Yang Huang- A Proof of the Classification Theorem of Overtwisted Contact Structures via Convex...

8/3/2019 Yang Huang- A Proof of the Classification Theorem of Overtwisted Contact Structures via Convex Surface Theory

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arXiv:1102

.5398v2[math.GT

]13May2011

A PROOF OF THE CLASSIFICATION THEOREM OF OVERTWISTED CONTACTSTRUCTURES VIA CONVEX SURFACE THEORY

YANG HUANG

ABSTRACT. I n [2], Y. Eliashberg proved that two overtwisted contact structures on a closed oriented

3-manifold are isotopic if and only if they are homotopic as 2-plane fields. We provide an alternative

proof of this theorem using the convex surface theory and bypasses.

CONTENTS

1. Preliminaries 2

2. Outline of the proof 4

3. Local properties of bypass attachments 4

4. Isotoping contact structures up to the 2-skeleton 7

5. Bypass triangle attachments 9

6. Overtwisted contact structures on S 2 [0, 1] induced by isotopies. 137. Classification of overtwisted contact structures on S 2 [0, 1] 238. Proof of the main theorem 26

References 28

A contact manifold (M, ) is a smooth manifold with a contact structure , i.e., a maximallynon-integrable codimension 1 tangent distribution. In particular, if the dimension of the manifold

is three, it was realized through the work of D. Bennequin and Y. Eliashberg in [1], [3] that contact

structures fall into two classes: tight or overtwisted. Since then, dynamical systems and foliation

theory of surfaces embedded in contact 3-manifolds have been studied extensively to analyze this

dichotomy. Based on these developments, Eliashberg gave a classification of overtwisted contact

structures in [2], which we now explain.

Let M be a closed oriented manifold and M be an oriented embedded disk. Furthermore,

we fix a point p . We denote by Contot(M, ) the space of cooriented, positive, overtwisted

contact structures on M which are overtwisted along , i.e., the contact distribution is tangent to along . Let Distr(M, ) be the space of cooriented 2-plane distributions on Mwhich are tangentto at p. Both spaces are equipped with the C-topology.

Theorem 0.1 (Eliashberg). Let M be a closed, oriented 3-manifold. Then the inclusion j :

Contot(M, ) Distr(M, ) is a homotopy equivalence.

In particular, we have:

Theorem 0.2. Let M be a closed, oriented 3-manifold. If and are two positive overtwistedcontact structures on M, then they are isotopic if and only if they are homotopic as 2-plane fields.

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2 YANG HUANG

Consequently, overtwisted contact structures are completely determined by the homotopy classes

of the underlying 2-plane fields. On the other hand, the classification of tight contact structures is

much more subtle and contains more topological information about the ambient 3-manifold.The goal of this paper is to provide an alternative proof of Theorem 0.2 based on convex sur-

face theory. Convex surface theory was introduced by E. Giroux in [7] building on the work

of Eliashberg-Gromov [4]. Given a closed oriented surface , we consider contact structures on

[0, 1] such that {0, 1} is convex. By studying the film picture of the characteristic foli-ations on {t} as t goes from 0 to 1, Giroux showed in [8] that, up to an isotopy, there are only

finitely many levels {ti}, 0 < t1 < < tn < 1, which are not convex. Moreover, for small > 0, the characteristic foliations on {ti } and {ti + }, i = 1, 2, , n, change by abifurcation. In [9], K. Honda gave an alternative description of the bifurcation of characteristic

foliations in terms ofdividing sets. Namely, he defined an operation, called the bypass attachment,

which combinatorially acts on the dividing set. It turns out that a bypass attachment is equivalent

to a bifurcation on the level of characteristic foliations. Hence, in order to study contact structureson [0, 1] with convex boundary, it suffices to consider the isotopy classes of contact structuresgiven by sequences of bypass attachments. In particular, we will study sequences of (overtwisted)

bypass attachments on S 2 [0, 1], which is the main ingredient in our proof of Theorem 0.2.This paper is organized as follows. In Section 1 we recall some basic knowledge in contact

geometry, in particular, convex surface theory and the definition of a bypass. Section 2 gives an

outline of our approach to the classification problem. Section 3 is devoted to establishing some

necessary local properties of the bypass attachment. Using techniques from previous sections, we

show in Section 4 that how to isotop homotopic overtwisted contact structures so that they agree in

a neighborhood of the 2-skeleton. Section 5, 6 and 7 are devoted to studying overtwisted contact

structures on S 2 [0, 1] which is the technical part of this paper. We finally finish the proof of

Theorem 0.2 in Section 8.

1. PRELIMINARIES

Let M be a closed, oriented 3-manifold. Throughout this paper, we only consider cooriented,

positive contact structures on M, i.e., those that satisfy the following conditions:

(1) there exists a global 1-form such that = ker().(2) d > 0, i.e., the orientation induced by the contact form agrees with the orientation

on M.

A contact structure is overtwisted if there exists an embedded disk D2 M such that istangent to D2 on D2. Otherwise, is said to be tight. We will focus on overtwisted contactstructures for the rest of this paper.

Let M be a closed, embedded, oriented surface in M. The characteristic foliation on

is by definition the integral of the singular line field (x) x Tx. One way to describe thecontact structure near is to look at its characteristic foliation.

Proposition 1.1 (Giroux). Let0 and 1 be two contact structures which induce the same charac-teristic foliation on . Then there exists an isotopy t, t [0, 1] relative to such that0 = id and(1)0 = 1.

Possibly after a C-small perturbation, we can always assume that M is convex, i.e., there

exists a vector field v transverse to such that the flow ofv preserves the contact structure. Using


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A PROOF OF THE CLASSIFICATION THEOREM OF OVERTWISTED CONTACT STRUCTURES 3

this transverse contact vector field v, we define the dividing seton to be {x | vx x}.Note that the isotopy class of does not depend on the choice of v. We refer to [7] for a more

detailed treatment of basic properties of convex surfaces. The significance of dividing sets incontact geometry is made clear by Girouxs flexibility theorem.

Theorem 1.2 (Giroux). Assume is convex with characteristic foliation , contact vector field v,

and dividing set . LetF be another singular foliation on divided by . Then there exists an

isotopy t, t [0, 1] in M such that

(1) 0 = id andt| = id for all t.(2) v is transverse to t() for all t.(3) 1() has characteristic foliation F.

We now look at contact structures on [0, 1] with convex boundary. The first important resultrelating to this problem is the following theorem due to Giroux.

Theorem 1.3 (Giroux). Letbe a contact structure on W = [0, 1] so that {0, 1} is convex.There exists an isotopy relative to the boundary s : W W, s [0, 1] , such that the surfaces1( {t}) are convex for all but finitely many t [0, 1] where the characteristic foliations satisfythe following properties:

(1) The singularities and closed orbits are all non-degenerate.

(2) The limit set of any half-orbit is either a singularity or a closed orbit.

(3) There exists a single retrogradient saddle-saddle connection, i.e., an orbit from a nega-

tive hyperbolic point to a positive hyperbolic point.

In the light of Girouxs flexibility theorem, one should expect a corresponding film picture of

dividing sets on convex surfaces. It turns out that the correct notion corresponding to a bifurcation

is the bypass attachment, which we now describe.

Definition 1.4. Let be a convex surface and be a Legendrian arc in which intersects inthree points, two of which are endpoints of . A bypass is a convex half-disk D with Legendrianboundary, where D = , D , and tb(D) = 1. We call an admissible arc, and D abypass along on .

Remark 1.5. The admissible arc in the above definition is also known as the arc of attachmentfor a bypass in literature.

Remark1.6. We do not distinguish isotopic admissible arcs 0 and 1, i.e., if there exists a path ofadmissible arcs t, t [0, 1] connecting them.

The following lemma shows how a bypass attachment combinatorially acts on the dividing set.

Lemma 1.7 (Honda). Following the terminology from Definition 1.4, let D be a bypass along on . There exists a neighborhood of D M diffeomorphic to [0, 1], such that {0, 1}are convex, and {1} is obtained from {0} by performing the bypass attachment operation as

depicted in Figure 1 in a neighborhood of.

It is worthwhile to mention that there are two distinguished bypasses, namely, the trivial bypass

and the overtwisted bypass as depicted in Figure 2. The effect of a trivial bypass attachment is iso-

topic to an I-invariant contact structure where no bypass is attached, while the overtwisted bypass


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4 YANG HUANG

(a) (b)

FIGURE 1. A bypass attachment along . (a) The dividing set on {0} before thebypass is attached. (b) The dividing set on {1} after the bypass is attached.

attachment immediately introduces an overtwisted disk in the local model, hence, for example, is

disallowed in tight contact manifolds.

(a) (b)

FIGURE 2. (a) The trivial bypass attachment. (b) The overtwisted bypass attachment.

2. OUTLINE OF THE PROOF

Let and be two overtwisted contact structures on M, homotopic to each other as 2-planefield distributions. Our approach to Theorem 0.2 has three main steps.

Step 1. Fix a triangulation T of M. Isotop and such that T becomes an overtwisted contact tri-angulation in the sense that the 1-skeleton T(1) is a Legendrian graph, the 2-skeleton T(2) is convex

and each 3-cell is an overtwisted ball with respect to both contact structures. We first show that

ife() = e() H2(M;Z), then one can isotopand so that they agree in a neighborhood ofT(2).

Step 2. We can assume that there exists a ball B3 M such that and agree on M \ B3. Ob-serve that |B3 and

|B3 can both be realized as sequences of bypass attachments. In section 5,

we will define the notion of a stable isotopy. Then we show that both of sequences of bypass are

stable isotopic to some power of the bypass triangle attachment. Moreover, the Hopf invariants of

|B3 and |B3 are uniquely determined by the number of bypass triangles attached according to [10].

Step 3. By elementary obstruction theory, the Hopf invariants of |B3 and |B3 are not necessarily

the same, but they can at most differ by an integral multiple of the divisibility of the Euler class

of either or . We show that this ambiguity can be resolved by further isotoping the contactstructures in a neighborhood ofT(2). This finishes the proof of Theorem 0.2.

3. LOCAL PROPERTIES OF BYPASS ATTACHMENTS

Let Mbe an overtwisted contact 3-manifold. Let Mbe a closed convex surface with dividing

set . For convenience, we choose a metric on M and denote M\ the metric closure of the open


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manifold M . We may assume without loss of generality each connected component of M\ is

overtwisted. In order to isotop convex surfaces through bypasses freely, we must show that there

are enough bypasses. In fact, bypasses exist along any admissible Legendrian arc on providedthat the contact structure is overtwisted. This is the content of the following lemma.

Lemma 3.1. Suppose that M \ is overtwisted. For any admissible arc , there exists abypass along in M\ . If separates M into two overtwisted components, then there exists sucha bypass in each component.

Proof. The technique for proving this lemma is essentially due to Etnyre and Honda [5], and inde-

pendently Torisu [11]. Let D M \ be an overtwisted disk. We push the interior of slightlyinto M \ with the endpoints of unmoved to obtain another Legendrian arc , i.e., tb() = 1and = = . Take a Legendrian arc connecting to D, namely, the two endpoints ofare contained in and D respectively and the interior ofis disjoint from D. Then we form

the Legendrian connected sum = # D via . By results in [5] and [11], tb( ) = 1 and it iseasy to see that these two arcs together bound a desired bypass D along .

We then show the triviality of the trivial bypass, i.e, attaching a trivial bypass does not change the

isotopy class of the contact structure in a neighborhood of the convex surface. The proof essentially

follows the lines of the proof of Proposition 4.9.7 in Geiges [6]. Here the contact structure may be

either overtwisted or tight.

Lemma 3.2. Let ( [0, 1], ) be a contact manifold with the contact structure obtained byattaching a trivial bypass on ( {0}, |{0}). Then there exists another contact structure , whichis isotopic to relative to the boundary, such that {t} is convex with respect to for all t [0, 1].

Proof. Since this is a local problem, we may assume that [0, 1] is a neighborhood of the

trivial bypass attachment. By Theorem 1.2, any Morse-Smale type characteristic foliation adapted

to {0} can be realized as the characteristic foliation of a contact structure isotopic to in aneighborhood of {0}. In particular, we can assume that the characteristic foliation on {0}

looks exactly the same as in Figure 3(a) such that e does not connect to any negative hyperbolic

point other than h along the flow line.

Look at the characteristic foliations on {t} as tgoes from 0 to 1. Generically we can assume

that the Morse-Smale condition fails at one single level, say, {1/2}, where an unstable saddle-saddle connection has to appear as shown in Figure 4(a).

Let {1/2} be an open neighborhood of the flow line from h to e as depicted inFigure 4(a). Observe that the characteristic foliation inside is of Morse-Smale type, and therefore

stable in the t-direction. According to the proof of Proposition 4.9.71 in Geiges [6], for a small

> 0, there exists an isotopy s, s [0, 1], compactly supported in (2, 2) [0, 1] and0 = id, such that = (1) satisfies the following:

(1) The characteristic foliation on {t} with respect to is isotopic to the one in Figure 4(b)for t [1/2 , 1/2 + ]. In particular, it is nonsingular.

(2) For t (1/2 2, 1/2 ) (1/2 + , 1/2 + 2), The characteristic foliation on {t}with respect to is almost Morse-Smale except that there may exist a half-elliptic-half-hyperbolic point.

1This is a stronger version of the usual Elimination Lemma.


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6 YANG HUANG

e h h+ e h h+

(a) (b)

FIGURE 3. (a) The characteristic foliation on {0}. The trivial bypass is attached

along the Legendrian arc in dash line. (b) The characteristic foliation on {1}

after attaching the trivial bypass. Here e (resp. h) denote the -elliptic (resp.

-hyperbolic) singular points of the foliation.

e h h+

(a) (b)

FIGURE 4. (a) The characteristic foliation on {1/2}, where a saddle-saddle con-nect from h to h+ exists. The region contains exactly two singular points {e, h}which are in elimination position. (b) The nonsingular characteristic foliation on

after the elimination.

Now we can make {t} convex for t [1/2 , 1/2 + ] because the only unstable saddle-saddle connection is eliminated and therefore the characteristic foliation becomes Morse-Smale.

For t [1/2 , 1/2 + ], there may exist half-elliptic-half-hyperbolic singular points, but we canas well construct a contact structure realizing this type of singularity so that each {t} stays

convex. Hence constructed above is as required.

Remark 3.3. Let ( [0, 1], ) be a contact manifold such that |0 = |1 and {t} is convex forall t [0, 1]. If S 1 S 1 and is tight, then it is a standard fact that is isotopic to an I-invariantcontact structure relative to the boundary. However, if either = S 1 S 1 or is overtwisted, thenthe above fact is not true anymore. We will study this phenomenon in detail in the case when

= S 2 and is overtwisted in Section 6.


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4. ISOTOPING CONTACT STRUCTURES UP TO THE 2-SKELETON

We are now ready to take the first main step towards the proof of Theorem 0.2. Since we will

isotop contact structures skeleton by skeleton, we start with the following definition.

Definition 4.1. Let(M, ) be an overtwisted contact manifold, and T be a triangulation of M. Thetriangulation T is called an overtwisted contact triangulation if the following conditions hold:

(1) The 1-skeleton is a Legendrian graph.

(2) Each 2-simplex is convex with Legendrian boundary.

(3) Each 3-simplex is an overtwisted ball.

Remark4.2. The overtwisted contact triangulation defined above is different from the usual contact

triangulation where the 3-simplexes are assumed to be tight.

The goal for this section is to prove the following Proposition.

Proposition 4.3. Let M be a closed, oriented 3-manifold with a fixed triangulation T. Let and be homotopic overtwisted contact structures on M. Then they are isotopic up to the 2-skeleton,i.e., there exists an isotopy t : M M, t [0, 1], 0 = id such that(1) =

in a neighborhood

of T(2).

Proof. Before we go into details of the proof, observe that if t : M M, t [0, 1], 0 = id is anisotopy, then (M, 1(), T) and (M, ,

11 (T)) carries the same contact information. In fact, we will

isotop the skeletons of the triangulation T and think of them as isotopies of contact structures.

By a C0-small perturbation of the 1-skeleton T(1), we can assume that T(1) is a Legendrian graph

with respect to and . Performing stabilizations to edges of T(1) if necessary, we can furtherassume that = in a neighborhood of T(1). For each 2-simplex 2 in T(2), we can alwaysstabilize the Legendrian unknot 2 sufficiently many times so that tb(2) < 0. Therefore a C-small perturbation of 2 relative to 2 makes it convex with respect to (resp. ) with dividing

set

2(resp.

2). Both

2and

2are proper 1-submanifolds of2 and generically the endpoints

are contained in the interior of the 1-simplexes. See Figure 5 for an example.

In order to make T an overtwisted contact triangulation for and , we still need to make surethat all 3-simplexes are overtwisted. We do this for , and the same argument applies to . Takean overtwisted disc D in (M, ). We can assume that D is contained in a 3-simplex 3

1. Let 3

2be

another 3-simplex which shares a 2-face with 31

, i.e., 31

32

= 2 is a 2-simplex. We claim that

by isotoping 2 relative to 2 if necessary, we can make both 31

and 32

overtwisted. The fact that

M is closed immediately implies that a finite steps of such isotopies will make T an overtwisted

contact triangulation. To prove the claim, we first take a parallel copy of the overtwisted disk Din an I-invariant neighborhood of D, denoted by D. Pick an arc connecting D to 2 inside 31

.

Let 2 be another 2-simplex obtained by isotoping 2 across D along , i.e., 2 satisfying thefollowing conditions:

(1) 2 = 2.(2) 2 2 bounds a neighborhood ofD .(3) 2 is convex.

By replacing 2 with 2, we obtain two new 3-simplexes, each of which contains an overtwisteddisk in the interior as claimed.


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8 YANG HUANG

FIGURE 5. An example of the dividing set on a 2-simplex.

Now by Girouxs flexibility theorem, it suffices to isotop and so that they induce isotopicdividing sets on each 2-simplex relative to T(1). To achieve this goal, we define the difference

2-cocycle by assigning to each oriented 2-simplex 2 an integer (R+(

2)) (R(

2))

(R+(

2)) + (R(

2)). Since is homotopic to as 2-plane fields, [] = e() e() = 0

H2(M,Z). Hence there exists an integral 1-cocycle so that 2d = since the Euler class is always

even.2 One should think ofas an element in Hom(C1(M),Z).Let 2 T(2) be an oriented convex 2-simplex and 1 2 be an oriented 1-simplex with the

induced orientation. We study the effect of stabilizing the 1-simplex 1 to the overtwisted contacttriangulation. If we positively stabilize 1 once and isotop 2 accordingly to obtain a new 2-

simplex 2, then the dividing set

2on 2 is obtained from

2by adding a properly embedded arc

contained in the negative region with both endpoints on the interior of1 as depicted in Figure 6.Similarly, if we negatively stabilize 1 once and isotop 2 accordingly as before, then the dividing

set on the isotoped 2 is obtained from

2by adding a properly embedded arc contained in the

positive region and with both endpoints on the interior of1.

+

+

+

+ +

(a) (b)

FIGURE 6. (a) The dividing set on 2 divides it into -regions. The bottom edgeis 1. (b) One possible dividing set on 2 after positively stabilizing 1 once.

Note that in general, the new overtwisted contact triangulation obtained by -stabilizing a 1-

simplex 1 is not unique. In fact, different choices may give non-isotopic dividing sets on theisotoped 2 in the new triangulation. However, for our purpose, we only care about the quantity

(R+) (R) on each 2-simplex and it is easy to see that different choices give the same valueto this quantity. Thus we will ignore this ambiguity by arbitrarily choosing an isotopy of the

2-simplex.

We denote the overtwisted contact triangulation obtained by -stabilizing 1 once in (M, ) byS

1(). As remarked at the beginning of the proof, one should think of S

1() as isotopies of. It

2More precisely, if we fix a trivialization of T M and consider the Gauss map associated to the contact distribution,

then the Euler class of the contact distribution is exactly twice the Poincare dual of the Pontryagin submanifold of the

Gauss map.


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is easy to see that S 1

() changes (R+(

2)) (R(

2)) by 1 for any 2-simplex 2 T(2) so

that 1 2 as an oriented boundary edge. The same holds for as well.

Now we argue that one can isotop and so that (R+(2 )) (R(2 )) = (R+(

2 ))

(R(

2)) on each 2-simplex 2. This can be done as follows. For each oriented 1-simplex

1 T(1), the 1-cocycle sends it to an integer n = (1). We perform n times the isotopy S +1

()to and n times the isotopy S

1() to at the same time. If we perform such operation to every

1-simplex in T, it is easy to see that the following properties are satisfied:

(1) = in a neighborhood ofT(1).

(2) (R+(

2)) (R(

2)) = (R+(

2)) (R(

2)), 2 T(2).

The second property implies that

2can be obtained from

2by attaching a sequence of by-

passes for each 2-simplex 2. Recall that T is an overtwisted contact triangulation and in particulareach 3-simplex is an overtwisted ball. Hence bypasses exist along any admissible arc in 2 inside

any 3-simplex with 2 as a 2-face by Lemma 3.1. Therefore by isotoping 2-simplexes throughbypasses, we can assume that and induce isotopic dividing sets on each 2-simplex relative toits boundary. The conclusion now follows immediately from Girouxs flexibility theorem.

5. BYPASS TRIANGLE ATTACHMENTS

In this section we study the effect of attaching a bypass triangle to the contact structure, in

particular, we give an alternative definition of the bypass triangle attachment. We start with the

definition of the bypass triangle attachment.

Notation: Let be a convex surface and be an admissible arc. We denote the bypassattachment along on by

. Let be another admissible arc on the convex surface obtained by

attaching the bypass along on . We denote the composition of bypass attachments by ,where the composition rule is to attach the bypass along first, then attach the bypass along .If (M, ) is a contact manifold with convex boundary, then denotes the contact structureobtained by attaching a bypass along to (M, ).

Remark 5.1. In general, bypass attachments are not commutative unless the attaching arcs are

disjoint.

Definition 5.2. Let be a convex surface and be an admissible arc. A bypass triangleattachment along is the composition of three bypass attachments along admissible arcs ,

and in a neighborhood of as depicted in Figure 7. We denote the bypass triangle attachmentalong by = .

Remark 5.3. The second admissible arc in the bypass bypass triangle is also known as the arcof anti-bypass attachmentto .

Warning: When we define a bypass attachment along on (, ), there are several choicesinvolved. Namely, we need to choose a multicurve, i.e., a 1-submanifold of , representing the iso-

topy class of, an admissible arc representing the isotopy class of, a neighborhood of where is supported. Since the space of choices of and its neighborhood is contractible according toTheorem 1.2, we can neglect this ambiguity. However the space of choices of multicurves repre-

senting is not necessarily contractible. This point will be made clear in the next section. For the


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10 YANG HUANG

(a) (b)

(c)

FIGURE 7. (a) A neighborhood of on , along which the first bypass is at-tached. (b) The second bypass is attached along the dotted arc

. (c) The third

bypass is attached along the dotted arc and finishes the bypass triangle.

rest of this paper, always means a multicurve on rather than its isotopy class.

Remark 5.4. If = S 2 and = S1, then the space of choices of multicurve is also contractible

since there is a unique tight contact structure in a neighborhood ofS 2 up to isotopy.

Observe that, up to an isotopy supported in a neighborhood of the admissible arc , the bypasstriangle attachment does not change .

In what follows we look at bypass triangle attachments along different admissible arcs, which

leads to our alternative definition of the bypass triangle attachment.

Lemma 5.5. Let and be two (overtwisted) contact structures on S2 [0, 1] , where and

are admissible arcs on S 2 {0}, such that

(1) S 2 {0, 1} is convex with respect to both and.

(2) = in a neighborhood of S2 {0} and#

S 2{0}= #

S 2{0}= 1.

(3) is obtained by attaching a bypass triangle to |S 2{0}, and is obtained by attachinga bypass triangle to |S 2{0}.

Then is isotopic to relative to the boundary.

Proof. Up to isotopy, there are only two different admissible arcs on (S 2 {0}, |S 2{0}) (or, (S2

{0}, |S 2{0})). Namely, one gives the trivial bypass and the other gives the overtwisted bypass. We

may assume without loss of generality that is not isotopic to , and is the trivial bypass and is the overtwisted bypass. We complete the bypass triangles and as depicted in Figure 8.Observe that is isotopic to , is isotopic to and bypass attachments along and are

trivial according to Lemma 3.2, we have the following isotopies:

=

= .


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FIGURE 8.

Since S 2 {0, 1} are convex, we can make sure that the isotopies above are supported in the interiorofS 2 [0, 1].

Definition 5.6. A minimal overtwisted ball (B3, ot) is an overtwisted ball where B3 has a tight

neighborhood, and the contact structure ot is obtained by attaching a bypass triangle to the stan-dard tight ball (B3, std).

Remark5.7. By Lemma 5.5, the minimal overtwisted ball is well-defined even if we do not specify

the admissible arc along which the bypass triangle is attached.

With the above preparation, we can now redefine the bypass triangle attachment which is more

convenient for our purpose. Let (M, ) be a contact 3-manifold with convex boundary M = .Identify a collar neighborhood of M with [1, 0] such that M = {0} and the contactvector field transverse to M is identified with the [1, 0]-direction. Let M be an admissiblearc along which the bypass triangle is attached. Push into the interior of M to obtain anotheradmissible arc, parallel to , contained in {1/2}, which we still denote by . Let N be a neigh-borhood of in {1/2}. Consider the ball with corners D [2/3, 1/3] M. By roundingthe corners, we get a smoothly embedded tight ball (B3

1, |B3

1) (M, ), in particular, B3

1has a tight

neighborhood in (M, ). Let (B32, ot) be a minimal overtwisted ball. We construct a new contact

manifold (M, ) = (M \ B31, ) (B

32, ot), where is an orientation-reversing diffeomorphism

identifying the standard tight neighborhoods of B31

and B32

. It is easy to see that is isotopic tothe contact structure obtained by attaching a bypass triangle to (M, ) along .

Remark5.8. The uniqueness of the tight contact structure on 3-ball, due to Eliashberg, guarantees

that the bypass triangle attachment described above is well-defined.

Using the above alternative description of the bypass triangle attachment, we prove the following

generalization of Lemma 5.5.

Lemma 5.9. Let(M, ) be a contact 3-manifold with convex boundary, and let , be two admis-sible arcs on M. Let (resp. ) be the contact structure on M obtained by attaching a bypasstriangle (resp. ) along (resp. ) to (M, ). Then is isotopic to relative to the boundary.


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12 YANG HUANG

Proof. Without loss of generality, we can assume that and are disjoint. If not, we take anotheradmissible arc which is disjoint from and . We then show that and , which

implies .As before, since M is convex, we can push and slightly into the manifold M, which we still

denote by and . Now let B3 M and B3 M be smoothly embedded tight balls containing

and respectively. Take a Legendrian arc connecting B3 and B3, i.e., the endpoints of are

contained in B3 and B3, respectively, and the interior of is disjoint from B

3 and B

3. Moreover,

we can assume that B3 B3 and B3 B3

. Let N() be a closed tubular neighborhood

of . By rounding the corners of B3 B3 N(), we get a smoothly embedded ball B

3 M

with tight convex boundary. Using our cut-and-paste definition of the bypass triangle attachment,

it is easy to see that (B3, |B3 ) and (B3, |B3) are isotopic, relative to the boundary, to the contact

boundary sums (B3, ot)#b(B3, std) and (B

3, std)#b(B3, ot), respectively. Hence both are isotopic

to the minimal overtwisted ball. One simply extends the isotopy by identity to the rest ofM

toconclude that on M.

According to Lemma 5.9, the isotopy class of the contact structure obtained by attaching a

bypass triangle does not depend on the choice of the attaching arcs. We shall write for a bypass

triangle attachment along an arbitrary admissible arc. An immediate consequence of this fact is

that the bypass triangle attachment commutes with any bypass attachment. This is the content of

the following corollary:

Corollary 5.10. Let (M, ) be contact 3-manifold with convex boundary, and be an admissiblearc on M. Then .

Proof. By Lemma 5.9, we can arbitrarily choose an admissible arc M along with the bypass

triangle is attached. In particular, we require that is disjoint from . Hence a neighborhood of where is supported in is also disjoint from . Thus we have the following isotopies:

.

which proves the commutativity.

Corollary 5.11. Let(S 2 [0, 1], ) be a contact manifold with convex boundary, where is isotopicto a sequence of bypass attachments 1 2 n, i.e., there exists 0 = t0 < t1 < < tn = 1such that S 2 {ti} are convex for 0 i n and S

2 [ti1, ti] with the restricted contact structure

is isotopic to the bypass attachment i. Then

is isotopic to k for 0

k

n, where k is thecontact structure isotopic to a sequence of bypass attachments 1 k k+1 n.

Proof. This is an iterated application of Corollary 5.10.

However, observe that subtracting a bypass triangle is in general not well-defined. So we need

the following definition.

Definition 5.12. Two contact structuresand on S 2[0, 1] are stably isotopic, denoted by ,if they become isotopic after attaching finitely many bypass triangles to S 2 {1} simultaneously,

i.e., n n for some n N.


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6. OVERTWISTED CONTACT STRUCTURES ON S 2 [0, 1] INDUCED BY ISOTOPIES.

Let be an overtwisted contact structure on S 2 [0, 1] such that S 2 {0} and S 2 {1} are convex

spheres. In general, any such can be represented by a sequence of bypass attachments. Moreprecisely, by Theorem 1.3, there exists an increasing sequence 0 = t0 < t1 < < tn = 1 suchthat S 2 {ti} is convex and |S 2[ti1,ti] is isotopic to a bypass attachment i for i = 1, , n. Inthis section, we consider a special class of overtwisted contact structures on S 2 [0, 1] such thatS 2 {t} is convex for t [0, 1], in other words, there is no bypass attached.

Let 0 be an I-invariant contact structure on S2 [0, 1]. Let t : S

2 S 2, t [0, 1], be anisotopy such that 0 = id. We define a new contact structure = (0) on S

2 [0, 1], where : S 2 [0, 1] S 2 [0, 1] is defined by (x, t) (t(x), t). Observe that S

2 {t} is convex

with respect to for all t [0, 1] by construction. Hence we get a smooth family of dividing setsS 2{t} for t [0, 1]. Conversely, a smooth family of dividing sets S 2{t}, t [0, 1] defines a uniquecontact structure on S 2 [0, 1], which is isotopic to constructed above for some isotopy t,t [0, 1]. In practice, it is usually easier to keep track of the dividing sets rather than the isotopy.

Definition 6.1. A contact structure on S 2 [0, 1] is induced by an isotopy if S 2 {t} is convexfor all t [0, 1], or, equivalently, there exists an isotopy : S 2 [0, 1] S 2 [0, 1] such that isisotopic to as constructed above.

It is convenient to have the following lemma.

Lemma 6.2. Let , be two contact structures on S 2 [0, 1] induced by isotopies and let t, t

be dividing sets on S 2 {t}, 0 t 1, with respect to , respectively. If 0 = 0, 1 =

1

and

there exists a path of smooth families of multicurves st, 0 s 1 satisfying the following:

(1) st is a multicurve, i.e., a finite disjoint union of simple closed curves, contained in S2 {t}

for0 s 1, 0 t 1.(2) 0t = t,

1t =

t for0 t 1,

(3) s0

= 0, s1

= 1 for0 s 1.

then is isotopic to relative to the boundary.

Proof. By Girouxs flexibility theorem, the path st, 0 s 1 of multicurves determines a path of

contact structures s on S 2 [0, 1] such that 0 = , 1 = . Hence is isotopic to relative to theboundary by Grays stability theorem.

We first consider a bypass attachment to the contact structures on S 2 [0, 1] induced by anisotopy.

Lemma 6.3. Let be a contact structure on S2 [0, 1/2] induced by an isotopy t, t [0, 1/2],

and(S 2 [1/2, 1], ) be a bypass attachment along an admissible arc S2 {1/2}. Then there

exists an admissible arc S 2 {0} such that (S 2 [0, 1], ) is isotopic, relative to theboundary, to (S 2 [0, 1], ).

Proof. We basically re-foliate the contact manifold (S 2 [0, 1], ). Recall that attaches abypass D on S 2 {1/2} so that D = is the union of two Legendrian arcs, where tb() = 1,tb() = 0. We extend D to a new bypass D on S 2 {0} through the isotopy t, t [0, 1/2], bydefining D = D ( [0, 1/2]), where = 1

1/2() S2 {0} is the new admissible arc along

which D is attached, and : S 2 [0, 1/2] S 2 [0, 1/2] is defined by (x, t) (t(x), t). By


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14 YANG HUANG

attaching the new bypass D on S 2 {0}, observe that the rest ofS 2 [0, 1] can be foliated by convexsurfaces, and the contact structure is also induced by the isotopy . Hence is isotopic to

as desired. Definition 6.4. The admissible arc constructed in Lemma 6.3 is called a push-down of. Con-versely, we call a pull-up of.

The rest of this section is rather technical and can be skipped at the first time reading. The only

result needed for our proof of Theorem 0.2 is Proposition 6.15.

We consider a subclass of the contact structures on S 2 [0, 1] induced by isotopies which wewill be mainly interested in. Fix a metric on S 2. Without loss of generality, we assume that there

exists a small disk D2(y) S2 centered at y of radius and a codimension 0 submanifold S 2{0}

of S 2 {0} such that S 2{0} D2(y) and D

2(y) S 2{0} = S 2{0}. Let (s) S

2 {0}, s [0, 1] bean embedded oriented loop such that (0) = (1) = y. Let A() be an annulus neighborhood of

containing D2(y) and disjoint from other components of the dividing set as depicted in Figure 9.

We define an isotopy t, t [0, 1], supported in A() which parallel transports D2(y) along in

A(). More precisely, by applying the stereographic projection map, we can identify A() with anannulus in R2. Then the parallel transportation is given by an affine map t : x x + (t) (0)for any x D2(y) and t [0, 1].

\ \

A()

FIGURE 9.

Definition 6.5. With the small disk D2(y) S 2{0}, the annulus A() and the isotopy t chosenas above, we say that the contact structure on S

2 [0, 1] is induced by a pure braid of thedividing set, where : S 2 [0, 1] S 2 [0, 1] is induced by t as before. We denote such contactstructures by ,(,D2(y),). When there is no confusion, we also abbreviate it by ,D2,.

Remark 6.6. For any simply connected region D S 2 {0} containing S 2{0}, one can isotop so

that D becomes a round disk with small radius as required in Definition 6.5. The isotopy class of

the contact structure on S 2 [0, 1] induced by a pure braid of the dividing set only depends on the

choice of D S 2{0} and the isotopy class of.

Remark 6.7. If is a contact structure on S 2 [0, 1] induced by a pure braid of the dividing set,then S 2{0} = S 2 {1}.

Before we give a complete classification of contact structures on S 2 [0, 1] induced by purebraids of the dividing set, we make a digression into the study of its homotopy classes using the

Pontryagin-Thom construction.

We can always assume that the isotopy (,D2(y), ) is supported in a disk D2 S 2. Trivialize

the tangent bundle of D2 [0, 1] by embedding it into R3 so that D2 is contained in the xy-plane.


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Consider the Gauss map G : (D2 [0, 1], ,D2,) S2. By Lemma 6.2, we can assume with-

out loss of generality that the dividing set is a disjoint union of round circles in D2 {t} for all

0 t 1, and p = (1, 0, 0) S2

R3

is a regular value. Suppose the number of connectedcomponents #D2{0} = m, then the Pontryagin submanifold B = G

1(p) is an oriented framed

monotone braid in the sense that B transversely intersects D2 {t} in m points for any 0 t 1,

and each connected component of the dividing set contains exactly one point. It is easy to check

that the pull-back framing is the blackboard framing, and consequently the self-linking number of

B is exactly writhe(B).

Example 6.8. If D2{0} is the disjoint union of two isolated circles, and D2{0} = S1 D2(y) is

the circle on the left as depicted in Figure 10. The isotopy t parallel transports D2(y) along the

oriented loop . We compute the homotopy class of the contact structure ,D2,.

(a) (b)

p1 p2

p1 p2

p1 p2

p1 p2

+

+

D2 [0, 1]

FIGURE 10. (a) The contact structure on S 2 [0, 1] induced by a full twist of thedividing circles, where {p1, p2} are pre-images of the regular value p = (1, 0, 0)

S2

. (b) The oriented braid with the blackboard framing B as the Pontryagin sub-manifold.

According to the Pontryagin-Thom construction, since writhe(B) = 2, the homotopy class of

,D2, is in general different from the I-invariant contact structure, and the difference is measured

by a decreasing the Hopf invariant by 2.3

Example 6.9. If D2{0} is the disjoint union of three circles, and D2{0} = S1 D2(y) is the circle

on the left as depicted in Figure 11. The isotopy t parallel transports D2(y) along the oriented loop

. We compute the homotopy class of the contact structure ,D2,.In this case, one computes that writhe(B) = 0, hence

,D2,is homotopic to the I-invariant

contact structure.

Now we are ready to classify the contact structures induced by pure braids of the dividing set

up to stable isotopy in the sense of Definition 6.5. One goal is to establish an isotopy equivalence

relation between a pure braid of the dividing set and the bypass triangle attachment. To start with,

we consider the contact structures induced by two special pure braid of the dividing set as depicted

in Figure 12. In Figure 12(a), the dividing set D2(y) is a single circle, and the dividing set

3However, if the divisibility of the Euler class is 2, then t gives a contact structure which is homotopic to the

I-invariant contact structure. We will discuss the divisibility of the Euler class in detail in Section 6.


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16 YANG HUANG

p1 p2 p3

p1 p2 p3

(a) (b)D2 [0, 1] p1 p2 p3

p1 p2 p3

+ +

+ +

FIGURE 11. (a) A braiding by a full twist of the left-hand side dividing circle along

, where {p1, p2, p3} = G1(p) is the pre-image of the regular value p = (1, 0, 0)

S 2. (b) The oriented framed braid B as the Pontryagin submanifold.

contained in the disk bounded by and disjoint from is also a single circle. In Figure 12(b),the dividing set D2(y) consists of m isolated circles nested in another circle, and the dividing

set contained in the disk bounded by and disjoint from consists ofn isolated circles nested inanother circle. We also assume that either m or n is not zero. For technical reasons, it is convenient

to have the following definitions.

Definition 6.10. Given two disjoint embedded circles , D2, < if and only ifis containedin the disk bounded by .

Definition 6.11. Let D2 be a finite disjoint union of embedded circles. The depth of is

the maximum length of chains 1 < 2 < < r , where i is a single circle for anyi {1, 2, , r}.

Observe that the depth of the dividing set in Figure 12(a) is 1, and the depth of the dividing set

in Figure 12(b) is 2. It turns out that to study the contact structure induced by an arbitrary pure

braid of the dividing set, it suffices to consider a finite composition of these two special cases.

(a) (b)

m n

FIGURE 12.

Lemma 6.12. If(S 2 [0, 1], ,D2,) is a contact manifold with contact structure induced by a pure

braid of the dividing set where D2 andare chosen as in Figure 12(a), then (S2 [0, 1], ,D2,)

is isotopic relative to the boundary to (S 2 [0, 1], 2) , where 2 denotes the contact structureobtained by attaching two bypass triangles on (S 2 {0}, ,D2,|S 2{0}).

Proof. Let be an admissible arc as depicted in Figure 13(b). Suppose that both bypass trianglesare attached along .

Observe that = , where , and are all trivial bypass attachments.Hence the contact manifold (S 2 [0, 1], 2) can be foliated by convex surfaces by Lemma 3.2.


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(a) (b)

,D2,

FIGURE 13. (a) The contact structure is induced by parallel transporting D2along . (b) Attaching two bypass triangles along the admissible arc .

In other words, it is induced by an isotopy. By Theorem 0.1 in [10], we know that attaching two

bypass triangles 2 decreases the Hopf invariant by 2. In Example 6.8, we checked by Pontryagin-Thom construction that ,D2, also decreases the Hopf invariant by 2. Observe that the isotopyclass relative to the boundary of a 2-strand oriented monotone braid with blackboard framing is

uniquely determined by its self-linking number, which is equal to the Hopf invariant. Hence 2is isotopic ,D2, in the region where both operations are supported. By extending the isotopy by

identity to the rest of S 2, we conclude that (S 2 [0, 1], ,D2,) is isotopic relative to the boundary

to (S 2 [0, 1], 2).

Lemma 6.13. If(S 2 [0, 1], ,D2,) is a contact manifold with contact structure induced by a pure

braid of the dividing set where D2 andare chosen as in Figure 12(b), then (S2 [0, 1], ,D2,)

is stably isotopic to (S 2 [0, 1], 2(m1)(n1)).

Proof. Let S 2 {1} be an admissible arc as depicted in the left-hand side of Figure 14(a).By Lemma 6.3, i f is the push-down of , then ,(,D2,) ,(D2,), where

is

obtained from by attaching a bypass along . We remark here that ,(,D2,) and ,(D2,) arecontact structures induced by the same isotopy, but are push-forward of different contact structures

on S 2 [0, 1]. Choose D2

to be the m isolated circles on the left and be an oriented loop

as depicted in the right-hand side of Figure 14(a). Let ,D2

,

be the contact structure induced

by an isotopy which parallel transports D2

along . Then Lemma 6.2 implies that ,(D2,)

is isotopic, relative to the boundary, to

,D2

,

, where is an isotopy rounding the outmost

dividing circle. An iterated application of Lemma 6.12 implies that ,D2

,

2m(n1).

We next isotop the contact structure . Consider the n isolated circles nested in a larger cir-

cle. Let D2

be the leftmost circle among the n circles and be an oriented loop as depictedin the right-hand side of Figure 14(b). We pull up through an isotopy which parallel transports

D2

along , and observe that the pull-up of is isotopic to . By using Lemma 6.3 one

more time, we get the isotopy of contact structures ,D2 , . It is left to determinethe isotopy class of the contact structure

,D2

, . Since is oriented counterclockwise, by ap-

plying Lemma 6.12 n 1 times, we get a stable isotopy ,D2

,

2(1n), i.e.,

,D2

,

2(n1)

is isotopic to the I-invariant contact structure.

To summarize what we have done so far, we have the following (stable) isotopies of contact

structures:


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18 YANG HUANG

(a)

(b)

,D2

,

,D2,

,D2

,

FIGURE 14. (a) Pushing down the bypass attachment . (b) Pulling up the bypassattachment .

,D2, = ,(,D2,)

,

(D

2

,)

,D2 ,

2m(n1)

,D2

,

2m(n1)

2(1n) 2m(n1)

2(m1)(n1)

= 2(m1)(n1) .


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Hence by definition, ,D2, is stably isotopic to 2(m1)(n1) as desired.

We now completely classify contact structures on S 2 [0, 1] induced by pure braids of thedividing set.

Proposition 6.14. If(S 2 [0, 1], ,D2,) is a contact manifold with contact structure induced by a

pure braid of the dividing set, then ,D2, is stably isotopic to (S2 [0, 1], l) for some l N.

Proof. Recall that D2 is a union of components of S 2 {0}, and is an oriented loop in the

complement of S 2{0} as in Definition 6.5. Let be the union of components of S 2{0} contained

in a disk bounded by and outside of A(). We may choose the disk so that is the orientedboundary. Since (,D2, ) is a pure braid of the dividing set, we have S 2{0} = S 2 {1}. Hence

we also view and as dividing sets on S 2 {1}. Choose pairwise disjoint admissible arcs

1, 2, , r, r+1, , k on S2 {1} such that the following conditions hold:

(1) 1, 2, , r1 are admissible arcs contained in D2 such that by attaching bypasses along

these arcs, the depth of becomes at most 2.

(2) r, r+1, , k are admissible arcs contained in the disk bounded by and outside ofA()such that by attaching bypasses along these arcs, the depth of becomes at most 2.

Observe that we choose 1, 2, , k so that the isotopy class of each i is invariant under thetime-1 map 1 which is supported in A() \D

2. Hence, by abuse of notation, we do not distinguish

i and its push-down through (,D2, ). By Lemma 6.3, we have the isotopy of contact structures

(,D2,) 1 k 1 k , where is the contact structure induced by a finitecomposition of special pure braids of the dividing set considered in Lemma 6.12 and Lemma 6.13,

Therefore is stable isotopic to a power of the bypass triangle attachment, say l for some

l N. To summarize, we have the following (stable) isotopies of contact structures, relative to the

boundary.

,D2, k (,D2,) 1 k

= (,D2,) (1 1 1 ) (k k k

)

((,D2,) 1 k) (1 1

) (k

k

)

(1 k ) (1 1 ) (k k

)

(1 k l) (

1

1) (

k

k)

l (1 1 1 ) (k k k

)

= l k

.

Hence ,D2, is stably isotopic to l by definition.

To conclude this section, we prove the following technical result which asserts that under certain

assumptions and up to possible bypass triangle attachments, one can separate two bypasses.

Proposition 6.15. Let(S 2, ) be a convex sphere with dividing set and (S 2, ) be an admis-sible arc such that the bypass attachment increases # by 1. Suppose that(S

2, ) is the newconvex sphere obtained by attaching to (S

2, ) and suppose (S 2, ) is another admissiblearc such that the bypass attachment decreases #

by 1. Then there exists an admissible arc


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20 YANG HUANG

(S 2, ) disjoint from , a map : S 2 [0, 1] S 2 [0, 1] induced by an isotopy, and aninteger l N such that

l relative to the boundary.

Proof. Let be the arc of anti-bypass attachment to contained in (S 2, ) as discussed in Re-mark 5.3. Then intersects in three points {p1, p2, p3} as depicted in Figure 15(b). Let 1 and2 be subarcs of from p1 to p2 and from p2 to p3 respectively. Observe that, in order to find anadmissible arc (S 2, ) which is disjoint from and satisfy all the conditions in the lemma,it suffices to find an admissible arc on (S 2, ), which we still denote by , and which is disjointfrom and also satisfies the conditions in the lemma. In fact, by symmetry, we only need to bedisjoint from 1. Without loss of generality, we can assume that intersects transversely and theintersection points are different from p1, p2 and p3.

p1

p2

p3

(a) (b)

FIGURE 15. (1) The convex sphere (S 2, ) with an admissible arc . (2) The con-vex sphere (S 2, ) obtained by attaching a bypass along , where is the arc of theanti-bypass attachment.

Claim: Up to isotopy and possibly a finite number of bypass triangle attachments, one can arrange

so that and1 do not cobound a bigon B on S2 as depicted in Figure 16(1).

1

(1) (2) (3)FIGURE 16. (1) The admissible arc together with 1 bound a minimal bigon,which contains other components of the dividing set in the interior. (2) Choose a

disk D2 containing all the dividing sets in the bigon and an oriented loop sothat it intersects in exactly one point. (3) The pull-up of through the contactstructure ,D2, bounds a trivial bigon with 1.

To verify the claim, note that if B is a trivial bigon, i.e., it contains no component of the dividing

set in the interior, then we can easily isotop to eliminate B. If otherwise, we consider a minimal


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bigon bounded by and 1 in the sense that the interior of the bigon does not intersect with .Take a disk D2 B containing all components of the dividing set in B, namely,

D2 =

and

(B \ D2

) = . By our assumption, the bypass attachment decreases #

by 1, so must intersect in three points which are contained in three different connected components of

respectively. One can find an oriented loop : [0, 1] S 2 \ with (0) = (1) D2 such that intersects in one point. Orient in such a way that it goes from to (1) in the interior of Bas depicted in Figure 16(b). Suppose that : S 2 [0, 1] S 2 [0, 1] is induced by an isotopy twhich parallel transports D2 along . By pulling up the the bypass attachment through , weget the following isotopy of contact structures (cf. proof of Lemma 6.13):

,(D2,) ,(,D2,)

where is obtained from by attaching a bypass along , and is the pull-up of which isisotopic to the one depicted in Figure 16(c).

Since and 1 cobound a trivial bigon, a further isotopy of will eliminate the bigon so that does not intersect 1 in this local picture. By Proposition 6.14, the contact structure ,(,D2,) is

stably isotopic to n for some n N. Define 1 : S 2 [0, 1] S 2 [0, 1] by (x, t) (1t (x), t),then it is easy to see that ,(D2,) ,1(D2,) is isotopic, relative to the boundary, to an I-invariantcontact structure. Since we will use this trick many times, we simply write1 for,1(D2,) whenthere is no confusion. To summarize, we have

,(,D2,) ,1(D2,)

n ,1(D2,)

n ,1(D2,)

By applying the above argument finitely many times, we can eliminate all bigons bounded by and 1. Hence the claim is proved.

Let us assume that intersects 1 nontrivially, and and 1 do not cobound any bigon on S2.

We consider the following two cases separately.

Case 1. Suppose does not intersect any of the three components of the dividing set generated bythe bypass attachment . Let 1, 2 and 3 be the three dividing circles which intersect with .If intersects 1 in exactly one point as depicted in Figure 17(a), then we choose a disk D

2 1

and an oriented loop in the complement of the dividing set as depicted in Figure 17(b) such that ,(1,D2,) 1

m 1 by arguments as before for some m N, where

intersects 1 in exactly two points and cobound a trivial bigon as depicted in Figure 17(c). Hencean obvious further isotopy of makes it disjoint from 1 as desired.

If intersects 1 in more than one point, we orient so that it starts from the point q = 1as depicted in Figure 18(a). Let q1 and q2 be the first and the second intersection points of with 1 respectively. Note that since we assume and 1 do not cobound any bigon, there is no

more intersection point 1 between q1 and q2. Letqq1,

q1q andq1q2 be oriented subarcs of

and q2q1 be an oriented subarc of 1. We obtain a closed, oriented (but not embedded) loop

= qq1 q1q2

q2q1 q1q by gluing the arcs together. To apply Proposition 6.14 in this case,

we take an embedded loop close to as depicted in Figure 18(2), which we still denote by . Let


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22 YANG HUANG

1

2

3

(a) (b) (c)

FIGURE 17. (1) The convex sphere (S 2, ) with an admissible arc intersecting1 in exactly one point. (2) Choose a disk D

2 containing 1 and an oriented loop ,

along which we apply the isotopy. (3) The pull-up of through the contact structure

1,D2,

bounds a trivial bigon with 1.

D2 be a small disk containing 1 as usual. Again by pulling up the bypass attachment through,(1,D2,), we have (stable) isotopies of contact structures ,(1,D2,) 1

r 1

for some r N, where and 1 bound a trivial bigon. Hence an obvious further isotopy eliminatesthe trivial bigon and decreases #( 1) by 1. By applying the above argument finitely many times,we can reduce to the case where intersects 1 in exactly one point, but we have already solved theproblem in this case. We conclude that under the hypothesis at the beginning of this case, there ex-

ists a disjoint with 1 such that l for some isotopy and an integer l N.

1

qq1

q2

(a) (b) (c)

FIGURE 18. (1) The convex sphere (S 2, ) with an admissible arc intersecting1 in at least two points, say, q1 and q2. (2) The embedded, oriented loop approx-imating the broken loop qq1 q1q2 q2q1 q1q. (3) The pull-up of through thecontact structure 1,D2, bounds a trivial bigon with 1.

Case 2. Suppose nontrivially intersects the union of the three components of the dividing setgenerated by the bypass attachment . Without loss of generality, we pick an intersection pointr as depicted in Figure 19(a). Orient so that it starts from r. Let r1 be the first intersection pointof and 1. Then , 1 and

bound a triangle rr1p1. By the assumption that there exists no

bigon bounded by and 1, the interior of the triangle rr1p1 does not intersect with . If theinterior of the triangle rr1p1 contains no components of the dividing set, then it is easy to isotop

so that #( 1) decreases by 1. If otherwise, take a small disk D2 rr1p1 containing all

components of the dividing set in rr1p1, i.e., rr1p1 \ D2 does not intersect with the dividing


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r

p1

r1

(a) (b) (c)

FIGURE 19. (1) The admissible arc , the dividing set and 1 cobound a topolog-ical triangle rr1p1, which may contain other components of the dividing set in the

interior. (2) Choose the disk D2 to contain all the components of the dividing set in

the topological triangle rr1p1, and an oriented loop which intersects in exactlyone point. (3) By applying the isotopy along , the admissible arc becomes

which bounds a trivial triangle with the dividing set and 1.

set . Let be an oriented loop based at a point in D2 intersecting exactly once. By pullingup the bypass attachment through (,D2,), we have (stable) isotopies of contact structures

,(,D2,) 1 n 1 so that , 1 and

bound a trivial triangle in the sense

that the interior of the triangle does not intersect with the dividing set. Hence we can further isotop to eliminate the trivial triangle and hence decrease #( 1) by 1. By applying such isotopiesfinitely many times, we get an admissible arc such that #( 1) = 0 and satisfy all the conditionsof the proposition.

7. CLASSIFICATION OF OVERTWISTED CONTACT STRUCTURES ON S 2 [0, 1]

We have established enough techniques to classify overtwisted contact structures on S 2 [0, 1].

Proposition 7.1. Let be an overtwisted contact structure on S 2 [0, 1] such that S 2 {0, 1} isconvex with S 2{0} = S 2{1} = S

1. Then n for some n N , where n denotes the contactstructure on S 2 [0, 1] obtained by attaching n bypass triangles to S 2 {0} with the standard tightneighborhood.

Proof. By Girouxs criterion of tightness, both S 2 {0} and S 2 {1} have neighborhoods which are

tight. Take an increasing sequence 0 = t0 < t1 < < tn = 1 such that is isotopic to a sequenceof bypass attachments 0 1 n1 , where i S

2 {ti} are admissible arcs along which

a bypass is attached. Define the complexity of a bypass sequence to be c = max0in #S 2{ti}. The

idea is to show that if c > 3, then we can always decrease c by 2 by isotoping the bypass sequenceand suitably attaching bypass triangles.

To achieve this goal, we divide the admissible arcs on (S 2, ) into four types (I), (II), (III) and(IV), according to the number of components of intersecting the admissible arc as depicted in

Figure 20, where we only draw the dividing set which intersects the admissible arc. Observe that

bypass attachment of type (I) increases # by 2, bypass attachment of type (II) and (III) do not

change #, and bypass attachment of type (IV) decreases # by 2. Hence the complexity of a

sequence of bypass attachments changes only if the types of bypasses in the sequence change. By

repeated application of Lemma 6.3, we may assume that contact structures induced by isotopies


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24 YANG HUANG

are contained in a neighborhood of S 2 {1}. By assumption, S 2 {1} has a tight neighborhood.

Hence according to Remark 5.4, we shall only consider sequences of bypass attachments modulo

contact structures induced by isotopies.

(I) (II) (III) (IV)

FIGURE 20. Four types of admissible arcs on (S 2, ).

Claim: We can isotop the sequence of bypass attachments such that only bypasses of type (I) and(IV) appear.

To prove the claim, we first show that a bypass attachment of type (III) can be eliminated. Take

an admissible arc of type (III). If the bypass attachment along is trivial, then by Lemma 3.2,the bypass attachment is induced by an isotopy. Otherwise there exists an admissible arc disjoint from as depicted in Figure 21(a)4 such that if one attaches a bypass along , followed bya bypass attached along , then the later bypass attachment is trivial.

(a) (b)

FIGURE 21.

By the disjointness of admissible arcs and , we get the following isotopies of contact struc-tures,

.

Observe that is a composition of type (I) and type (IV) bypass attachments. Hence a finitenumber of such isotopies will eliminate all bypass attachments of type (III) in a sequence.

Similarly suppose that is the bypass attachment of type (II) in a sequence and is nontrivial.Choose an admissible arc disjoint from as depicted in Figure 21(b) such that if one attaches abypass along , followed by a bypass attached along , then the later bypass attachment is trivial.

4In literature, we say is obtained from by left rotation.


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By the disjointness of and again, we get the following isotopies of contact structures:

.

Observe that is a composition of bypass attachments both of type (III), hence by a furtherisotopy will turn into a composition of bypass attachments of type (I) and (IV). A finite numberof such isotopies will eliminate bypasses of type (II). The claim follows.

From now on, we assume that any bypass attachment in 0 1 n1 either increasesor decreases # by 2.

Assume that the complexity of the bypass sequence is achieved at level S 2 {tr} for some

r {0, 1, , n} and is at least 5, i.e., #S 2{tr} = c 5. Then it is easy to see that r1 istype (I) and r is type (IV). By Proposition 6.15, we can always assume that r is disjoint from

r1 modulo finitely many bypass triangle attachments. Hence we can view both r1 and r asadmissible arcs on S 2 {tr1}. Let S 2{tr1} be the dividing circle which nontrivially intersectsr1. We do a case-by-case analysis depending on the number of points r intersecting with .

Case 1: If r intersects in at most one point, then one easily check that by applying isotopyr1 r r r1 to the sequence of bypass attachments, #S 2{tr} decreases by 2.

Case 2: Ifr intersects in exactly two points, then once again we apply the isotopy r1 r r r1 to the sequence of bypass attachments. Now observe that r r1 is a composition ofbypass attachments of type (III). In the proof of the claim above, we see that any bypass attachment

of type (III) is isotopic to a composition of a bypass attachment of type (IV) followed by a bypass

attachment of type (I). Such an isotopy also decreases the local maximum of # by 2.

Case 3: Ifr also intersects in three points, we consider a disk D bounded by and r1 as de-picted in Figure 22(a). IfD contains no component of the dividing set in the interior, then r1 ris isotopic to a bypass triangle attachment, more precisely, there exists a trivial bypass along an

admissible arc on S 2 {tr} such that r1 r is a bypass triangle attachment along r1.Suppose D contains at least one connected component of the dividing set. Let be an admissiblearc on S 2 {tr1} disjoint from r1 and r such that it intersects in two points and the dividingset contained in D in one point as depicted in Figure 22(b).

We have the following isotopies of contact structures due to Lemma 5.9 and the disjointness of

admissible arcs:

r1 r r1 r

= r1 r

r1 r

One can check that the last five bypass attachments above are all of type (III). Hence we can

isotop further as before to eliminate type (III) bypass attachments to decrease the complexity of

the bypass sequence.


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26 YANG HUANG

r1 r1r r

D

(a) (b)

FIGURE 22.

To summarize, we proved that any sequence of bypass attachments 0 1 n1 onS 2 [0, 1] is stably isotopic to another sequence of bypass attachments whose complexity is at most

three, which is clearly isotopic to a power of bypass triangle attachments. Thus the proposition isproved.

8. PROOF OF THE MAIN THEOREM

Now we are ready to finish the proof of Theorem 0.2.

Proof of Theorem 0.2. By Proposition 4.3, we can isotop and so that they agree in a neighbor-hood of the 2-skeleton. Without loss of generality, we can furthermore assume that there exists an

embedded closed ball B3 M such that

(1) B3 is convex and has a tight neighborhood in M with respect to both and .(2) = in M\ B3.(3) The restriction ofand to M\ B3 and to B3 are all overtwisted.

Take a small ball B3 B3 in a Darboux chart so that both |B3 and

|B3 are tight. We identify

B3 \ B3 with S2 [0, 1] and represent the contact structures |B3\B3 and

|B3\B3 by two sequences

of bypass attachments. By Proposition 7.1, both |B3\B3 and |B3\B3 are stably isotopic to some

power of the bypass triangle attachment, in other words, there are isotopies of contact structures

|B3\B3 r n+r and |B3\B3

s m+s for some n, m, r, s N. By assumption, the restriction of

and to M\B3 are overtwisted, so there exist bypass triangle attachments along any admissible arcon B3 according to Lemma 3.1. By simultaneously attaching sufficiently many bypass trianglesto |B3\B3 and

|B3\B3, we can further assume that |B3\B3 n, |B3\B3

m and = on M\ B3.

Let d be the divisibility of the Euler class e() = e() H2(M;Z). By elementary obstruction

theory and Theorem 0.1 in [10], we have d|(mn). To complete the proof of the theorem, we need toshow that |M\B3 is isotopic to|M\B3

d relative to the boundary. Since d = g.c.d.{e()| H2(M)},it suffices to prove the following more general fact.

Lemma 8.1. Let be a closed surface of genus g and be an I-invariant contact structure on [0, 1]. Then l is stably isotopic to relative to the boundary, where l = e()().

Proof. Since we only consider stable isotopies of contact structures, one can prescribe any dividing

set on such that the Euler class evaluates on is equal to l. In particular, we consider the

dividing set on as depicted in Figure 23, namely, there are g + 1 circles 1 g+1 dividing


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into two punctured disks, in each of which there are p and q isolated circles respectively. We call

the left most circles in the sets of p and q isolated circles 0 and 1 respectively. We also choose

admissible arcs {1, 2, , p1} and {1, 2, , q1}, and orient i, 1 i g + 1, in a way asdepicted in Figure 23.

1 2 g+1

1 2p1

. . .

1

q1

. . .

. . .

+

+ + + +

0

1

FIGURE 23.

An easy calculation shows that l = 2(p q). Choose small disks D2,0, D2,1 in such that

D2,0 = 0 and D

2,1 = 1. Observe that the bypass triangle attachment along any i

and j consists of three trivial bypass attachments, hence is isotopic to contact structures inducedby a pure braid of the dividing set. More precisely, let i , i = 1, 2, , g + 1, be an orientedloop in the negative region which is parallel to i. We have the following isotopies of contactstructures 21

2p1

(0,D2,0,1

g+1

) (0,D2,0,1

) (0,D2,0,g+1

), where we think of

1

g+1

as an oriented loop homologous to the union of the is. Similarly one can study the

bypass triangle attachments along the js, but with an opposite orientation. Let +

ibe an oriented

loop in the positive region which is parallel to i for 1 i g + 1. We have the following (stable)isotopies of contact structures 21

2q1

(1,D2,1,+

1+

g+1) (1,D2,1,

+

1) (1,D2,1,

+

g+1).

To summarize the computations above, we get the following (stable) isotopies of contact structures:

l (21 2p1

) (21 2q1

)

((0 ,D2,0,1

) (0,D2,0,g+1

)) ((1 ,D2,1,+

1) (1,D2,1,

+

g+1))

((0 ,D2,0,1

) (1,D2,1,+

1)) ((0,D2,0,

g+1

) (1,D2,1,+

g+1))

where the last step follows from the fact that isotopies that parallel transport D2,0 and D2,1 are

disjoint.

Now it suffices to prove that (0,D2,0,i

) (1,D2,1,+

i) is stably isotopic to an I-invariant contact

structure for 1 i g + 1. To see this, take an annular neighborhood Ai of i containing D2,0

and D2

,1and an admissible arc i which intersects 0, 1, and i as depicted in Figure 24. We can

assume that the isotopies (0,D2,0,

i ) and (1,D

2,1,

+

i ) are supported in Ai. For simplicity of

notation, we denote the composition (0,D2,0,i

) (1,D2,1 ,+

i) by i .

By pushing down the bypass attachment i through i , we have the following isotopies ofcontact structures:

i i = i i i i

i (i) i i

i i i

= i


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28 YANG HUANG

0 1

i

i

+

+

FIGURE 24. An annulus neighborhood Ai ofi containing 0 and 1.

where i is the push-down ofi which is isotopic to i, and the (i) is easily seen to be isotopic toan I-invariant contact structure. The argument works for all i {1, 2, , g +1}, hence we establishthe stable isotopy as desired.

Acknowledgements. The author is very grateful to Ko Honda for inspiring conversations throughoutthis work. The author also thank MSRI for providing an excellent environment for mathematical

research during the academic year 2009-2010.

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[1] D. Bennequin, Entrelacements etequations de Pfaff, Asterisque, 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. Martinets work, Ann. Inst. Fourier 42 (1992), 165-192.

[4] Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Proceeding of Symposium Pure Math., vol.52, Amer.

Math. Soc., Providence, RI, (1991), 165-192

[5] J. Etnyre and K. Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003), 59-74.

[6] H. Geiges, An Introduction to Contact Topology, Cambridge University Press, (2008)

[7] E. Giroux, Convexit e en topologie de contact, Comm. Math. Helv. 66 (1991), 637-677.

[8] E. Giroux, Sur les transformations de contact au-dessus des surfaces, Essays on geometry and related topics, Vol.

1,2, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, (2001), 329350.

[9] K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309368 (electronic).

[10] Y. Huang, Bypass attachments and homotopy classes of 2-plane fields in contact topology, preprint 2011.

arXiv:1105.2348v1 [math.GT]

[11] I. Torisu, On the additivity of the Thurston-Bennequin invariant of Legendrian knots, Pacific J. Math. 210 (2003)

359-365

UNIVERSITY OF SOUTHERN CALIFORNIA, LOS ANGELES, CA 90089

E-mail address: [email protected]
http://arxiv.org/abs/1105.2348http://arxiv.org/abs/1105.2348

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