2 and we get an overtwisted disk.
3. If #Γ
T
0
= 1 and−1 < r <−
1
2
, then #Γ
T
1
= 1 and s(T
1
) =−
3
4
.
In Theorem 3.1.1, we have normalized the Legendrian rulings so that−1 < r < 0.
Proof. This follows from the Bypass Attachment Lemma. Refer to Figure 3.1 for the
case #Γ
T
0
> 1. Note that in the following representation of T
2
, the sides are identified
and the top and bottom are identified.
(A) (B)
Figure 3.1: Bypass attachment for #Γ > 1.
In the case where Γ
T
0
= 1, there are two possibilities depending on the slope of
the Legendrian arc of attachment of the bypass. If the bypass is attached on a ruling
curve of slope r with−
1
2
≤ r < 0, then the dividing set will have two components
and will contain a homotopically trivial curve. See Figure 3.2. This bypass leads to an
overtwisted disk and therefore cannot exist inside a tight contact manifold.
21
Figure 3.2: Bypass attachment for #Γ = 1 and−
1
2
≤ r < 0.
On the other hand, if the slope of the arc of attachment is < −
1
2
, then the new
dividing set consists of one homotopically essential curve of slope−
3
4
. Refer to Figure
3.3.
Figure 3.3: Bypass attachment for #Γ = 1 and−1 < r <−
1
2
.
We are now going to interpret this result in terms of the Farey tessellation on the
hyperbolic unit disk H
2
= {(x,y)|x
2
+ y
2
≤ 1}. Recall that if∞ >
p
q
> 0 (p,q
relatively prime) and∞ >
p
!
q
!
> 0 (p
$
,q
$
relatively prime) are such that (p,q),(p
$
,q
$
)
22
form a Z-basis of Z
2
, then the point labeled
p+p
!
q+q
!
sits halfway on S
1
between
p
q
and
p
!
q
!
on
the arc for which y is always positive. We have the following restatement of Case (2)
and Case (3) of the above theorem in terms of the Farey tessellation by transforming it
using the action of SL(2,Z).
Proposition 3.1.2. LetΣ = T
2
be a convex surface with #Γ
Σ
= 1 and slope s =−
p
q
.
Let us denote r =−
p
!
q
!
with 0 < p
$
< p and 0 < q
$
< q (p
$
,q
$
relatively prime) such that
−pq
$
+p
$
q =−1. Assume a bypass is attached from the back along a curve of slope m.
1. On the Farey tessellation, if m is sitting on the arc from−
p−2p
!
q−2q
!
to r going clock-
wise, then the resulting convex surfaceΣ
$
will have #Γ
Σ
! = 1 and the new slope
s
$
will be obtained as follows: Take the arc from s to r moving clockwise. On this
arc, let s
$
be the fourth point having an edge with r.
2. If m is sitting on the arc from−
p−p
!
q−q
!
to−
p−2p
!
q−2q
!
going clockwise, then the resulting
convex surfaceΣ
$
will have #Γ
Σ
! = 2.
Figure 3.4 is an illustration of the proposition when s = −∞,r = −1, m is any-
where on the arc (−
1
2
,−1) and s
$
=−
3
4
.
Proof. Note that the bypass attachment performed in the Case (3) actually performs 4
twists around the Legendrian curve of slope−1. Thus in the Farey tessellation, moving
clockwise along the arc from−∞ to−1, we are doing a sequence of 4 hops on the
points having an edge with−1. We can now transform the previous situation through
SL(2,Z) to the case where s =−
p
q
and r =−
p
!
q
!
as described in Proposition 3.1.2.
23
1
0
0
1
-
1
1
-
1
2
-
2
3
-
3
4
-
Figure 3.4: Interpretation of Case (3) on the Farey tessellation.
Corollary 3.1.3. LetΣ = T
2
be a convex torus with #Γ = 1 and slope s =−
p
q
, (p and
q relatively prime integers > 1). If a bypass D is attached from the back along a curve
of slope m sitting on the arc from−
p−2p
!
q−2q
!
to r going counterclockwise where r =−
p
!
q
!
with 0 < p
$
< p and 0 < q
$
< q (p
$
,q
$
relatively prime) such that−pq
$
+p
$
q =−1, then
the new dividing set will have one component with slope s
$
=−
p−4p
!
q−4q
!
.
Proof. This follows from the definition of s
$
in the above proposition and the fact that
if you have two points−
p
q
and−
p
!
q
!
having an edge together, then the next point going
24
clockwise on the arc going from−
p
q
to−
p
!
q
!
having an edge with−
p
!
q
!
is given by−
p−p
!
q−q
!
.
Repeat the operation 4 times and get the desired slope.
3.2 Analysis of a basic slice
We now consider nonorientable tight contact structures on T
2
×I.
Definition 3.2.1. (T
2
× [0,1],ξ) will be called a basic slice if the following conditions
are satisfied:
1. ξ is tight;
2. The boundary T
0
and T
1
are convex and #Γ
0
= #Γ
1
= 1;
3. The minimal integer representatives of Z
2
corresponding to s
i
= s(Γ
i
) form a
Z-basis of Z
2
.
In the following propositions, we describe (T
2
× I,ξ) with boundary conditions
#Γ
0
= #Γ
1
= 1; s
0
=−
3
4
,s
1
=−∞
Proposition 3.2.2. (T
2
×I,ξ) is tight.
Proof. Let us consider the double cover of the convex torus with dividing curve and arc
of bypass attachment as in Theorem 2.1.1(3). We have a convex torus with dividing
setΓ satisfying #Γ = 2 and slope s
0
=−∞ and with two arcs of attachment . After
25
attaching both bypasses, we obtain the following sequence of slopes (−∞,−2,−
3
2
). We
now need to use a result from the Classification of orientable tight contact structures on
T
2
×I,(see [10]).
Lemma 3.2.3. Let (T
2
× [0,1],ξ) be a contact manifold which admits a factorization
T
2
×I =∪
k−1
i=0
N
i
, where eachN
i
= T
2
×[
i
k
,
i+1
k
] is a basic slice, ands
0
> s i
k
> ... > s
1
is obtained by taking the shortest counterclockwise sequence of hops along edges from
s
1
to s
0
on∂H
2
. Then (T
2
× [0,1],ξ) is tight.
Applying the above lemma, we see that the orientation double cover is tight and
thus, so is T
2
×I of the proposition.
Proposition 3.2.4. Inside (T
2
× I,ξ) there is no convex torus T parallel to T
0
or (T
1
)
whose dividing set has slope%=−∞,−
3
4
.
Proof. Assume the existence of a torus T
$
parallel to T
1
with slope s in (−∞,−1). The
orientation double cover is given by T
2
× [0,1] with the following convex boundary
conditions: #Γ
0
= #Γ
1
= 2, s
1
=∞,s
0
=−
3
2
. Using the Imbalance Principle, there
exists a bypass along T
1
. Factoring T
2
× I, we get the following sequence of slopes:
−
3
2
,−2,∞. But T
$
is now a convex torus of slope 2s∈ (−∞,−2). Now this point 2s
is not on the shortest path from−
3
2
to∞. This leads to a contradiction.
Remark 3.2.5. One should notice the difference with a basic slice (T
2
× I,ξ) in the
case whenξ is oriented. In that case, for any rational slope s, there is a convex torus T
$
parallel to the boundary whose dividing set attains slope s.
26
Chapter 4
Nonorientable contact structures on
the solid torus
In this chapter, we give the first examples of nonorientable tight contact structures which
become overtwisted when pulled back to the orientation double cover. In this chapter,
we fix an identification of T
2
= ∂(S
1
×D
2
) with R
2
/Z
2
, where (1,0)
T
is the meridian
circle (slope is 0) and (0,1)
T
the longitude (slope is∞). Let (S
1
× D
2
,ξ) be the solid
torus with convex boundary T
2
such that #Γ
T
2 = 1 and slope s(Γ) = −
p
q
satisfying
−∞ <−
p
q
≤−1.
4.1 First Example
Theorem 4.1.1. Consider the solid torus S
1
× D
2
with convex boundaryΣ = T
2
and
dividing set Γ
Σ
which consists of one curve of slope s(Γ) = −
6
5
. Then there exists
a unique nonorientable tight contact structure ξ. Its orientation double cover is over-
twisted.
Proof. First let us find tight contact structures on our solid torus. To achieve this goal,
we will use the state transition method and Eliashberg’s classification theorem of tight
27
contact structures on the ball B
3
. Let D be a compressing disk for S
1
× D
2
. Apply-
ing the Legendrian realization principle toΣ, we make ∂D Legendrian and then per-
turb D so that it becomes convex with Legendrian boundary. We then cut S
1
× D
2
along D to obtain a 3-ball B
3
. Now the setC of potentially allowable configurations is
{(1),(2),(3)} as given in Figure 4.1.
(1) (2)
(3)
Figure 4.1: Potentially allowable configurations.
Note that the configurations whereΓ
D
is∂-parallel are disallowed since they lead to
#Γ
∂B
3 > 1 after edge-rounding. (See Figure 4.2).
One can easily compute the state transitions (1)↔ (2)↔ (3)↔ (1) where each
state transition is given by a single bypass move which leaves #Γ
B
3 = 1. We describe
the transition from (1), to (2) in Figure 4.3.
28
Figure 4.2: Disallowed Configuration.
Thus, applying Theorem 2.4.11, we found a tight contact structure on S
1
×D
2
with
#Γ
T
2 = 1. This contact structure is unique up to isotopy relative to the boundary, since
the three states are in the only allowable ones and are connected to each other.
Now the oriented double cover of (S
1
× D
2
,ξ) is a solid torus (S
1
× D
2
,ξ
$
) for
which the dividing curveΓ
$
has 2 components of slope s
$
= −
3
5
. The configurations
(1),(2) and (3) will lead to #Γ
$
∂B
3
> 1, implying the overtwistedness ofξ.
Remark 4.1.2. In the oriented case, on the solid torus S
1
× D
2
with convex boundary
Σ = T
2
and dividing setΓ
Σ
which consists of two parallel curves of slope s(Γ) =−
6
5
,
29
(1) (2)
Figure 4.3: First state transition.
there are two tight contact structures up to isotopy fixing the boundary, and they are
universally tight.
4.2 Analysis of the solid torus with boundary slope−
p
q
Let (S
1
× D
2
,ξ) be the solid torus with convex boundary T
2
such that #Γ
T
2 = 1 and
slope s(Γ) =−
p
q
satisfying−∞ <−
p
q
≤−1. Recall that (p,q) are relatively prime
and p is even. Our hope is to be able to assess the tightness (or not) of (S
1
× D
2
,ξ),
given the slope−
p
q
.
Recall from the previous chapter: LetΣ = T
2
be a convex torus with slope s(Γ) =
−
p
q
, #Γ = 1 and assume a bypass D is attached from the back on a Legendrian curve of
slope m, which lies on the clockwise arc on the boundary if the Farey tessellation from
−
p−2p
!
q−2q
!
to−
p
!
q
!
. Then the new dividing curve will have slope s(Γ
1
) = −
|p−4p
!
|
|q−4q
!
|
where
p
$
and q
$
are the smallest relatively prime positive integers such that pq
$
−qp
$
= 1.
30
LetΓ
1
be the dividing curve obtained by attaching a bypass from the back along a
curve of slope 0.Γ
1
is then isotopic to one of the following:
(A) (B)
(C) (D)
Figure 4.4: Bypass Attachment for curve of slope−
p
q
along a curve of slope 0.
Let us denote by N the number of twists in the p-direction necessary to reach those
positions. Then
• (A) occurs when qN =−1 mod p. Therefore, N = p
$
.
• (B) will occur when qN = 0 mod p and thus N = p.
31
• (C) will occur whenqN = 1 modp. Therefore,N = p−p
$
. In that case, attaching
the bypass leads to a homotopically trivial curve and therefore cannot exist in a
tight contact manifold.
• (D) will occur when qN = 2 mod p. Therefore, N = p− 2p
$
.
We have the same argument in the vertical direction: (A),(B),(C),(D) correspond
respectively to N
$
= q
$
,q,q− q
$
,q
$
− 2q
$
where N
$
is the number of twists in the q-
direction.
Lemma 4.2.1. If p
1
≥ p or if q
1
≥ q, then the new surface Σ
1
has a homotopically
trivial curve.
Proof. Case 1:−p+4p
$
> p . (Note thatp
$
being positive, we cannot havep−4p
$
> p).
This implies
• (i) p− 2p
$
< 0 so D cannot happen.
• (ii) p−p
$
< p
$
so we are in case (C) andΓ
1
has a homotopically trivial curve.
Case 2:−p + 4p
$
= p. Then p
$
=
p
2
which leads to case and (C) again. We can repeat
the same argument in the vertical direction.
The previous lemma therefore allows us by a simple computation to eliminate some
solid tori (for instance in the case when p = 2n for n ≥ 2 and q = 1) as obviously
overtwisted.
Lemma 4.2.2. If p
1
< p and q
1
< q, then the new dividing curve has slope−
|p−4p
!
|
|q−4q
!
|
.
32
Proof. This is a consequence of Corollary 3.1.3.
Example 4.2.3. Consider the solid torus (S
1
× D
2
,ξ) with convex boundary T
2
such
that #Γ
T
2 = 1 and slope s(Γ) =−
4n
4n−1
, for n≥ 1. There is no tight contact structure
on S
1
×D
2
.
Proof. First we show this is true for n = 2 (i.e., the slope s(Γ) = −
4
3
). Using again
the state transition method, we see that the only possible configurations forΓ
D
are ∂-
parallel (see Figure 4.5 ) and lead toΓ
∂B
3 > 1. Thus, the corresponding solid torus is
overtwisted.
Figure 4.5: Configuration for curve of slope−
4
3
.
33
We can then prove the statement by induction on n. Assume there is no tight contact
structure on the solid torus (S
1
× D
2
,ξ) with convex boundary T
2
such that #Γ
T
2 = 1
and slope s(Γ) = −
4n
4n−1
. Consider then the solid torus (S
1
× D
2
,ξ) with boundary
slope−
4(n+1)
4(n+1)−1
. Let D be a meridian disk with Legendrian boundary and tb(∂D) =
−4(n + 1). Using the Imbalance Principle, there exists a bypass along ∂D. Attaching
the bypass to the ∂(S
1
× D
2
) and applying Corollary 3.1.3, we know that attaching a
bypass to the torus of slope−
4(n+1)
4(n+1)−1
will lead to a new slope−
4n
4n−1
. We can then
prove the theorem by induction on n.
Remark 4.2.4. Again, in the oriented case, on the solid torus S
1
× D
2
with convex
boundaryΣ = T
2
and dividing setΓ
Σ
which consists of two parallel curves of slope
s(Γ) =−
4n
4n−1
, there are two tight contact structures up to isotopy fixing the boundary.
Example 4.2.5. Consider the solid torus S
1
× D
2
with convex boundaryΣ = T
2
and
dividing setΓ
Σ
which consists of one curve of slope s(Γ) =−2. There exists a unique
nonorientable tight contact structure. Its orientation double cover is also tight.
Proof. In this case, there is only one configuration since there is only one arc, it is poten-
tially allowable. Thus, there is a unique nonorientable tight contact structure onS
1
×D
2
.
Since there is only one configuration, the contact structure is actually universally tight
and thus its oriented double cover, namely S
1
×D
2
with convex boundaryΣ = T
2
and
dividing setΓ
Σ
which consists of 2 parallel curves of slopes(Γ) =−1, is also tight.
34
Theorem 4.2.6. Consider the solid torus (S
1
× D
2
,ξ) with convex boundary T
2
such
that #Γ
T
2 = 1 and slope s(Γ) = −
4n+2
4n+1
, for n ≥ 0. Then there exists a unique (up
to isotopy fixing the boundary) nonorientable tight contact structure such that, when
pulled back to its orientation double cover, it becomes overtwisted.
Proof. We are going to use the state transition method combined with induction on the
number of arcs on the meridian diskD along which we cut the solid torusS
1
×D
2
to get
B
3
. Let us denote byC the set of potentially allowable configurations. C is not empty
since all the configurations with all arcs parallel to each other are potentially allow-
able by edge rounding lemma. From the same argument, the configurations with only
boundary-parallel arcs are not potentially allowable. Let us associate to each configu-
ration a graph where each vertex denotes a connected component of the configuration
bounded by different arcs and/or the boundary of D. Thus the following configuration
for the graph is disallowed if it contains a branching as in Figure 4.6.
Figure 4.6: Graph leading to a disallowed configuration.
A bypass attachment will transform the graph as described in Figure 4.7 .
35
Figure 4.7: Graph of a bypass attachment.
Starting with a potentially allowable configuration, a single bypass move is allow-
able if it does not lead to a configuration containing Figure 4.6. Let us denote by 2n + 1
the number or arcs on D (hence 2n + 2 is the number of vertices on any correspond-
ing configuration). Suppose (S
1
× D
2
,ξ) with #Γ
T
2 = 1 and slope s(Γ) =−
2(2n+1)
4n+1
is tight. Consider each allowable configuration, then you can always attach two arcs
such that the graph for the new disc does not contain the branching from Figure 4.6.
(Note that there may be more than one way of doing this and thus we get more poten-
tially allowable configurations). Now take two allowable configurations C
1
and C
2
with
2n + 1 arcs such that there exists a single allowable bypass move from C
1
to C
2
, then
by the previous process, attached two new arcs the same way on C
1
and C
2
. We get two
potentially allowable configurations C
$
1
and C
$
2
. Next, on C
$
1
, attach the same bypass
that was attached on C
1
. This bypass attachment leads to C
$
2
. Therefore C
$
1
and C
$
2
are
allowable. Hence, given a class [C] in π
0
(C), repeating the process described above to
each element of [C], we obtain a class [C
$
] inπ
0
(C
$
).
36
Figure 4.8: State transition for slope−
10
9
obtained from state for slope−
6
5
.
Therefore applying Theorem 1.4.6 , we obtain a tight contact structure on the solid
torus of slope−
2(2(n+1)+1)
4(n+1)+1
.
The next step is to find a tight contact structure such that, when pulled back to its
orientation double cover, it becomes overtwisted. Assume ξ is a tight contact structure
on (S
1
× D
2
,ξ) with #Γ
T
2 = 1 and slope s(Γ) = −
2(2n+1)
4n+1
corresponding to [C],
where C is the configuration with all arcs parallel to each other. Then by the process
described previously, we can obtain a tight contact structureξ
$
on the solid torus of slope
−
2(2(n+1)+1)
4(n+1)+1
which can also be represented by a configuration with all arcs parallel
to each other. We now prove that the orientation double cover of such tight contact
structures is overtwisted. This will end the proof of Theorem 4.2.6. The orientation
37
double of cover of ξ on (S
1
× D
2
,ξ) with #Γ
T
2 = 1 and slope s(Γ) = −
2(2n+1)
4n+1
is
a contact structureΞ on (S
1
× D
2
,ξ) with #Γ
T
2 = 2 and slope s(Γ) = −
2n+1
4n+1
. On
such a solid solid torus, the configurations with all arcs parallel to each other will lead
after edge rounding to an overtwisted contact structure on B
3
. This implies thatΞ is
overtwisted.
The uniqueness can also be shown by induction: let ξ be a tight contact structure
on M / S
1
× D
2
with #Γ
T
2 = 1 and slope s(Γ) =−
2(2(n+1)+1
4(n+1)+1
. We can then factor
M into M / N ∪ (M\N) where N is the solid torus with #Γ
T
2 = 1 and slope
s(Γ) = −
2(2n+1)
4n+1
and M\N = T
2
× I with boundary conditions #Γ
0
= #Γ
1
= 1;
s
0
=−
2(2n+1)
4n+1
,s
1
=−
2(2(n+1)+1
4(n+1)+1
. By induction hypothesis, there exists a unique contact
structure on N, and by Proposition 3.2.2 and 3.2.3, M\N being a basic slice, it also
admits a unique contact structure. This implies the uniqueness of a tight contact structure
on M.
38
Chapter 5
Example on the torus bundle over S
1
A T
2
-bundle M over S
1
can be viewed as T
2
× [0,1] with coordinates (x,t), whose
boundary components are glued via the monodromy map A : T
2
×{1}→ T
2
×{0},
where (x,1)0→ (Ax,0). The T
2
-bundle isomorphism type only depends on the conju-
gacy class [A] in SL(2,Z).
Theorem 5.0.7. Let M be the T
2
-bundle over S
1
with monodromy map A given by




5 4
−4 −3




. Then there exists a nonorientable tight contact structure on M such that
its orientation double cover is overtwisted.
Proof. In this proof we will apply the state traversal method [11] which can be described
as follows: decompose your contact manifold (M,ξ) as M = M
1
∪...∪M
k
where each
M
i
is irreducible, each boundary component of ∂M
I
is incompressible and the union
of all the boundaries is convex. Assume that we can determine whether ξ|
M
i
is tight.
A state is a collection{(M
i
,ξ
i
)|i = 1,...,k} such that the contact structures ξ
i
glue to
form a contact structure isotopic toξ. The state is called a tight state if each of theξ
i
is
tight. ξ will be tight if, starting with a tight state, we can show that all states that can be
obtained from from this one are also tight.
39
Let M be as described in Theorem 5.0.7. The initial state is obtained by cutting
M along the fiber torus T
2
×{1}, taken to be convex and with boundary conditions
#Γ
T
1
= 1 and slopes
1
=−∞. Hence the initial state consists ofT
2
×I is with #Γ
T
0
=
#Γ
T
1
= 1 and respective slopes s
0
=−
3
4
, s
1
=−∞, namely a basic slice which is tight
according to Proposition 3.2.2. The initial state is thus tight. By Proposition 3.2.4, this
is the only possible state. We use the monodromy map A to obtain M. Hence by the
state traversal method, we obtain a nonorientable tight contact structure on M.
The oriented double cover of (M,ξ) is given by T
2
× I with monodromy map
A
$
=




5 2
−8 −3




where #Γ
T
!
0
= #Γ
$
T
!
1
= 2, s
$
0
= −
3
2
, s
$
1
= −∞. Factoring
T
2
× [0,1] into (T
2
× [0,
1
2
])∪ (T
2
× [
1
2
,1]), we get the following sequence of bound-
ary slopes−∞,−2,−
3
2
. Moving (T
2
× [
1
2
,1]) to the back via A, leads to the sequence
(−2,−
3
2
,−2). Using Lemma 3.2.3 from the classification of oriented tight contact struc-
tures on T
2
× I, the contact structure on T
2
× [−
1
2
,
1
2
] is overtwisted. Hence, (M,ξ) is
overtwisted. (Referring to the classification of oriented contact structures on T
2
-bundle
over S
1
, A
$
is in the case where the rotation is large, namely, at leastπ).
40
Reference List
[1] D. Bennequin, Entrelacements et ´ equations de Pfaff, Ast´ erisque 107,108 (1983)
87-161.
[2] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds,
Invent. Math. 98 (1989) 623-637.
[3] Y. Eliashberg, Contact 3-manifolds, twenty years since J. Martinet’s work, Ann.
Inst. Fourier 42 (1992) 165-192.
[4] Y. Eliashberg, Filling by holomorphic discs and its applications, London Math.
Soc. Lecture Notes Series 151 (1991) 45-67.
[5] J. Etnyre, K. Honda, On the non-existence of tight contact structures, Annals of
Math. 153 (2001) 749-766.
[6] Y. Eliashberg, W. Thurston, Confoliations, Amer. Math. Soc., Providence, Uni-
versity Lectures Series 13 (1998)
[7] E. Giroux, Convexit´ e en topologie de contact, Comm. Math. Helv. 66 (1991) 637-
677.
[8] E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages
de surfaces, Invent. Math. 141 (2000) 615-689.
[9] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math.
82 (1985) 307-347.
[10] K. Honda, On the classification of tight contact structures. I, Geom. Topol. 4
(2000) 309-368.
[11] K. Honda, On the classification of tight contact structures. II, J. Differential Geom.
55 (2000) 83-143.
41
[12] K. Honda, Gluing tight contact structures, Duke Math. Journ. 115 (2002) 435-
478.
[13] S. Makar-Limanov, Tight contact structures on solid tori, Trans. Amer. Math. Soc.
350 (1998) 1045-1078.
[14] J. Martinet, Formes de contact sur les vari´ et´ es de dimension 3, Springer Lectures
Notes in Math. 209 (1971) 142-163.
[15] M. Peixoto, Structural stability on two-dimensional manifolds, Topology. 1 (1962)
101-120.
42

Asset Metadata

Creator Crombecque, David (author)

Core Title Nonorientable contact structures on 3-manifolds

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 12/11/2006

Defense Date 10/16/2006

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

Tag nonorientable contact structures,OAI-PMH Harvest

Language English

Advisor Honda, Ko (committee member)

Creator Email crombecq@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m223

Unique identifier UC1475112

Identifier etd-Crombecque-20061211 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-157169 (legacy record id),usctheses-m223 (legacy record id)

Legacy Identifier etd-Crombecque-20061211.pdf

Dmrecord 157169

Document Type Dissertation

Rights Crombecque, David

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract Since Bennequin's work, it has been well-known that in 3-dimensional topology, there is a dichotomy between tight and overtwisted contact structures. Overtwisted contact structures are well-understood through their classification due to Y. Eliashberg but the study of tight contact structures from a 3-dimensional perspective is still at its early stage. In 1991 though, E. Giroux introduced convex surfaces which enjoy two essential proprerties: flexibility and genericity. In 2000, K. Honda introduced the notion of bypasses, which he combined with convex surfaces to classify tight contact structures on such manifolds as the solid torus, lens spaces and torus bundles which fiber over the circle.