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On the homotopy class of 2-plane fields and its applications in contact topology
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On the homotopy class of 2-plane fields and its applications in contact topology
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Content
ONTHEHOMOTOPYCLASSOF2-PLANEFIELDSANDITSAPPLICATIONSIN
CONTACTTOPOLOGY
by
YangHuang
ADissertationPresentedtothe
FACULTYOFTHEUSCGRADUATESCHOOL
UNIVERSITYOFSOUTHERNCALIFORNIA
InPartialFulfillmentofthe
RequirementsfortheDegree
DOCTOROFPHILOSOPHY
(MATHEMATICS)
August2012
Copyright2012 YangHuang
Dedication
Tomyparents.
ii
Acknowledgments
First of all, I would like to thank my advisor, Ko Honda, for teaching me much of the
backgroundforthisthesis,suggestingthedirectionofresearch,andgivingsubstantialhelp
alongtheway. Bothhismathematicalandprofessionalguidancehavebeeninvaluable. My
graduate career also benefits a lot from the excellent teaching of many of the mathematics
faculty at the University of Southern California. Especially I learned algebraic topology
from Francis Bonahon and homological algebra from Eric Friedlander although, unfortu-
nately, this fascinating algebraic machinery is not directly related to the material in this
thesis.
Duringmygraduatecareer,theacademicyear2009-2010thatIspendatMathematical
Sciences Research Institute to participate in the Symplectic and Contact Geometry and
Topology program has been particularly important for me. Not only there are various
research talks and working group discussions which covers most of the active areas of
research in contact and symplectic topology, but also I make a lot of friends who share
common interests in more than mathematics. In particular I would like to mention Keon
Choi, Vinivius Gripp, Chung-Jun Tsai, Vera Vértesi, and of course my academic brothers
Roman Golovko, Russell Avdek, and Yin Tian. I learned various aspects of mathematics
fromconversationswithyouandIamalwaysgratefultothat.
Havingtoliveandstudyabroadindependentlyinthepastfiveyearswasnoteasy,espe-
ciallywhenIevenhadtroublecommunicatinginEnglishattheverybeginning. Howeverit
turnsoutthatthisrelativelyshortperiodofmylifeworksoutinaparticularlyfruitfulway.
All of these would never happen without the consistent support from my family, as well
iii
as all graduate students and staff in the mathematics department at USC, who successfully
makethisdepartmentwarmlikeabigfamily.
iv
TableofContents
Dedication................................................................. ii
Acknowledgments.......................................................... iii
ListofFigures ............................................................. viii
Abstract................................................................... ix
Chapter1 Introduction .................................................... 1
Chapter2 ContactTopologyPreliminaries................................... 4
2.1 Convexsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Bypasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Bypasstriangleattachment . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter3 HomotopyClassesof2-planeFields............................... 10
3.1 ThePontryagin-Thomconstructionforclosedmanifolds . . . . . . . . . . 10
3.2 ThePontryagin-Thomconstructionformanifoldswithboundary . . . . . . 12
3.3 The3-dimensionalobstructionclass o
3
(ξ,ξ
′
)oforiented2-planefields . . . 14
Chapter4 BypassAttachmentsandHomotopyClassesofContactStructures.... 17
4.1 Computationofthehomotopyclassofabypassattachment . . . . . . . . . 17
4.2 Computationofthehomotopyclassofabypasstriangleattachemnt . . . . . 20
Chapter5 ClassificationofOvertwistedContactStructuresviaConvexSurface
Theory.................................................................... 27
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 OutlineoftheproofofTheorem1.0.2 . . . . . . . . . . . . . . . . . . . . 28
5.3 Localpropertiesofbypassattachments . . . . . . . . . . . . . . . . . . . . 29
5.4 Isotopingcontactstructuresuptothe2-skeleton . . . . . . . . . . . . . . . 34
5.5 Bypasstriangleattachmentrevisited . . . . . . . . . . . . . . . . . . . . . 38
5.6 Overtwistedcontactstructureson S
2
×[0,1]inducedbyisotopies . . . . . 43
5.7 Classificationofovertwistedcontactstructureson S
2
×[0,1] . . . . . . . . 60
5.8 ProofofTheorem1.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
References................................................................. 69
v
ListofFigures
2.1 Abypass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Abypassattachmentalongα. . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 (a)Thetrivialbypassattachment. (b)Theovertwistedbypassattachment. . 7
2.4 Abypasstriangleattachment. . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 ThePontryaginsubmanifoldG
−1
ξ
V
∗σ
α
(p)inV. Thebluearcisaparallelcopy
ofG
−1
ξ
V
∗σ
α
(p)whichdefinestheframing. . . . . . . . . . . . . . . . . . . . 18
4.2 (a) The Pontryagin submanifoldG
−1
ξ
Vα
(p) contained in V
α
. (b) The Pontrya-
gin submanifold G
−1
ξ
V
∗σ
α
(p) contained in V. The blue arc is a parallel copy
ofG
−1
ξ
V
∗σ
α
(p)whichdefinestheframing. . . . . . . . . . . . . . . . . . . . 20
4.3 (a) The Pontryagin submanifold G
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p) contained in U. (b) The
PontryaginsubmanifoldG
−1
τ
(p)containedin N(γ)×[2,3]. . . . . . . . . . 23
4.4 ThePontryaginsubmanifoldG
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p)containedin U afteranisotopy. 24
4.5 The Pontryagin submanifoldG
−1
η∗△
α
(p). The blue circle is a parallel copy of
G
−1
η∗△
α
(p)whichdefinestheframing. . . . . . . . . . . . . . . . . . . . . . 25
5.1 TheLegendrianarcγ connecting∂Band∂
˜
D. . . . . . . . . . . . . . . . . 30
5.2 Thecharacteristicfoliationon N(γ)withcornersrounded. . . . . . . . . . . 30
5.3 Thepreferredcharacteristicfoliationon
˜
D∪N(γ)∪ B. . . . . . . . . . . . 30
vi
5.4 Thebypass Dalongα. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 (a) The characteristic foliation on Σ×{0}. The trivial bypass is attached
along the Legendrian arc in dash line. (b) The characteristic foliation on
Σ×{1} after attaching the trivial bypass. Here e
±
(resp. h
±
) denote the
±-elliptic(resp.±-hyperbolic)singularpointsofthefoliation. . . . . . . . . 32
5.6 (a)ThecharacteristicfoliationonΣ×{1/2},whereasaddle-saddleconnect
from h
−
to h
+
exists. The region Ω contains exactly two singular points
{e
−
,h
−
}whichareineliminationposition. (b)The nonsingularcharacteris-
ticfoliationonΩaftertheelimination. . . . . . . . . . . . . . . . . . . . . 32
5.7 Anexampleofthedividingsetona2-simplex. . . . . . . . . . . . . . . . . 36
5.8 (a)Thedividingsetonσ
2
dividesitinto±-regions. Thebottomedgeisσ
1
.
(b)Onepossibledividingseton ˜ σ
2
afterpositivelystabilizingσ
1
once. . . . 36
5.9 Bypasstriangleattachmentson S
2
. . . . . . . . . . . . . . . . . . . . . . . 39
5.10 Contactstructureinducedbyanisotopy. . . . . . . . . . . . . . . . . . . . 45
5.11 (a)ThecontactstructureonS
2
×[0,1]inducedbyafulltwistofthedividing
circles,where{p
1
,p
2
}arepre-imagesoftheregularvalue p= (1,0,0)∈ S
2
.
(b) The oriented braid with the blackboard framingB as the Pontryagin
submanifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.12 (a) A braiding by a full twist of the left-hand side dividing circle along
γ, where{p
1
,p
2
,p
3
} = G
−1
(p) is the pre-image of the regular value p =
(1,0,0)∈ S
2
. (b)TheorientedframedbraidBasthePontryaginsubmanifold. 48
5.13 Twospecialisotopiesofdividingsets. . . . . . . . . . . . . . . . . . . . . 49
5.14 (a) The contact structure is induced by parallel transporting
˜
Γ⊂ D
2
ϵ
along
γ. (b)Attachingtwobypasstrianglesalongtheadmissiblearcα. . . . . . . 49
5.15 (a) Pushing down the bypass attachment σ
α
. (b) Pulling up the bypass
attachmentσ
˜ α
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.16 (a) The convex sphere (S
2
,Γ) with an admissible arc α. (b) The convex
sphere (S
2
,Γ
′
) obtained by attaching a bypass alongα, whereδ is the arc
oftheanti-bypassattachment. . . . . . . . . . . . . . . . . . . . . . . . . . 55
vii
5.17 (a) The admissible arc β together with δ
1
bound a minimal bigon, which
contains other components of the dividing set in the interior. (b) Choose a
disk D
2
ϵ
containingallthedividingsets
˜
Γin the bigon and an oriented loop
γ so that it intersectsβ in exactly one point. (c) The pull-up ofβ through
thecontactstructureξ˜
Γ,D
2
ϵ
,γ
boundsatrivialbigonwithδ
1
. . . . . . . . . . . 56
5.18 (a) The convex sphere (S
2
,Γ
′
) with an admissible arc β intersecting δ
1
in
exactly one point. (b) Choose a disk D
2
ϵ
containing Γ
1
and an oriented
loopγ, along which we apply the isotopy. (c) The pull-up ofβ through the
contactstructureξ
Γ
1
,D
2
ϵ
,γ
boundsatrivialbigonwithδ
1
. . . . . . . . . . . . 57
5.19 (a) The convex sphere (S
2
,Γ
′
) with an admissible arc β intersecting δ
1
in
at least two points, say, q
1
and q
2
. (b) The embedded, oriented loop γ
approximatingthebrokenloop ⃗ qq
1
∪ ⃗ q
1
q
2
∪ ⃗ q
2
q
1
∪ ⃗ q
1
q. (c)Thepull-upof
βthroughthecontactstructureξ
Γ
1
,D
2
ϵ
,γ
boundsatrivialbigonwithδ
1
. . . . . 59
5.20 (a) The admissible arcβ, the dividing setΓ
′
andδ
1
cobound a topological
triangle△rr
1
p
1
, which may contain other components of the dividing set
in the interior. (b) Choose the disk D
2
ϵ
to contain all the components of
the dividing set in the topological triangle△rr
1
p
1
, and an oriented loopγ
which intersectsβ in exactly one point. (c) By applying the isotopy along
γ, the admissible arcβ becomesβ
′
which bounds a trivial triangle with the
dividingsetandδ
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.21 Fourtypesofadmissiblearcsαon(S
2
,Γ). . . . . . . . . . . . . . . . . . . 61
5.22 Addingtrivialbypasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.23 Addingatrivialbypasstodecreasethecomplexity. . . . . . . . . . . . . . 64
5.24 ThedividingsetonΣ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.25 Anannulusneighborhood A
i
ofγ
i
containingΓ
0
andΓ
1
. . . . . . . . . . . . 67
viii
Abstract
Thegoalofthisthesisistostudyhomotopyclassesoforiented2-planefieldsin3-manifolds
and apply them to 3-dimensional contact topology. Using a generalized version of the
Pontryagin-Thom construction, we compute the homotopy class of a bypass attachment.
Basedonthiscomputation,wegiveanewprooftotheclassificationtheoremofovertwisted
contactstructures,originallyduetoEliashberg,usingconvexsurfacetheory.
ix
Chapter1
Introduction
Acontactmanifold(M,ξ)isasmoothmanifoldwithacontactstructureξ,i.e.,amaximally
non-integrable codimension 1 tangent distribution. In particular, if the dimension of the
manifold is 3, it was realized through the work of Bennequin and Eliashberg [1], [2] that
contact structures fall into two classes: tight or overtwisted. Since then, dynamical sys-
tems and foliation theory of surfaces embedded in contact 3-manifolds have been studied
extensivelytoanalyzethisdichotomy. Basedonthesedevelopments,Eliashberg[2]gavea
classificationofovertwistedcontactstructures,whichwenowexplain.
Let M be a closed oriented manifold and△⊂ M be an oriented embedded disk. Fur-
thermore, we fix a point p ∈ △. We denote by Cont
ot
(M,△) the space of co-oriented,
positive, overtwisted contact structures on M which are overtwisted along△, i.e., the con-
tact distribution is tangent to△ along ∂△. Let Distr(M,△) be the space of co-oriented
2-plane distributions on M which are tangent to△ at p. Both spaces are equipped with the
C
∞
-topology.
Theorem1.0.1(Eliashberg). Let M be a closed, oriented 3-manifold. Then the inclusion
j : Cont
ot
(M,△)→ Distr(M,△)isahomotopyequivalence.
Inparticular,wehave:
Theorem 1.0.2 (Eliashberg). Let M be a closed, oriented 3-manifold. Ifξ andξ
′
are two
positive overtwisted contact structures on M, then they are isotopic if and only if they are
homotopicas2-planefields.
1
Consequently, overtwisted contact structures are completely determined by the homo-
topy 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
theambient3-manifold.
It turns out that to understand homotopy classes of contact structures, viewed as ori-
ented 2-plane fields, it is essential to study how bypass attachments affect the homotopy
type of the contact structure on a given contact manifold with convex boundary. A bypass
attachment, initially defined by Honda [10], can be viewed as the fundamental building
block of a contact 3-manifold. Combined with convex surface theory, bypasses have been
broadlyusedinvariousclassificationproblemsin3-dimensionalcontactgeometrybymany
authors. Howeverithasnotbeenclearuntilnowhowthisoperationchangesthehomotopy
class of the underlying 2-plane field, which is one of the questions we shall address in
this thesis. In particular, we shall examine how a special sequence of bypass attachments,
namely, a bypass triangle attachment, affects the homotopy type of the contact structure.
This computation is particularly interesting because it plays an important role in the anal-
ysis of the universal cover of a contact categoryC(Σ) defined in [9], which we denote by
e
C(Σ). HereΣisaclosed,orientedsurface. NotethattheobjectsofC(Σ)aredividingsetson
Σ, and the morphisms are isotopy classes of tight contact structures onΣ×[0,1]. Loosely
speaking
e
C(Σ) in addition keeps track of the homotopy class of the contact structure as a
2-plane field, which defines a grading on
e
C(Σ). It is observed in [9] that
e
C(Σ) is a category
equipped with distinguished triangles given by the bypass triangle attachment. Then our
computationassertsthattheshiftfunctorin
e
C(Σ)indeeddecreasesthegradingby1.
Another benefit of being able to understand homotopy classes of contact structures is
that we can provide an alternative proof of Theorem 1.0.2 based on convex surface the-
ory. Namely, given two homotopic overtwisted contact structuresξ andξ
′
on M, we shall
constructanexplicitisotopyof M whichcarriesξ toξ
′
.
2
Thethesisisorganizedasfollows: Chapter2givesabriefintroductionto3-dimensional
contact topology with an emphasis on convex surface theory and bypasses. Chapter 3 is
devoted to a systematic study of homotopy classes of 2-plane fields in 3-manifolds, in
particular, we develop the Pontryagin-Thom construction for 3-manifolds with boundary
whichisadaptedtoourapplicationsinlaterchapters. Chapter4presentsthecomputations
of the effect of a bypass attachment (resp. bypass triangle attachment) to the homotopy
classofthecontactstructure. Chapter5givesanewproofofTheorem1.0.2.
3
Chapter2
ContactTopologyPreliminaries
2.1 Convexsurfaces
Let M beaclosed,oriented3-manifold. Acontactstructureξ on M isrank2subbundleof
TM suchthatthefollowingconditionsaresatisfied:
1. thereexistsaglobal1-formαsuchthatξ= ker(α).
2. α∧ dα > 0, i.e., the orientation induced by the contact form α agrees with the
orientationon M.
A contact structureξ is overtwisted if there exists an embedded disk D
2
⊂ M such that
ξ istangentto D
2
on∂D
2
. Otherwise,ξ issaidtobe tight.
LetΣ⊂ M be a closed, embedded, oriented surface in M. The characteristic foliation
Σ
ξ
onΣisbydefinitiontheintegralofthesingularlinefieldΣ
ξ
(x)Bξ
x
∩T
x
Σ. Onewayto
describethecontactstructurenearΣistolookatitscharacteristicfoliation.
Proposition2.1.1(Giroux). Letξ
0
andξ
1
betwocontactstructureswhichinducethesame
characteristic foliation onΣ. Then there exists an isotopyϕ
t
: M→ M, t∈ [0,1] fixingΣ
suchthatϕ
0
= id and(ϕ
1
)
∗
ξ
0
=ξ
1
.
Possibly after a C
∞
-small perturbation, we can always assume thatΣ⊂ M is convex,
i.e.,thereexistsavectorfieldvtransversetoΣsuchthattheflowofvpreservesthecontact
4
structure. Using this transversecontact vector field v, we define the dividing set onΣ to be
Γ
Σ
B{x∈ Σ| v
x
∈ ξ
x
}. Note that the isotopy class ofΓ
Σ
does not depend on the choice
of v. We refer to [6] for a more detailed treatment of basic properties of convex surfaces.
The significance of dividing sets in contact geometry is made clear by Giroux’s flexibility
theorem.
Theorem 2.1.2 (Giroux). Assume Σ is convex with characteristic foliation Σ
ξ
, contact
vectorfield v,anddividingsetΓ
Σ
. LetF beanother singular foliation onΣ dividedbyΓ
Σ
.
Thenthereexistsanisotopyϕ
t
: M→ M,t∈ [0,1]suchthat
1. ϕ
0
= id andϕ
t
|
Γ
Σ
= id forall t.
2. vistransversetoϕ
t
(Σ)forall t.
3. ϕ
1
(Σ)hascharacteristicfoliationF.
Remark 2.1.3. In the light of Theorem 2.1.2, any adapted characteristic foliation can be
realized by aC
0
-small isotopy of the convex surface, or equivalently, it can be realized by
a C
0
-small isotopy of the contact structure near a fixed convex surface. This allows us to
only specify the dividing set in order to determine the isotopy class of a contact structure
nearaconvexsurface.
2.2 Bypasses
We now look at contact structures onΣ×[0,1] with convex boundary. The first important
resultrelatingtothisproblemisthefollowingtheoremduetoGiroux.
Theorem2.2.1(Giroux). Letξ beacontactstructureonW =Σ×[0,1]sothatΣ×{0,1}is
convex. There exists an isotopy relative to the boundaryϕ
s
: W→ W, s∈ [0,1], such that
thesurfacesϕ
1
(Σ×{t})areconvexforallbutfinitelymanyt∈ [0,1]wherethecharacteristic
foliationssatisfythefollowingproperties:
5
1. Thesingularitiesandclosedorbitsareallnon-degenerate.
2. Thelimitsetofanyhalf-orbitiseitherasingularityoraclosedorbit.
3. There exists a single “retrogradient” saddle-saddle connection, i.e., an orbit from a
negativehyperbolicpointtoapositivehyperbolicpoint.
According to Theorem 2.2.1 one should expect a corresponding “film picture” of di-
viding sets on convex surfaces. It turns out that the correct notion corresponding to a
bifurcationisthe bypassattachment,whichwenowdescribe.
Aproperlyembeddedarcγin(M,ξ)isLegendrianifT
x
γ⊂ξ
x
forany x∈γ. Following
[11],wehave:
Definition2.2.2. Abypass DonΣisaconvexdiskwithLegendrianboundary∂D=α∪β
suchthatthefollowingconditionshold:
1. α=Σ∩D.
2. Γ
Σ
∩α={p
1
,p
2
,p
3
},where p
1
,p
2
,p
3
aredistinctpoints.
3. α∩β={p
1
,p
3
}.
4. foranappropriateorientationofD, p
1
and p
3
arebothpositiveellipticsingularpoints
of D, p
2
is a negative elliptic singular point of D, and all the singular points alongβ
arepositiveandalternatebetweenellipticandhyperbolic.
WecalltheLegendrianarcαasaboveanadmissiblearc,and DabypassalongαonΣ.
Remark 2.2.3. The admissible arc α in the above definition is also known as the arc of
attachmentforabypassinliterature.
Givenaconvexsurfaceandabypassasabove,wenowdescribeabypassattachment.
6
p
1
p
2
p
3
+ +
−
+
+
+
β
α
Figure2.1: Abypass
Lemma2.2.4(Honda). Usingtheterminologyfromabove,letDbeabypassalongαonΣ.
ThereexistsaneighborhoodofΣ∪D⊂ MdiffeomorphictoΣ×[0,1],suchthatΣ×{0,1}are
convex, andΓ
Σ×{1}
is obtained fromΓ
Σ×{0}
by performing the bypass attachment operation
asdepictedinFigure2.2inaneighborhoodofα.
α
Figure2.2: Abypassattachmentalongα.
Itisworthwhiletomentionthattherearetwodistinguishedbypasses,namely,thetrivial
bypass and the overtwisted bypass as depicted in Figure 2.3. The effect of a trivial bypass
attachmentisisotopictoan I-invariantcontactstructurewherenobypassisattached,while
the overtwisted bypass attachment immediately introduces an overtwisted disk in the local
model,hence,forexample,isdisallowedintightcontactmanifolds.
−→ −→
(a) (b)
Figure2.3: (a)Thetrivialbypassattachment. (b)Theovertwistedbypassattachment.
Note that a bypass attachment can be viewed in the following two different ways: (1)
Insideacontactmanifold,abypassattachmentisotopesaconvexsurfaceacrossthebypass
7
to obtain a new convex surface, where the dividing set changes in the way as depicted in
Figure 2.2. (2) For a contact manifold with convex boundary, a bypass attachment on the
boundary changes the isotopy class of the contact structure. In particular the dividing set
on the boundary changes in the same way as depicted in Figure 2.2. In this thesis we will
use both points of view when necessary and the reader should not get confused when we
applybypassattachmentsindifferentsituations.
2.3 Bypasstriangleattachment
Thebypasstriangleattachmentisaspecialcompositionofthreebypassattachments,which
will be important for our applications. To define a bypass triangle attachment, we first set
upthefollowingnotationsjustforconvenience.
Notation: LetΣbeaconvexsurfaceandα⊂Σbeanadmissiblearc. Wedenotethebypass
attachment along α on Σ by σ
α
. Let β be another admissible arc on the convex surface
obtained by attaching the bypass along α on Σ. We denote the composition of bypass
attachments by σ
α
∗σ
β
, where the composition rule is to attach the bypass along α first,
then attach the bypass alongβ in the same direction. If (M,ξ) is a contact manifold with
convex boundary, thenξ∗σ
α
denotes the contact structure obtained by attaching a bypass
alongαto(M,ξ).
Remark 2.3.1. In general, bypass attachmentsσ
α
andσ
β
are not commutative unless the
attachingarcsaredisjoint. Infactβmaynotevenbeanarcofattachmentbeforethebypass
attachmentσ
α
.
Definition2.3.2. LetΣ be a convex surface andα⊂Σ be an admissible arc. A bypass tri-
angle attachmentalongαisthecompositionofthreebypassattachmentsalongadmissible
arcsα,α
′
andα
′′
in a neighborhood ofα as depicted in Figure 2.4. We denote the bypass
8
triangleattachmentalongαby△
α
=σ
α
∗σ
α
′∗σ
α
′′.
Remark 2.3.3. The second admissible arcα
′
in the bypass bypass triangle is also known
asthe arc of anti-bypass attachmenttoσ
α
.
α
α
′
α
′′
σ
α
σ
α
′ σ
α
′′
Figure2.4: Abypasstriangleattachment.
It is important for our purposes to notice that the bypass triangle attachment does not
changetheisotopyclassofthecontactstructureinaneighborhoodofΣbyTheorem2.1.2.
This is because the configuration of the dividing set is preserved by the bypass triangle
attachment up to an isotopy supported in a neighborhood of α. One can easily visualize
thisfromFigure2.4.
9
Chapter3
HomotopyClassesof2-planeFields
3.1 ThePontryagin-Thomconstructionforclosedmanifolds
The Pontryagin-Thom construction is designed to study homotopy types of smooth maps
f : M → S
n
, where M is a closed manifold. The idea is that instead of working with
maps between manifolds, we study framed submanifolds of M associated with these maps
and framed cobordism between them. Throughout this paper, we always assume M is
3-dimensionaland n= 2.
Fix a Riemannian metric on M. Let L⊂ M be a link. A framing of L is the homotopy
class of a smooth functionσ which assigns to each point x∈ L a basis{v
1
(x),v
2
(x)} of the
orthogonal complement of T
x
L in T
x
M. We call the pair (L,σ) a framed link. Two framed
links (L,σ) and (L
′
,σ
′
) are framed cobordant if there exists a framed surface (Σ,δ) in the
4-manifold M× [0,1] such that (Σ,δ)|
M×0
= (L,σ) and (Σ,δ)|
M×1
= (L
′
,σ
′
), where the
framingδ is the homotopy class of a smooth function which assigns to each point y∈ Σ a
basisoftheorthogonalcomplementof T
y
Σin T
y
(M×[0,1]).
Remark3.1.1. TheaboveconstructionisindependentofthechoiceofaRiemannianmetric
on M sincethespaceofRiemannianmetricson M ispath-connected.
The main result of the Pontryagin-Thom construction is the following theorem. See
Chapter7of[13]formoredetails.
10
Theorem 3.1.2. Let M be a closed 3-manifold. Then there exists a one-to-one correspon-
dence
{smoothmaps f : M→ S
2
uptohomotopy}
1−1
←−−−→{framedlinksin M uptoframed
cobordism}.
Sketch of proof. To construct a framed link in M from a smooth map f : M → S
2
, let
p∈ S
2
be a regular value of f. By choosing a basis{v
1
,v
2
} of T
p
S
2
, we obtain a framed
link (L
f,p
,σ
f,p
) in M, where L
f,p
= f
−1
(p) and σ
f,p
(x) is the pull-back of{v
1
,v
2
} via the
isomorphism f
∗
: T
x
L
⊥
→ T
p
S
2
,∀x∈ L.
Conversely, let (L,σ) be a framed link in M. Identify an open tubular neighborhood
N(L)of Lwith L×R
2
viaσ. Chooseasmoothmapϕ :R
2
→ S
2
whichmapsevery xwith
||x||≥ 1toabasepointy∈ S
2
,andmapstheopenunitdisk||x||< 1diffeomorphicallyonto
S
2
\{y}. Wedefineasmoothmap f : M→ S
2
intwosteps. Firstwedefine f|
N(L)
: N(L)≃
L×R
2
π
2
−→R
2
ϕ
− → S
2
,whereπ
2
: L×R
2
→R
2
istheprojectionontothesecondfactor. Then
weextend f|
N(L)
to f : M→ S
2
bytheconstantmap f|
M\N(L)
≡ y∈ S
2
.
One can show that the above construction in both directions establishes the desired
one-to-onecorrespondence.
Definition 3.1.3. Given a smooth map f : M→ S
2
, we call the framed link (L
f,p
,σ
f,p
)
constructedabovethe Pontryaginsubmanifold associated with f.
Remark 3.1.4. Although the construction of (L
f,p
,σ
f,p
) depends on the choice of p, its
framedcobordismclassdoesnot. ComparewiththerelativePontryagin-Thomconstruction
discussedinSection3.2.
However, Theorem 3.1.2 is still not satisfactory for our purposes because we will be
workingwithcontactmanifoldswithboundary. BeforewegeneralizethePontryagin-Thom
constructiontomanifoldswithboundary,welookatasimpleapplicationofTheorem3.1.2
11
whichcomputesthehomotopygroupπ
3
(S
2
),whichwasfirstconsideredbyHopf[12]. We
willuseasimilarcomputationinarelativecaseinSection3.3.
Lemma3.1.5. Thereexistsanisomorphism h :π
3
(S
2
)
∼
− →Z
Sketch of proof. Sinceanycontinuousmap f : S
3
→ S
2
canbeapproximatedbyasmooth
map, we can assume that the elements inπ
3
(S
2
) are represented by smooth maps. Now it
followsimmediatelyfromTheorem3.1.2thatπ
3
(S
2
)={(L,σ)}/∼,where(L,σ)∼ (L
′
,σ
′
)
if and only if they are framed cobordant. The group structure on{(L,σ)}/∼ is defined by
[(L
1
,σ
1
)]+ [(L
2
,σ
2
)] = [(L
1
⊔ L
2
,σ
1
⊔σ
2
)] and−[(L,σ)] = [(L,−σ)]. If
˜
L is a parallel
copy of L given by the framing σ, then we define n(L,σ) to be the self-linking number
lk(L,
˜
L). Now we define the group homomorphism h : π
3
(S
2
)→ Z by sending [(L,σ)] to
n(L,σ). Itiseasytoverifythat hiswell-definedandisanisomorphism.
3.2 ThePontryagin-Thomconstructionformanifoldswithboundary
Let M be a compact 3-manifold with boundary. Let f : M→ S
2
be a smooth map and
p∈ S
2
be a regular value of f. The Pontryagin submanifold (f
−1
(p),σ
f,p
) associated with
the pair (f,p) is a framed 1-dimensional submanifold of M, i.e., it is the disjoint union
of a framed link and a finite collection of framed arcs with endpoints contained in ∂M.
Two framed 1-dimensional submanifolds (L,σ) and (L
′
,σ
′
) of M are relatively framed
cobordant if there exists a framed surface (Σ,δ) in M× [0,1] such that (i) (Σ,δ)|
M×{0}
=
(L,σ), (ii) (Σ,δ)|
M×{1}
= (L
′
,σ
′
), and (iii) (Σ,δ)|
∂M×{t}
= (L,σ)|
∂M×{0}
= (L
′
,σ
′
)|
∂M×{1}
for
anyt∈ [0,1]. Wehavethefollowingtheoremwhichcanbeviewedastherelativeanalogue
ofTheorem3.1.2.
Theorem 3.2.1. Let M be a compact 3-manifold with boundary. If f, f
′
: M→ S
2
are
smooth maps such that f|
∂M
= f
′
|
∂M
, then f is homotopic to f
′
relative to the boundary if
12
and only if for any p∈ S
2
which is a common regular value of f and f
′
, (f
−1
(p),σ
f,p
) is
relativelyframedcobordantto(f
′−1
(p),σ
f
′
,p
).
Proof. Let H : M×[0,1]→ S
2
beahomotopybetween f and f
′
relativetotheboundary.
Generically we can assume p∈ S
2
is also a regular value of H. Hence the Pontryagin
submanifold (H
−1
(p),δ
H
) defines a relative framed cobordism between (f
−1
(p),σ
f,p
) and
(f
′−1
(p),σ
f
′
,p
).
Conversely,let(Σ,δ)⊂ M×[0,1]bearelativeframedcobordismbetween(f
−1
(p),σ
f,p
)
and(f
′−1
(p),σ
f
′
,p
). Let∂M×[−1,0]⊂ M beacollarneighborhoodof∂M where∂M×{0}
isidentifiedwith∂M,and
˜
M bethemetricclosureof M\(∂M×[−1,0]). Abusingnotation,
we shall write Σ for Σ∩ (
˜
M× [0,1]). As in the proof of Theorem 3.1.2, we identify
an open tubular neighborhood N(Σ) of Σ in
˜
M× [0,1] with Σ×R
2
via δ, and define a
smooth map H
1
:
˜
M× [0,1]→ S
2
by (i) H
1
|
N(Σ)
: N(Σ)≃ Σ×R
2
π
2
−→ R
2
ϕ
− → S
2
where
π
2
: Σ×R
2
→ R
2
is the projection onto the second factor, and (ii) H
1
| ˜
M\N(Σ)
≡ y∈ S
2
.
Observe that H
1
|
∂
˜
M×{t}
: ∂
˜
M×{t} → S
2
is homotopic to f : ∂M×{t} → S
2
for any
t∈ [0,1], and let H
t
2
: ∂M×[−1,0]×{t}→ S
2
be the homotopy, i.e., H
t
2
|
s=−1
= H
1
|
∂
˜
M×{t}
andH
t
2
|
s=0
= f,where s∈ [−1,0]. DefineH
2
:∂M×[−1,0]×[0,1]byH
2
(x,s,t)= H
t
2
(x,s)
for x∈∂M, s∈ [−1,0] and t∈ [0,1]. We construct a map H : M×[0,1]→ S
2
by gluing
H
1
andH
2
along∂M×{−1}×[0,1]whichsatisfiesH|
∂M×{t}
= f|
∂M
= f
′
|
∂M
foranyt∈ [0,1].
Onecanverifythat H|
M×{0}
and H|
M×{1}
arehomotopicto f and f
′
relativetotheboundary,
respectively,asintheclosedcase. Hencetheconclusionfollows.
Remark3.2.2. TheproofofTheorem3.2.1aboveisslightlymoreinvolvedthantheproof
in the closed case, namely, we used the submanifold
˜
M⊂ M onto which M deformation
retracts. This is because the restriction of the Pontryagin submanifold to ∂M does not
determine f|
∂M
: ∂M→ S
2
(resp. f
′
|
∂M
). It only determines the homotopy type of f|
∂M
(resp. f
′
|
∂M
).
13
Corollary3.2.3. Let f, f
′
: M→ S
2
besmoothmapssuchthat f|
∂M
= f
′
|
∂M
. If(f
−1
(p),σ
f,p
)
isrelativelyframedcobordantto(f
′−1
(p),σ
f
′
,p
)forsomecommonregularvalue pof f and
f
′
,thenthesameholdsforallcommonregularvaluesof f and f
′
.
Proof. ThisfollowsimmediatelyfromtheproofofTheorem3.2.1.
Hence in practice, in order to verify that f is homotopic to f
′
relative to the boundary,
itsufficestochecktheframedcobordantconditionforapreferredcommonregularvalue.
Remark 3.2.4. One can easily generalize Theorem 3.2.1 to arbitrary dimension using the
sameproof.
3.3 The 3-dimensional obstruction class o
3
(ξ,ξ
′
) of oriented 2-plane
fields
Let M be a compact oriented 3-manifold, and ξ and ξ
′
be two oriented 2-plane fields on
M such that ξ = ξ
′
on M\ B
3
for a 3-ball B
3
⊂ int(M). The goal of this section is to
define a difference class betweenξ andξ
′
. Fix a trivialization of TM. Let G
ξ
: M→ S
2
and G
ξ
′ : M → S
2
be the Gauss maps associated with ξ and ξ
′
, respectively. Take a
common regular value p∈ S
2
of G
ξ
and G
ξ
′, and let (L,σ) and (L
′
,σ
′
) be the Pontryagin
submanifolds associated with (G
ξ
,p) and (G
ξ
′,p), respectively, i.e., L = G
−1
ξ
(p) and L
′
=
G
−1
ξ
′
(p). By assumption, (L,σ) = (L
′
,σ
′
) on M\ B
3
. Hence we may focus on the relative
framed cobordism classes of (L,σ)|
B
3 and (L
′
,σ
′
)|
B
3. Since B
3
is contractible, L is always
relatively cobordant to L
′
but the framing may not extend to the cobordism. To fix this
issue, let C⊂ int(B
3
) be a trivial loop which does not link with L
′
. Observe that (L,σ) is
relatively framed cobordant to (L
′
⊔C,σ
′
⊔δ) in B
3
for some framing δ of C. If C
′
is a
parallelcopyofC givenbyδ,thenwedefinen(C,δ)tobetheself-linkingnumberlk(C,C
′
)
withrespecttotheorientationof B
3
inheritedfromtheorientationof M.
14
Definition3.3.1. Letξ andξ
′
be oriented 2-plane field distributions on M such thatξ=ξ
′
on M\ B
3
for a 3-ball B
3
⊂ M. We define the 3-dimensional obstruction class o
3
(ξ,ξ
′
)∈
Z/d(ξ) to be n(C,δ) as constructed above modulo d(ξ), where d(ξ) is the divisibility of the
Eulerclass e(ξ)∈ H
2
(M,Z).
Remark 3.3.2. One can think of o
3
(ξ,ξ
′
) as a relative version of π
3
(S
2
) discussed in
Lemma3.1.5.
Itiseasytoseethatthedefinitionofo
3
(ξ,ξ
′
)isindependentofvariouschoicesinvolved,
namely, the trivialization of TM, the 3-ball B
3
⊂ M, the trivial loop C and the common
regularvalue p∈ S
2
. Theindependenceofthechoiceofcommonregularvaluesisslightly
nontrivial,soweprovethisinthefollowinglemma.
Lemma 3.3.3. The obstruction class o
3
(ξ,ξ
′
) ∈ Z/d(ξ) is independent of the choice of
p∈ S
2
.
Proof. Let
ˆ
M = M∪
∂M
(−M) be a closed oriented 3-manifold, where−M is M with the
oppositeorientation. GlueG
ξ
andG
ξ
′ along∂M toobtainasmoothmap
ˆ
G :
ˆ
M→ S
2
given
by:
ˆ
G(x)=
G
ξ
(x) if x∈ M,
G
ξ
′(x) if x∈−M.
If q∈ S
2
isanothercommonregularvalueofG
ξ
andG
ξ
′, then pand qareboth regular
values of
ˆ
G. We write o
p
3
(ξ,ξ
′
) (resp. o
q
3
(ξ,ξ
′
)) for the obstruction class to indicate the
potential dependence on the choice of p (resp. q). According to Proposition 4.1 in [8], we
have o
p
3
(ξ,ξ
′
)− o
q
3
(ξ,ξ
′
) = 0∈ Z/d(ξ). Hence o
3
(ξ,ξ
′
) is independent of the choice of p
modulo d(ξ).
Using the same argument as in proof of Proposition 4.1 in [8], we also obtain the fol-
lowingresult.
15
Proposition3.3.4. Ifξ andξ
′
aretwocontactstructureson M suchthatξ|
M\B
3 =ξ
′
|
M\B
3 for
some 3-ball B
3
⊂ int(M), thenξ is homotopic toξ
′
relative to the boundary if and only if
o
3
(ξ,ξ
′
)= 0∈Z/d(ξ).
16
Chapter4
BypassAttachmentsandHomotopy
ClassesofContactStructures
4.1 Computationofthehomotopyclassofabypassattachment
The goal of this section is to understand the effect of attaching a bypass to the homotopy
class of the contact structure. To achieve this, we first compute the homotopy class of a
bypassattachmentinalocalmodelwhichwedescribenow. LetV = [−3/4,3/4]×[−1,1]×
[0,1]⊂R
3
be a 3-manifold with boundary equipped with the standard coordinates, andξ
V
be a contact structure on V defined by ξ
V
= ker λ, where λ = cos(2πx)dy− sin(2πx)dz.
Letα = [−1/2,1/2]×{0}×{1}⊂ V be a Legendrian arc. Letξ
V
∗σ
α
denote the contact
structure given by attaching a bypass to (V,ξ
V
) alongα (c.f. Lemma 2.2.4). Trivialize TV
bythestandardembeddingV⊂R
3
andlookattheassociatedGaussmapG
ξ
V
∗σ
α
: V→ S
2
.
Observe that p= (1,0,0)∈ S
2
is a regular value ofG
ξ
V
∗σ
α
by construction. We will prove
thefollowingtheorem.
Theorem 4.1.1. Let (V,ξ
V
∗σ
α
) be the contact manifold described above. Then the Pon-
tryagin submanifold G
−1
ξ
V
∗σ
α
(p)⊂ V is a properly embedded framed arc with framing as
depictedinFigure4.1.
Remark 4.1.2. The Pontryagin submanifoldG
−1
ξ
V
∗σ
α
(p) in Theorem 4.1.1 depends on vari-
ous choices including the trivialization of TV and the regular value p. For example, it will
be clear from the proof of Theorem 4.1.1 that q= (−1,0,0)∈ S
2
is also a regular value of
17
x
y
z
Figure 4.1: The Pontryagin submanifoldG
−1
ξ
V
∗σ
α
(p) in V. The blue arc is a parallel copy of
G
−1
ξ
V
∗σ
α
(p)whichdefinestheframing.
G
ξ
V
∗σ
α
,butG
−1
ξ
V
∗σ
α
(q)istheemptyset.
The following corollary follows immediately from Theorem 4.1.1 by the local nature
ofthebypassattachment.
Corollary4.1.3. Let(M,ξ)beacontact3-manifoldwithconvexboundaryandα⊂∂M be
a Legendrian arc along which a bypass can be attached. Then there exists a trivialization
of TM and a common regular value p ∈ S
2
of G
ξ
and G
ξ∗σ
α
such that the Pontryagin
submanifoldG
−1
ξ∗σ
α
(p)= G
−1
ξ
(p)∪γ,whereG
−1
ξ
(p)isthePontryaginsubmanifoldassociated
withξ andγ⊂ M isaproperlyembeddedframedarcasdepictedinFigure4.1whichdoes
notlinkG
−1
ξ
(p).
Proof. Let N(α) ⊂ ∂M be a small neighborhood of α. Consider a thickening N(α)×
[−ϵ,0]⊂ M of N(α) for small ϵ > 0, such that N(α)×{0}⊂ ∂M. Since ∂M is convex
by assumption, we may assume that N(α)×[−ϵ,0] with the restricted contact structure is
contactomorphicto(V,ξ
V
)asconstructedabove. Moreoverwecanchooseatrivializationof
TM suchthatitsrestrictiontoN(α)×[−ϵ,0]ispushedforwardtothestandardtrivialization
of TV under the contactomorphism (which is, in particular, a diffeomorphism). With this
setup,theconclusionisanimmediateconsequenceofTheorem4.1.1.
18
Now we are ready to compute the relative Pontryagin submanifold associated with the
contact3-manifold(V,ξ∗σ
α
)asconstructedinTheorem4.1.1.
Proof of Theorem4.1.1. LetΣ
t
= [−3/4,3/4]×[−1,1]×{t}⊂ V be a foliation by convex
surfaces with respect to the contact vector field ∂/∂z for t ∈ [0,1]. The dividing set Γ
t
on Σ
t
, t∈ [0,1], is the disjoint union of three parallel intervals ({1/2}× [−1,1]×{t})∪
({0}× [−1,1]×{t})∪ ({−1/2}× [−1,1]×{t}) which divide Σ
t
into positive and negative
regions. Let α = [−1/2,1/2]×{0}×{1}⊂ Σ
1
be the Legendrian arc along which an I-
invariant neighborhood of the bypass D
α
={(x,y,z)| 1≤ z≤ 1+
√
1/4− x
2
,y = 0} is
attached. We choose the characteristic foliation on D
α
so that it is half of an overtwisted
disk with one negative elliptic singular point at the center and alternating positive elliptic
and hyperbolic singular points on the boundary, and the dividing setΓ
D
α
is a semi-circle
centeredat(0,0,1)withradius1/4. Bygluinga∂/∂y-invariantneighborhood D
α
×[−ϵ,ϵ]
of D
α
for smallϵ > 0 to (V,ξ
V
), we obtain a contact manifold (V
α
,ξ
V
α
) with corners where
V
α
= V∪(D
α
×[−ϵ,ϵ]). Abusingnotation,wealsodenotethecontactmanifoldobtainedby
roundingcornersonD
α
×[−ϵ,ϵ]⊂ V
α
by(V
α
,ξ
V
α
). ByslightlytiltingD
α
×{−ϵ}andD
α
×{ϵ},
wecanfurtherassumethatthe∂/∂z-directionistransverseto∂
+
V
α
,thetopboundaryofV
α
.
Observethat,uptoisotopy,Γ
∂
+
V
α
isasdepictedintheright-hand-sideofFigure2.2.
Choose a non-positive smooth function g : V
α
→ R
≤0
supported in a neighborhood of
D
α
×[−ϵ,ϵ] such that the time-1 mapϕ
1
X
of the flow of X = g∂/∂z sends V
α
diffeomorphi-
cally onto V. We identify V
α
with V viaϕ
1
X
, and we denote the contact structure (ϕ
1
X
)
∗
(ξ
V
α
)
byξ
V
∗σ
α
, whereξ
V
∗σ
α
is known as the contact structure obtain by attaching a bypass
alongαtoξ
V
.
Next,westudythehomotopytypeof(V,ξ
V
∗σ
α
)usingthePontryagin-Thomconstruc-
tion. Let p = (1,0,0)∈ S
2
be a regular value of the Gauss map G
ξ
V
∗σ
α
associated with
ξ
V
∗σ
α
,whereTV istrivializedbythestandardembeddingV⊂R
3
. Observethat pisalso
19
a regular value of the Gauss map G
ξ
Vα
: V
α
→ S
2
associated with ξ
V
α
. In order to keep
trackoftheframingofG
−1
ξ
Vα
(p),wefixanotherregularvalue p
′
= (1−δ,
√
2δ−δ
2
,0)∈ S
2
near p for smallδ> 0. It is easy to see thatG
−1
ξ
Vα
(p) andG
−1
ξ
Vα
(p
′
) are two parallel arcs with
endpointscontainedin D
α
×{−ϵ,ϵ}asdepictedinFigure4.2(a). Withoutlossofgenerality,
we can assume that the endpoints ofG
−1
ξ
Vα
(p) andG
−1
ξ
Vα
(p
′
) are contained in the dividing set
Γ
D
α
×{−ϵ,ϵ}
. Note that G
ξ
Vα
(x) is contained in the unit circle S
1
={z = 0}⊂ S
2
if and only
if the same holds for G
ξ
V
∗σ
α
(ϕ
1
X
(x)). By applying the diffeomorphism ϕ
1
X
: V
α
→ V, we
obtain the Pontryagin submanifold G
−1
ξ
V
∗σ
α
(p) associated with ξ
V
∗σ
α
with framing given
byG
−1
ξ
V
∗σ
α
(p
′
)asdepictedinFigure4.2(b). ThisfinishestheproofofTheorem4.1.1.
−
+
+
−
+
−
(a) (b)
Figure 4.2: (a) The Pontryagin submanifold G
−1
ξ
Vα
(p) contained in V
α
. (b) The Pontryagin
submanifold G
−1
ξ
V
∗σ
α
(p) contained in V. The blue arc is a parallel copy of G
−1
ξ
V
∗σ
α
(p) which
definestheframing.
4.2 Computationofthehomotopyclassofabypasstriangleattachemnt
We are now ready to compute the homotopy class of a bypass triangle attachment defined
in2.3.2. Themainresultofthissectionisthefollowingtheorem.
Theorem4.2.1. If(M,ξ)isacontactmanifoldwithconvexboundary,andξ
′
isthecontact
structure obtained fromξ by attaching a bypass triangle on∂M, thenξ
′
is isotopic toξ in
20
thecomplementofaball,ando
3
(ξ,ξ
′
)=−1. Inparticular,ξ
′
isnothomotopictoξ relative
totheboundaryas2-planefields.
Remark4.2.2. Recallthato
3
(ξ,ξ
′
)definedin(3.3.1)isthe3-dimensionalobstructionclass
which distinguishes homologous ξ and ξ
′
. However it is straightforward to see that ξ is
isotopictoξ
′
inthecomplementofaballbecauseofthelocalnatureofbypassattachments
and the observation that the bypass triangle attachment does not change the dividing set.
Therefore o
3
(ξ,ξ
′
)iswell-definedinthesituationofTheorem4.2.1.
Remark 4.2.3. Theorem 4.2.1 is an important ingredient in the analysis of the universal
coverofacontactcategoryC(F)definedin[9],i.e.,theshiftfunctoractuallydecreasesthe
gradingby1.
Remark 4.2.4. We will make heavy use of Theorem 4.2.1 in our proof of Theorem 1.0.2
inChapter5.
Ourstrategyistofirstconstructalocalmodelforthebypass triangleattachmentinR
3
,
andcomputetheassociatedPontryaginsubmanifoldbasedonessentiallythesamemethods
used in the proof of Theorem 4.1.1. Next we identify a neighborhood of the arc of attach-
ment α in M where the bypass triangle is attached with our previously constructed local
model (c.f. proof of Corollary 4.1.3), and conclude that the bypass triangle attachment
drops o
3
by1.
Wefirstestablishatechnicallemmawhichenablesustoisotopcharacteristicfoliations
onadiskadaptedtoafixeddividingsetwithoutaffectingthePontryaginsubmanifold.
Lemma 4.2.5. Let (D
2
× [0,1],ξ) be a contact 3-manifold with T(D
2
× [0,1]) trivialized
bythestandardembedding D
2
×[0,1]⊂R
3
,i.e., D
2
iscontainedinthe xy-planeand[0,1]
isinthedirectionofthe z-axis. Supposethefollowingconditionshold:
21
1. There exists a contact vector field on D
2
×[0,1], with respect to which D
2
×{t} are
convexandthedividingsetsΓ
D
2
×{t}
agreeforall t∈ [0,1].
2. The characteristic foliationsF
D
2
×{t}
agree in a neighborhood of Γ
D
2
×{t}
for all t ∈
[0,1].
3. The Gauss map G
ξ
satisfies: (i) G
ξ
(Γ
D
2
×{t}
)⊂{z = 0}⊂ S
2
, (ii) G
ξ
(R
+
(D
2
×{t}))⊂
{z> 0}⊂ S
2
,and(iii)G
ξ
(R
−
(D
2
×{t}))⊂{z< 0}⊂ S
2
foranyt∈ [0,1].
4. p= (1,0,0)∈ S
2
isaregularvalueofG
ξ
,andG
−1
ξ
(p)isdisjointfrom∂D
2
×[0,1].
Then G
−1
ξ
(p) is framed cobordant to G
−1
ξ
0
(p) relative to the boundary, where ξ
0
is the
I-invariantcontactstructureon D
2
×[0,1]withξ
0
|
D
2
×{0}
=ξ|
D
2
×{0}
.
Proof. The conclusion follows from the observation that G
−1
ξ
(p)∩(D
2
×{t})⊂ Γ
D
2
×{t}
for
all t∈ [0,1],andξ is I-invariantinaneighborhoodofΓ
D
2
×{0}
×[0,1]in D
2
×[0,1].
The following proposition constructs a local model for the bypass triangle attachment
explicitlyandcomputesitsPontryaginsubmanifold.
Proposition 4.2.6. Let T = [−3/4,3/4]× [−1,1]× [0,3] ⊂ R
3
be a 3-manifold, η =
ker(cos(2πx)dy−sin(2πx)dz) be a contact structure on T, andα ={−1/2≤ x≤ 1/2,y =
z= 0}beaLegendrianarc. Thenthereexistsacontact3-manifold(T,η∗△
α
)whereη∗△
α
is the contact structure obtained from η by attaching a bypass triangle along α, such that
the Pontryagin submanifold G
−1
η∗△
α
(p) is the unknot with framing−1 with respect to the
standardorientation. Here p= (1,0,0)∈ S
2
isaregularvalueofG
−1
η∗△
α
.
Proof. We construct (T,η∗△
α
) and compute its Pontryagin submanifold in three steps
correspondingtothreebypassattachmentsσ
α
,σ
α
′ andσ
α
′′ respectively.
22
STEP1. Wesimplyusetheconstructionof(V,η∗σ
α
)
1
intheproofofTheorem4.1.1. Recall
thatthePontryaginsubmanifoldG
−1
η∗σ
α
(p)isaframedarcinV asdepictedinFigure4.2(b).
STEP 2. We compute the Pontryagin submanifold associated with the second bypass at-
tachmentσ
α
′ intwosubsteps.
SUBSTEP 2.1. Weattachthesecondbypassinasimilarmanner. LetU = [−3/4,3/4]×
[−1,1]× [1,2]⊂ R
3
be a contact 3-manifold with contact structure obtained by a ∂/∂z-
invariant extension of η∗σ
α
|
Σ
1
, where Σ
1
= [−3/4,3/4]× [−1,1]×{1}. Recall that the
second bypass is attached along the Legendrian arc α
′
as depicted in Figure 2.4(b). Let
D
α
′ bethebypassalongα
′
,and(U
α
′,η
α,α
′)bethecontact3-manifoldobtainedbyrounding
the corners of U∪ (D
α
′× [−ϵ,ϵ]) with the glued contact structure for small ϵ > 0. By
Lemma 4.2.5, we can choose a Legendrian representativeα
′
within its isotopy class such
that p < G
η
α,α
′
(D
α
′× [−ϵ,ϵ]), the image of D
α
′× [−ϵ,ϵ] under the associated Gauss map
G
η
α,α
′
. Since the contact structure remains I-invariant away from a neighborhood ofα
′
, by
pushing D
α
′×[−ϵ,ϵ]intoU,weobtainthecontact3-manifold(U,(η∗σ
α
|
Σ
1
)∗σ
α
′)whose
PontryaginsubmanifoldG
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p)isasdepictedinFigure4.3(a).
γ
−
+
−
+
(a) (b)
Figure 4.3: (a) The Pontryagin submanifold G
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p) contained in U. (b) The Pon-
tryaginsubmanifoldG
−1
τ
(p)containedin N(γ)×[2,3].
1
Thecontactstructureηhereisthesameasξ
V
inthenotationofTheorem4.1.1.
23
SUBSTEP 2.2 Let γ⊂ Γ
Σ
2
be the arc containing the endpoints of G
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p) on
Σ
2
= [−3/4,3/4]× [−1,1]×{2}, and N(γ) be a neighborhood of γ on Σ
2
. It is easy to
see that there exists an isotopy ϕ
t
: N(γ) → N(γ), t ∈ [0,1], ϕ
0
= id, such that p is
not contained in the image of N(γ) under the Gauss map G
(ϕ
1
)
∗
(η∗σ
α
∗σ
α
′|
N(γ)
)
. If we define
Φ : N(γ)× [2,3]→ N(γ)× [2,3] byΦ(x,t) = (ϕ
t
(x),t) for x∈ N(γ), t∈ [2,3], then we
canpush-forwarda∂/∂z-invariantcontactstructureη∗σ
α
∗σ
α
′|
N(γ)
on N(γ)×[2,3]viaΦ
to obtain a new contact structure on N(γ)×[2,3], which we denote byτ. The Pontryagin
submanifold G
−1
τ
(p) in N(γ)× [2,3] is a framed arc as depicted in Figure 4.3(b). To see
how G
−1
τ
(p) is linked with the blue arc in Figure 4.3(b) which determines the framing, we
note that the endpoints of the blue arc are in between of the endpoints of G
−1
τ
(p). If we
suppose the interval [2,3] is parameterized by time, then the endpoints of the blue arc will
merge and disappear before the endpoints of G
−1
τ
(p) do as time goes from 2 to 3. Hence
weobtainacontactmanifold(U∪(N(γ)×[2,3]),((η∗σ
α
)|
Σ
1
∗σ
α
′)∪τ). Byroundingthe
corners of N(γ)× [2,3] and pushing it into U as usual, we obtain the contact 3-manifold
which we still denote by (U,(η∗σ
α
)|
Σ
1
∗σ
α
′) whose associated Pontryagin submanifold
G
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p)isaframedarcasdepictedinFigure4.4.
− +
−
+
Figure4.4: ThePontryaginsubmanifoldG
−1
(η∗σ
α
|
Σ
1
)∗σ
α
′
(p)containedin U afteranisotopy.
STEP 3. We finish the bypass triangle by attaching the third bypass D
α
′′ along α
′′
as
depictedinFigure2.4(c). Asinprevioussteps,let W = [−3/4,3/4]×[−1,1]×[2,3]⊂R
3
24
be a contact 3-manifold with contact structure obtained by a ∂/∂z-invariant extension of
η∗ σ
α
∗ σ
α
′|
Σ
2
. Again by Lemma 4.2.5, we can choose α
′′
so that p is not contained
in the image of D
α
′′× [−ϵ,ϵ] under the Gauss map. Hence the same argument as before
producesthethirdcontact3-manifold(W,(η∗σ
α
∗σ
α
′|
Σ
2
)∗σ
α
′′)whoseassociatedPontryagin
submanifoldG
−1
(η∗σ
α
∗σ
α
′|
Σ
2
)∗σ
α
′′
(p)istheemptyset.
Finally,inordertoconstruct(T,η∗△
α
)withthedesiredproperties,wesimplylet(T,η∗
△
α
)= (V,η∗σ
α
)∪(U,(η∗σ
α
|
Σ
1
)∗σ
α
′)∪(W,(η∗σ
α
∗σ
α
′|
Σ
2
)∗σ
α
′′)gluedalongadjacentfaces.
ItiseasytoseethattheassociatedPontryaginsubmanifoldG
−1
η∗△
α
(p)obtainedbygluingthe
framedarcsfromSteps1,2,and3istheunknotwithframing−1. SeeFigure4.5.
α
Figure 4.5: The Pontryagin submanifold G
−1
η∗△
α
(p). The blue circle is a parallel copy of
G
−1
η∗△
α
(p)whichdefinestheframing.
Proof of Theorem4.2.1. Letα⊂ ∂M be the Legendrian arc such thatξ
′
≃ ξ∗△
α
relative
to the boundary, and N(α) be a neighborhood of α on ∂M. Let ∂M× [−1,0]⊂ M be a
collar neighborhood of∂M with an I-invariant contact structure such that∂M is identified
with∂M×{0}. Assume up to a boundary relative isotopy that△
α
is supported in N(α)×
[−2/3,−1/3]⊂ int(M), i.e., ξ = ξ
′
on M\ (N(α)× [−2/3,−1/3]), and that there exists
a contactomorphismψ : (N(α)× [−2/3,−1/3],ξ
′
)→ (T,η∗△
α
) where (T,η∗△
α
) is the
local model for a bypass triangle attachment constructed in Proposition 4.2.6. Without
loss of generality, we also choose the trivialization of TM so that its restriction to N(α)×
[−2/3,−1/3] coincides with the pull-back of TR
3
via ψ. Let p = (1,0,0) ∈ S
2
be a
25
common regular value of G
ξ
and G
ξ
′. Observe that the Pontryagin submanifold G
−1
ξ
′
(p)
restrictedto N(α)×[−2/3,−1/3]istheunknotwithframing−1. SinceG
−1
ξ
(p)restrictedto
N(α)×[−2/3,−1/3]istheemptyset,itfollowsfromDefinition3.3.1thato
3
(ξ,ξ
′
)=−1as
desired.
Inparticular,ξisnothomotopictoξ
′
relativetotheboundarybyProposition3.3.4since
d(ξ)isalwayseven.
26
Chapter5
ClassificationofOvertwistedContact
StructuresviaConvexSurfaceTheory
5.1 Introduction
Thegoal ofthispaper istoprovideanalternativeproof ofTheorem 1.0.2 based on convex
surface theory. Convex surface theory was introduced by Giroux in [6] building on the
work of Eliashberg-Gromov [3]. Given a closed oriented surfaceΣ, we consider contact
structuresonΣ×[0,1]suchthatΣ×{0,1}isconvex. Bystudyingthe“filmpicture”ofthe
characteristic foliations onΣ×{t} as t goes from 0 to 1, Giroux showed in [7] that, up to
an isotopy, there are only finitely many levelsΣ×{t
i
}, 0< t
1
<···< t
n
< 1, which are not
convex. Moreover,forsmallϵ > 0,thecharacteristicfoliationsonΣ×{t
i
−ϵ}andΣ×{t
i
+ϵ},
i = 1,2,··· ,n, change by a bifurcation. In [10], Honda gave an alternative description of
the bifurcation of characteristic foliations in terms of dividing sets. Namely, he defined an
operation, called the bypass attachment, which combinatorially acts on the dividing set. It
turnsoutthatabypassattachmentisequivalenttoabifurcationonthelevelofcharacteristic
foliations. Hence,inordertostudycontactstructuresonΣ×[0,1]withconvexboundary,it
suffices to consider the isotopy classes of contact structures given by sequences of bypass
attachments. Inparticular,wewillstudysequencesof(overtwisted)bypassattachmentson
S
2
×[0,1],whichisthemainingredientinourproofofTheorem1.0.2.
27
5.2 OutlineoftheproofofTheorem1.0.2
Letξandξ
′
betwoovertwistedcontactstructureson M,homotopictoeachotheras2-plane
fielddistributions. OurapproachtoTheorem1.0.2hasthreemainsteps.
Step 1. Fix a triangulation T of M. Isotop ξ and ξ
′
through contact structures such that
T becomes an overtwisted contact triangulation 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 if e(ξ) = e(ξ
′
)∈ H
2
(M;Z), then one
canisotopξ andξ
′
sothattheyagreeinaneighborhoodof T
(2)
.
Step 2. We can assume that there exists a ball B
3
⊂ M such thatξ andξ
′
agree on M\ B
3
.
Taking a small Darboux ball B
3
std
⊂ B
3
, observe thatξ|
B
3 andξ
′
|
B
3 can both be realized as
attaching sequences of bypasses to B
3
std
. In section 5, we will define the notion of a stable
isotopy. Thenweshowthatbothofsequencesofbypassarestablyisotopictosomepower
of the bypass triangle attachment. Moreover, the boundary relative homotopy classes of
ξ|
B
3 andξ
′
|
B
3,measuredbyo
3
(ξ,ξ
′
)definedin(3.3.1),areuniquelydeterminedbythenum-
berofbypasstrianglesattachedaccordingtoTheorem4.2.1.
Step 3. By elementary obstruction theory, the homotopy classes of ξ|
B
3 and ξ
′
|
B
3 are not
necessarily the same, but they can at most differ by an integral multiple of the divisibility
of the Euler class of eitherξ orξ
′
. See Section 8 for the definition of the divisibility. We
show that this ambiguity can be resolved by further isotoping the contact structures in a
neighborhoodof T
(2)
. ThisfinishestheproofofTheorem1.0.2.
28
5.3 Localpropertiesofbypassattachments
Let M be an overtwisted contact 3-manifold. LetΣ⊂ M be a closed convex surface with
dividing setΓ
Σ
. For convenience, we choose a metric on M and denote M\Σ the metric
closureoftheopenmanifold M\Σ. Herewerestrictourselftothecasethateachconnected
componentof M\Σisovertwisted,butingeneralitispossiblethatallcomponentsof M\Σ
are tight even if 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 onΣ provided that the contact structure is overtwisted. This is
thecontentofthefollowinglemma.
Lemma5.3.1. Supposethat M\Σisovertwisted. Foranyadmissiblearcα⊂Σ,thereexists
abypassalongαin M\Σ. Inparticular,ifΣseparates M intotwoovertwistedcomponents,
thenthereexistsabypassalongαineachcomponent.
Proof. The technique for proving this lemma is essentially due to Etnyre-Honda [4], and
independentlyTorisu[14]. Weconstructabypass Dalongαasfollows. Let
˜
D⊂ M\Σbe
anovertwisteddisk.
Firstwepushtheinteriorofαslightlyinto M\Σwiththeendpointsofαfixedtoobtain
another Legendrian arc ˜ α, such thatα and ˜ α cobound a convex bigon Bwith tb(∂B)=−2.
Next, take a Legendrian arc γ connecting ˜ α and ∂
˜
D in the complement of Σ∪
˜
D∪ B,
namely, the two endpoints ofγ are contained in ˜ α and∂
˜
D respectively and the interior of
γ is disjoint fromΣ∪
˜
D∪ B as depicted in Figure 5.1. Suppose N(γ) γ× [−ϵ,ϵ] is a
band with the core γ×{0} identified with γ, such that the characteristic foliation is non-
singular and is given by γ×{t}, t∈ [−ϵ,ϵ]. In particular γ×{−ϵ} and γ×{ϵ} are both
Legendrian. In order to glue N(γ) to
˜
D and B smoothly without corners, we may assume,
up to a small perturbation near∂γ× [−ϵ,ϵ], that the characteristic foliation has two half-
hyperbolicsingularitiesalong∂γ×[−ϵ,ϵ]asdepictedinFigure5.2. Moreimportantly,we
29
B
˜
D
γ
α
Figure5.1: TheLegendrianarcγ connecting∂Band∂
˜
D.
assume that the characteristic foliation flows from the negative half-hyperbolic singularity
tothepositivehyperbolicsingularity. Inparticular N(γ)isnotconvex.
− +
Figure5.2: Thecharacteristicfoliationon N(γ)withcornersrounded.
Nowwecanglue
˜
D, N(γ),and Btogethersuchthatthecharacteristicfoliationmatches
asshowninFigure5.3. Tomakethesurface
˜
D∪N(γ)∪BconvexwithLegendrianboundary,
α
p
1
p
2
+ −
+
+
−
−
−
−
−
−
−
−
−
Figure5.3: Thepreferredcharacteristicfoliationon
˜
D∪N(γ)∪ B.
wenotethattherearetwohalf-elliptic-half-hyperbolicsingularities p
1
and p
2
inFigure5.3.
We perform a small perturbation in a neighborhood of p
1
and p
2
to eliminate these two
singularities so that the resulting characteristic foliation on D=
˜
D∪ N(γ)∪ B is given by
30
Figure 5.4, which is easily seen to be of Morse-Smale type. Therefore D is convex with
Legendrian boundary. The dividing set Γ on D has to separate the positive and negative
singularities and to be transverse to the characteristic foliation. SoΓ is, up to isotopy, the
half-circleasdepictedinFigure5.4asdesired,andtherefore Disabypassalongα.
α
+ +
−
−
−
−
−
−
−
−
−
Figure5.4: Thebypass Dalongα.
Wethenshowthetrivialityofthetrivialbypass,i.e.,attachingatrivialbypassdoesnot
change the isotopy class of the contact structure in a neighborhood of the convex surface.
TheproofessentiallyfollowsthelinesoftheproofofProposition4.9.7inGeiges[5]. Here
thecontactstructuremaybeeitherovertwistedortight.
Lemma5.3.2. Let(Σ×[0,1],ξ)beacontactmanifoldwiththecontactstructureξobtained
byattachingatrivialbypasson(Σ×{0},ξ|
Σ×{0}
). Thenthereexistsanothercontactstructure
˜
ξ,whichisisotopictoξ relativetotheboundary,suchthatΣ×{t}isconvexwithrespectto
˜
ξ forall t∈ [0,1].
Proof. Since this is a local problem, we may assume that Σ× [0,1] is a neighborhood
of the trivial bypass attachment. By Theorem 2.1.2, any Morse-Smale type characteristic
foliationadaptedtoΓ
Σ×{0}
canberealizedasthecharacteristicfoliationofacontactstructure
isotopictoξinaneighborhoodofΣ×{0}. Inparticular,wecanassumethatthecharacteristic
foliationonΣ×{0}looksexactlythesameasinFigure5.5(a)suchthate
−
doesnotconnect
toanynegativehyperbolicpointotherthan h
−
alongtheflowline.
31
e− h− h+
e− h− h+
(a) (b)
Figure 5.5: (a) The characteristic foliation onΣ×{0}. The trivial bypass is attached along
the Legendrian arc in dash line. (b) The characteristic foliation onΣ×{1} after attaching
the trivial bypass. Here e
±
(resp. h
±
) denote the±-elliptic (resp. ±-hyperbolic) singular
pointsofthefoliation.
LookatthecharacteristicfoliationsonΣ×{t}ast goesfrom0to1. Genericallywecan
assume that the Morse-Smale condition fails at one single level, say,Σ×{1/2}, where an
unstablesaddle-saddleconnectionhastoappearasshowninFigure5.6(a).
e− h− h+
Ω Ω
(a) (b)
Figure 5.6: (a) The characteristic foliation on Σ×{1/2}, where a saddle-saddle connect
from h
−
to h
+
exists. The region Ω contains exactly two singular points{e
−
,h
−
} which
are in elimination position. (b) The nonsingular characteristic foliation on Ω after the
elimination.
LetΩ⊂Σ×{1/2} be an open neighborhood of the flow line from h
−
to e
−
as depicted
in Figure 5.6(a). Observe that the characteristic foliation insideΩ is of Morse-Smale type,
and therefore stable in the t-direction. According to the proof of Proposition 4.9.7
1
in
1
ThisisastrongerversionoftheusualEliminationLemma.
32
Geiges[5],forasmallδ> 0,thereexistsanisotopyϕ
s
:Σ×[0,1]→Σ×[0,1], s∈ [0,1],
compactlysupportedinΩ×(1/2−2δ,1/2+2δ)⊂Σ×[0,1]andϕ
0
= id,suchthat
˜
ξ= (ϕ
1
)
∗
ξ
satisfiesthefollowing:
1. The characteristic foliation on Ω×{t} with respect to
˜
ξ is isotopic to the one in
Figure5.6(b)for t∈ [1/2−δ,1/2+δ]. Inparticular,itisnonsingular.
2. Fort∈ (1/2−2δ,1/2−δ)∪(1/2+δ,1/2+2δ),ThecharacteristicfoliationonΩ×{t}
with respect to
˜
ξ is almost Morse-Smale except that there may exist a half-elliptic-
half-hyperbolicpoint.
Weremarkherethattheaboveconditionsareachievedin[5]byisotopingsurfacesΣ×{t},
t∈ [1/2−2δ,1/2+2δ]whilefixingthecontactstructureξ,butthisisequivalenttoisotoping
ξ while fixingΣ×{t}. We will switch between these two equivalent point of view again in
theproofofProposition5.4.3.
Now we can makeΣ×{t} convex for t∈ [1/2−δ,1/2+δ] because the only unstable
saddle-saddle connection is eliminated and therefore the characteristic foliation becomes
Morse-Smale. Fort< [1/2−δ,1/2+δ],theremayexisthalf-elliptic-half-hyperbolicsingu-
larpoints,butwecanaswellconstructacontactstructurerealizingthistypeofsingularity
sothateachΩ×{t}staysconvex. Hence
˜
ξ constructedaboveisasrequired.
Remark 5.3.3. Let (Σ× [0,1],ξ) be a contact manifold such that ξ|
Σ
0
= ξ|
Σ
1
andΣ×{t}
is convex for all t∈ [0,1]. IfΣ , S
1
× S
1
and ξ is tight, then it is a standard fact that ξ
is isotopic to an I-invariant contact structure relative to the boundary. However, if either
Σ= S
1
×S
1
orξ isovertwisted,thentheabovefactisnottrueanymore. Wewillstudythis
phenomenonindetailinthecasewhenΣ= S
2
andξ isovertwistedinSection6.
33
5.4 Isotopingcontactstructuresuptothe2-skeleton
WearenowreadytotakethefirstmainsteptowardstheproofofTheorem1.0.2. Sincewe
willisotopcontactstructuresskeletonbyskeleton,westartwiththefollowingdefinition.
Definition 5.4.1. Let (M,ξ) be an overtwisted contact manifold, and T be a triangulation
of M. The triangulation T is called an overtwisted contact triangulation if the following
conditionshold:
1. The1-skeletonisaLegendriangraph.
2. Each2-simplexisconvexwithLegendrianboundary.
3. Each3-simplexisanovertwistedball.
Remark 5.4.2. The overtwisted contact triangulation defined above is different from the
usual contacttriangulationwherethe3-simplexesareassumedtobetight.
ThegoalforthissectionistoprovethefollowingProposition.
Proposition 5.4.3. Let M be a closed, oriented 3-manifold with a fixed triangulation T.
Letξ andξ
′
be homotopic overtwisted contact structures on M. Then they are isotopic up
to the 2-skeleton, i.e., there exists an isotopy ϕ
t
: M→ M, t∈ [0,1], ϕ
0
= id such that
(ϕ
1
)
∗
ξ=ξ
′
inaneighborhoodof T
(2)
.
Proof. Beforewegointodetailsoftheproof,observethatifϕ
t
: M→ M,t∈ [0,1],ϕ
0
= id
isanisotopy,then(M,ϕ
1
(ξ),T)and(M,ξ,ϕ
−1
1
(T))carriesthesamecontactinformation. In
fact, we will isotop the skeletons of the triangulation T and think of them as isotopies of
contactstructures.
By a C
0
-small perturbation of the 1-skeleton T
(1)
, we can assume that T
(1)
is a Legen-
driangraphwithrespecttoξ andξ
′
. PerformingstabilizationstoedgesofT
(1)
ifnecessary,
34
we can further assume that ξ = ξ
′
in a neighborhood of T
(1)
. For each 2-simplex σ
2
in
T
(2)
, we can always stabilize the Legendrian unknot ∂σ
2
sufficiently many times so that
tb(∂σ
2
) < 0. Therefore a C
∞
-small perturbation of σ
2
relative to ∂σ
2
makes it convex
with respect toξ (resp. ξ
′
) with dividing setΓ
ξ
σ
2
(resp. Γ
ξ
′
σ
2
). BothΓ
ξ
σ
2
andΓ
ξ
′
σ
2
are proper
1-submanifolds of σ
2
and generically the endpoints are contained in the interior of the
1-simplexes. SeeFigure5.7foranexample.
In order to make T an overtwisted contact triangulation for ξ and ξ
′
, we still need to
make sure that all 3-simplexes are overtwisted. We do this forξ, and the same argument
appliestoξ
′
. Takeanovertwisteddisc Din(M,ξ). Wecanassumethat Discontainedina
3-simplexσ
3
1
. Letσ
3
2
beanother3-simplexwhichsharesa2-facewithσ
3
1
,i.e.,σ
3
1
∩σ
3
2
=σ
2
isa2-simplex. Weclaimthatbyisotopingσ
2
relativeto∂σ
2
ifnecessary,wecanmakeboth
σ
3
1
andσ
3
2
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 D in an I-invariant neighborhood of D,
denoted by D
′
. Pick an arcγ connecting D
′
toσ
2
insideσ
3
1
. Let ˜ σ
2
be another 2-simplex
obtainedbyisotopingσ
2
across D
′
alongγ,i.e., ˜ σ
2
satisfyingthefollowingconditions:
1. ∂˜ σ
2
=∂σ
2
.
2. σ
2
∪ ˜ σ
2
boundsaneighborhoodof D
′
∪γ.
3. ˜ σ
2
isconvex.
By replacing σ
2
with ˜ σ
2
, we obtain two new 3-simplexes, each of which contains an
overtwisteddiskintheinteriorasclaimed.
Now by Giroux’s flexibility theorem, it suffices to isotopξ andξ
′
so that they induce
isotopicdividingsetsoneach2-simplexrelativetoT
(1)
. Toachievethisgoal,wedefinethe
difference 2-cocycleδ by assigning to each oriented 2-simplexσ
2
an integerχ(R
+
(Γ
ξ
′
σ
2
))−
35
Figure5.7: Anexampleofthedividingsetona2-simplex.
χ(R
−
(Γ
ξ
′
σ
2
))−χ(R
+
(Γ
ξ
σ
2
))+χ(R
−
(Γ
ξ
σ
2
)). Sinceξ is homotopic toξ
′
as 2-plane fields, [δ] =
e(ξ)−e(ξ
′
)= 0∈ H
2
(M,Z). Hencethereexistsanintegral1-cocycleθsothat2dθ=δsince
theEulerclassisalwayseven.
2
Oneshouldthinkofθ asanelementin Hom(C
1
(M),Z).
Letσ
2
∈ T
(2)
be an oriented convex 2-simplex andσ
1
⊂∂σ
2
be an oriented 1-simplex
with the induced orientation. We study the effect of stabilizing the 1-simplex σ
1
to the
overtwisted contact triangulation. If we positively stabilizeσ
1
once and isotopσ
2
accord-
ingly to obtain a new 2-simplex ˜ σ
2
, then the dividing setΓ
ξ
˜ σ
2
on ˜ σ
2
is obtained fromΓ
ξ
σ
2
by adding a properly embedded arc contained in the negative region with both endpoints
ontheinteriorofσ
1
asdepictedinFigure5.8. Similarly,ifwenegativelystabilizeσ
1
once
and isotop σ
2
accordingly as before, then the dividing set on the isotoped σ
2
is obtained
fromΓ
ξ
σ
2
byaddingaproperlyembeddedarccontainedinthepositiveregionandwithboth
endpointsontheinteriorofσ
1
.
−
+ −
+
−
−
+
−
+
+
−
(a) (b)
Figure5.8: (a)Thedividingsetonσ
2
dividesitinto±-regions. Thebottomedgeisσ
1
. (b)
Onepossibledividingseton ˜ σ
2
afterpositivelystabilizingσ
1
once.
Notethatingeneral,thenewovertwistedcontacttriangulationobtainedby±-stabilizing
2
More precisely, if we fix a trivialization of TM and consider the Gauss map associated to the contact
distribution,thentheEulerclassofthecontactdistributionisexactlytwicethePoincarédualofthePontryagin
submanifoldoftheGaussmap.
36
a1-simplexσ
1
isnotunique. Infact,differentchoicesmaygivenon-isotopicdividingsets
on the isotopedσ
2
in the new triangulation. However, for our purpose, we only care about
thequantityχ(R
+
)−χ(R
−
)oneach2-simplexanditiseasytoseethatdifferentchoicesgive
the same value to this quantity. Thus we will ignore this ambiguity by arbitrarily choosing
anisotopyofthe2-simplex.
We denote the overtwisted contact triangulation obtained by±-stabilizing σ
1
once in
(M,ξ) by S
±
σ
1
(ξ). As remarked at the beginning of the proof, one should think of S
±
σ
1
(ξ) as
isotopies ofξ. It is easy to see that S
±
σ
1
(ξ) changesχ(R
+
(Γ
ξ
σ
2
))−χ(R
−
(Γ
ξ
σ
2
)) by±1 for any
2-simplexσ
2
∈ T
(2)
so thatσ
1
⊂∂σ
2
as an oriented boundary edge. The same holds forξ
′
aswell.
Nowwearguethatonecanisotopξandξ
′
sothatχ(R
+
(Γ
ξ
σ
2
))−χ(R
−
(Γ
ξ
σ
2
))=χ(R
+
(Γ
ξ
′
σ
2
))−
χ(R
−
(Γ
ξ
′
σ
2
))oneach2-simplexσ
2
. Thiscanbedoneasfollows. Foreachoriented1-simplex
σ
1
∈ T
(1)
, the 1-cocycle θ sends it to an integer n = θ(σ
1
). We perform n times the iso-
topy S
+
σ
1
(ξ) toξ and n times the isotopy S
−
σ
1
(ξ
′
) toξ
′
at the same time. If we perform such
operationtoevery1-simplexinT,itiseasytoseethatthefollowingpropertiesaresatisfied:
1. ξ=ξ
′
inaneighborhoodof T
(1)
.
2. χ(R
+
(Γ
ξ
σ
2
))−χ(R
−
(Γ
ξ
σ
2
))=χ(R
+
(Γ
ξ
′
σ
2
))−χ(R
−
(Γ
ξ
′
σ
2
)),∀σ
2
∈ T
(2)
.
ThesecondpropertyimpliesthatΓ
ξ
′
σ
2
canbeobtainedfromΓ
ξ
σ
2
byattachingasequence
of bypasses for each 2-simplex σ
2
. Recall that T is an overtwisted contact triangulation
and in particular each 3-simplex is an overtwisted ball. Hence bypasses exist along any
admissible arc inσ
2
inside any 3-simplex withσ
2
as a 2-face by Lemma 5.3.1. Therefore
by isotoping 2-simplexes through bypasses, we can assume that ξ and ξ
′
induce isotopic
dividing sets on each 2-simplex relative to its boundary. The conclusion now follows im-
mediatelyfromGiroux’sflexibilitytheorem.
37
5.5 Bypasstriangleattachmentrevisited
In this section we study the effect of attaching a bypass triangle to the contact structure, in
particular,wegiveanalternativedefinitionofthebypasstriangleattachment.
Recall that a bypass triangle attachment alongα, denoted by△
α
, is the composition of
threebypassattachmentsσ
α
∗σ
α
′∗σ
α
′′ asdepictedinFigure2.4. Howeverwhenwedefine
abypassattachmentσ
α
alongαon(Σ,Γ
Σ
),thereareseveralchoicesinvolved. Namely,we
need to choose a multicurve, i.e., a 1-submanifold ofΣ, representing the isotopy class of
Γ
Σ
, an admissible arc representing the isotopy class ofα, a neighborhood ofα whereσ
α
is
supported. Sincethespaceofchoicesofαanditsneighborhoodiscontractibleaccordingto
Theorem2.1.2,wecanneglectthisambiguity. Howeverthespaceofchoicesofmulticurves
representingΓ
Σ
is not necessarily contractible. This point will be made clear in the next
section. Fortherestofthethesis,Γ
Σ
alwaysmeansamulticurveonΣratherthanitsisotopy
class.
Remark 5.5.1. IfΣ= S
2
andΓ
Σ
= S
1
, then the space of choices of multicurve is simply-
connected since there is a unique tight contact structure in a neighborhood of S
2
up to
isotopy.
Observethat,uptoanisotopysupportedinaneighborhoodoftheadmissiblearcα,the
bypasstriangleattachmentdoesnotchangeΓ
Σ
.
Inwhatfollowswelookatbypasstriangleattachmentsalongdifferentadmissiblearcs,
whichleadstoouralternativedefinitionofthebypasstriangleattachment.
Lemma 5.5.2. Letξ
α
andξ
β
be two (overtwisted) contact structures on S
2
×[0,1], where
αandβareadmissiblearcson S
2
×{0},suchthat
1. S
2
×{0,1}isconvexwithrespecttobothξ
α
andξ
β
.
2. ξ
α
=ξ
β
inaneighborhoodof S
2
×{0}and#Γ
ξ
α
S
2
×{0}
= #Γ
ξ
β
S
2
×{0}
= 1.
38
3. ξ
α
is obtained by attaching a bypass triangle△
α
to ξ
α
|
S
2
×{0}
, and ξ
β
is obtained by
attachingabypasstriangle△
β
toξ
β
|
S
2
×{0}
.
Thenξ
α
isisotopictoξ
β
relativetotheboundary.
Proof. Uptoisotopy,thereareonlytwodifferentadmissiblearcson(S
2
×{0},ξ
α
|
S
2
×{0}
)(or,
(S
2
×{0},ξ
β
|
S
2
×{0}
)). Namely,onegivesthetrivialbypassandtheothergivestheovertwisted
bypass. Wemayassumewithoutlossofgeneralitythatαisnotisotopictoβ,andσ
α
isthe
trivial bypass andσ
β
is the overtwisted bypass. We complete the bypass triangles△
α
and
△
β
asdepictedinFigure5.9.
α
α
′
α
′′
β
β
′
β
′′
σα σ
α
′ σ
α
′′
σ
β
σ
β
′ σ
β
′′
Figure5.9: Bypasstriangleattachmentson S
2
.
Observethatα
′
isisotopictoβ,α
′′
isisotopictoβ
′
andbypassattachmentsalongαand
β
′′
aretrivialaccordingtoLemma5.3.2,wehavethefollowingisotopies:
△
α
=σ
α
∗σ
α
′∗σ
α
′′
≃σ
α
′∗σ
α
′′
≃σ
β
∗σ
β
′
≃σ
β
∗σ
β
′∗σ
β
′′ =△
β
.
SinceS
2
×{0,1}areconvex,wecanmakesurethattheisotopiesabovearesupportedinthe
interiorof S
2
×[0,1].
39
Definition5.5.3. Aminimalovertwistedball(B
3
,ξ
ot
)isanovertwistedballwhere∂B
3
has
atightneighborhood,andthecontactstructureξ
ot
isobtainedbyattachingabypasstriangle
tothestandardtightball(B
3
,ξ
std
).
Remark 5.5.4. By Lemma 5.5.2, the minimal overtwisted ball is well-defined even if we
donotspecifytheadmissiblearcalongwhichthebypasstriangleisattached.
Withtheabovepreparation,wecannowredefinethebypasstriangleattachmentwhich
ismoreconvenientforourpurpose. Let(M,ξ)beacontact3-manifoldwithconvexbound-
ary∂M=Σ. Identifyacollarneighborhoodof∂M withΣ×[−1,0]suchthat∂M=Σ×{0}
and the contact vector field transverse to ∂M is identified with the [−1,0]-direction. Let
α⊂ ∂M be an admissible arc along which the bypass triangle is attached. Push α into
the interior of M to obtain another admissible arc, parallel toα, contained inΣ×{−1/2},
which we still denote by α. Let N be a neighborhood of α in Σ×{−1/2}. Consider the
ball with corners N× [−2/3,−1/3]⊂ M. By rounding the corners, we get a smoothly
embedded tight ball (B
3
1
,ξ|
B
3
1
) ⊂ (M,ξ), in particular, ∂B
3
1
has a tight neighborhood in
(M,ξ). Let (B
3
2
,ξ
ot
) be a minimal overtwisted ball. We construct a new contact manifold
(M,
˜
ξ)= (M\ B
3
1
,ξ)∪
ϕ
(B
3
2
,ξ
ot
), whereϕ is an orientation-reversing diffeomorphism iden-
tifying the standard tight neighborhoods of∂B
3
1
and∂B
3
2
. It is easy to see that
˜
ξ is isotopic
tothecontactstructureobtainedbyattachingabypasstriangleto(M,ξ)alongα.
Remark 5.5.5. The uniqueness of the tight contact structure on 3-ball, due to Eliashberg,
guaranteesthatthebypasstriangleattachmentdescribedaboveiswell-defined.
Usingtheabovealternativedescriptionofthebypasstriangleattachment,weprovethe
followinggeneralizationofLemma5.5.2.
Lemma 5.5.6. Let (M,ξ) be a contact 3-manifold with convex boundary, and let α,β be
two admissible arcs on ∂M. Let ξ
α
(resp. ξ
β
) be the contact structure on M obtained by
40
attaching a bypass triangle△
α
(resp.△
β
) alongα (resp. β) to (M,ξ). Thenξ
α
is isotopic to
ξ
β
relativetotheboundary.
Proof. Without loss of generality, we can assume thatα andβ are disjoint. If not, we take
another admissible arcγ which is disjoint fromα andβ. We then show thatξ
α
≃ ξ
γ
and
ξ
β
≃ξ
γ
,whichimpliesξ
α
≃ξ
β
.
Asbefore,since∂Misconvex,wecanpushαandβslightlyintothemanifold M,which
westilldenotebyαandβ. Nowlet B
3
α
⊂ M and B
3
β
⊂ M besmoothlyembeddedtightballs
containing α and β respectively. Take a Legendrian arc τ connecting B
3
α
and B
3
β
, i.e., the
endpoints ofτ are contained in∂B
3
α
and∂B
3
β
, respectively, and the interior ofτ is disjoint
from B
3
α
and B
3
β
. Moreover,wecanassumethatτ∩∂B
3
α
∈Γ
∂B
3
α
andτ∩∂B
3
β
∈Γ
∂B
3
β
. LetN(τ)
be a closed tubular neighborhood ofτ. By rounding the corners of B
3
α
∪ B
3
β
∪ N(τ), 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 (B
3
,ξ
α
|
B
3) and (B
3
,ξ
β
|
B
3)
areisotopic,relativetotheboundary,tothecontactboundarysums(B
3
,ξ
ot
)#
b
(B
3
,ξ
std
)and
(B
3
,ξ
std
)#
b
(B
3
,ξ
ot
), respectively. Hence both are isotopic to the minimal overtwisted ball.
One simply extends the isotopy by identity to the rest of M to conclude that ξ
α
≃ ξ
β
on
M.
According to Lemma 5.5.6, the isotopy class of the contact structure obtained by at-
taching 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. Thisisthecontentofthefollowingcorollary:
Corollary 5.5.7. Let (M,ξ) be contact 3-manifold with convex boundary, and α be an
admissiblearcon∂M. Thenξ∗σ
α
∗△≃ξ∗△∗σ
α
.
41
Proof. By Lemma 5.5.6, we can arbitrarily choose an admissible arcβ⊂∂M along which
the bypass triangle△ is attached. In particular, we require thatβ is disjoint fromα. Hence
a neighborhood of β where△
β
is supported in is also disjoint from α. Thus we have the
followingisotopies:
ξ∗σ
α
∗△≃ξ∗σ
α
∗△
β
≃ξ∗△
β
∗σ
α
≃ξ∗△∗σ
α
.
whichprovesthecommutativity.
Corollary5.5.8. Let(S
2
×[0,1],ξ)beacontactmanifoldwithconvexboundary,whereξis
isotopictoasequenceofbypassattachmentsσ
1
∗σ
2
∗···∗σ
n
,i.e.,thereexists0= t
0
< t
1
<
···< t
n
= 1suchthatS
2
×{t
i
}areconvexfor0≤ i≤ nandS
2
×[t
i−1
,t
i
]withtherestricted
contact structure is isotopic to the bypass attachment σ
i
. Then ξ∗△ is isotopic to ξ
k
for
0≤ k≤ n, whereξ
k
is the contact structure isotopic to a sequence of bypass attachments
σ
1
∗···∗σ
k
∗△∗σ
k+1
···∗σ
n
.
Proof. ThisisaniteratedapplicationofCorollary5.5.7.
However, observe that subtracting a bypass triangle is in general not well-defined. So
weneedthefollowingdefinition.
Definition5.5.9. Twocontactstructuresξ andξ
′
onS
2
×[0,1]arestablyisotopic,denoted
byξ∼ξ
′
,iftheybecomeisotopicafterattachingfinitelymanybypasstrianglestoS
2
×{1}
simultaneously,i.e.,ξ∗△
n
≃ξ
′
∗△
n
forsome n∈N.
42
5.6 Overtwistedcontactstructureson S
2
×[0,1]inducedbyisotopies
Let ξ 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 ξ can be represented by a sequence of bypass
attachments. More precisely, by Theorem 2.2.1, there exists an increasing sequence 0 =
t
0
< t
1
<··· < t
n
= 1 such that S
2
×{t
i
} is convex and ξ|
S
2
×[t
i−1
,t
i
]
is isotopic to a bypass
attachmentσ
i
for i = 1,··· ,n. In this section, we consider a special class of overtwisted
contact structures on S
2
×[0,1] such that S
2
×{t} is convex for t∈ [0,1], in other words,
thereisnobypassattached.
Letξ
0
bean I-invariantcontactstructureonS
2
×[0,1]withdividingsetΓ
0
onS
2
×{0}.
Let ϕ
t
: S
2
→ S
2
, t∈ [0,1], be an isotopy such that ϕ
0
= id. We define a new contact
structureξ
Γ
0
,Φ
= Φ
∗
(ξ
0
) on S
2
× [0,1], whereΦ : S
2
× [0,1]→ S
2
× [0,1] is defined by
(x,t)7→ (ϕ
t
(x),t). Observe that S
2
×{t} is convex with respect to ξ
Γ
0
,Φ
for all t∈ [0,1]
by construction. Hence we get a smooth family of dividing sets Γ
S
2
×{t}
for t ∈ [0,1].
Conversely, a smooth family of dividing sets Γ
S
2
×{t}
, t ∈ [0,1] defines a unique contact
structure on S
2
× [0,1], which is isotopic to ξ
Γ
0
,Φ
constructed above for some isotopyϕ
t
,
t∈ [0,1]. In practice, it is usually easier to keep track of the dividing sets rather than the
isotopy.
Definition5.6.1. Acontactstructureξ onS
2
×[0,1]isinducedby an isotopyifS
2
×{t}is
convexforallt∈ [0,1],or,equivalently,thereexistsanisotopyΦ : S
2
×[0,1]→ S
2
×[0,1]
suchthatξ isisotopictoξ
Γ
0
,Φ
asconstructedabove.
Itisconvenienttohavethefollowinglemma.
Lemma 5.6.2. Letξ,ξ
′
be two contact structures on S
2
×[0,1] induced by isotopies and
letΓ
t
,Γ
′
t
bedividingsetsonS
2
×{t},0≤ t≤ 1,withrespecttoξ,ξ
′
respectively. IfΓ
0
=Γ
′
0
,
Γ
1
=Γ
′
1
and there exists a path of smooth families of multicurvesΓ
s
t
, 0≤ s≤ 1 satisfying
thefollowing:
43
1. Γ
s
t
is a multicurve, i.e., a finite disjoint union of simple closed curves, contained in
S
2
×{t}for0≤ s≤ 1,0≤ t≤ 1.
2. Γ
0
t
=Γ
t
,Γ
1
t
=Γ
′
t
for0≤ t≤ 1,
3. Γ
s
0
=Γ
0
,Γ
s
1
=Γ
1
for0≤ s≤ 1.
thenξ isisotopictoξ
′
relativetotheboundary.
Proof. ByGiroux’sflexibilitytheorem, thepathΓ
s
t
, 0≤ s≤ 1of multicurves determines a
path of contact structuresξ
s
on S
2
×[0,1] such thatξ
0
=ξ,ξ
1
=ξ
′
. Henceξ is isotopic to
ξ
′
relativetotheboundarybyGray’sstabilitytheorem.
We first consider a bypass attachment to the contact structures on S
2
× [0,1] induced
byanisotopy.
Lemma 5.6.3. Let ξ
Γ
0
,Φ
be a contact structure on S
2
× [0,1/2] induced by an isotopy
ϕ
t
: S
2
→ S
2
, t ∈ [0,1/2], and (S
2
× [1/2,1],σ
α
) be a bypass attachment along an
admissible arc α⊂ S
2
×{1/2}. Then there exists an admissible arc ˜ α⊂ S
2
×{0} such
that (S
2
×[0,1],ξ
Γ
0
,Φ
∗σ
α
) is isotopic, relative to the boundary, to (S
2
×[0,1],σ
˜ α
∗ξ
Γ
′
0
,Φ
),
whereΓ
′
0
isthedividingsetobtainedbyattachingabypassalongαtoΓ
0
.
Proof. We basically re-foliate the contact manifold (S
2
×[0,1],ξ
Γ
0
,Φ
∗σ
α
). Recall thatσ
α
attaches a bypass D on S
2
×{1/2} so that∂D=α∪β is the union of two Legendrian arcs,
wheretb(α)=−1,tb(β)= 0. WeextendDtoanewbypass
˜
DonS
2
×{0}throughtheisotopy
ϕ
t
: S
2
→ S
2
,t∈ [0,1/2],bydefining
˜
D= D∪Φ(˜ α×[0,1/2]),where ˜ α=ϕ
−1
1/2
(α)⊂ S
2
×{0}
isthenewadmissiblearcalongwhich
˜
Disattached,andΦ : S
2
×[0,1/2]→ S
2
×[0,1/2]
is defined by (x,t)7→ (ϕ
t
(x),t). By attaching the new bypass
˜
D on S
2
×{0}, observe that
the rest of S
2
×[0,1] can be foliated by convex surfaces, and the contact structure is also
inducedbyΦ. Henceξ
Γ
0
,Φ
∗σ
α
isisotopictoσ
˜ α
∗ξ
Γ
′
0
,Φ
asdesired.
44
Definition 5.6.4. The admissible arc ˜ α constructed in Lemma 5.6.3 is called a push-down
ofα. Conversely,wecallαa pull-upof ˜ α.
The rest of this section is rather technical and can be skipped at the first time reading.
TheonlyresultneededforourproofofTheorem1.0.2isProposition5.6.15.
We consider a subclass of the contact structures on S
2
× [0,1] induced by isotopies
which we will be mainly interested in. Fix a metric on S
2
. Without loss of generality, we
assumethatthereexistsasmalldiskD
2
ϵ
(y)⊂ S
2
centeredatyofradiusϵ andacodimension
0 submanifold
˜
Γ
S
2
×{0}
ofΓ
S
2
×{0}
such that
˜
Γ
S
2
×{0}
⊂ D
2
ϵ
(y) and D
2
ϵ
(y)∩Γ
S
2
×{0}
=
˜
Γ
S
2
×{0}
. Let
γ(s)⊂ S
2
×{0}, s∈ [0,1]beanembeddedorientedloopsuchthatγ(0)=γ(1)= y. LetA(γ)
be an annulus neighborhood ofγ containing D
2
ϵ
(y) and disjoint from other components of
the dividing set as depicted in Figure 5.10. We define an isotopyϕ
t
: S
2
→ S
2
, t∈ [0,1],
supported in A(γ) which parallel transports D
2
ϵ
(y) along γ in A(γ). More precisely, by
applying the stereographic projection map, we can identify A(γ) with an annulus in R
2
.
Then the parallel transportation is given by an affine mapϕ
t
: x7→ x+γ(t)−γ(0) for any
x∈ D
2
ϵ
(y)and t∈ [0,1].
˜
Γ Γ\
˜
Γ Γ\
˜
Γ
γ
A(γ)
Figure5.10: Contactstructureinducedbyanisotopy.
Definition 5.6.5. With the small disk D
2
ϵ
(y)⊃
˜
Γ
S
2
×{0}
such that
˜
Γ
S
2
×{0}
∩∂D
2
ϵ
(y) =∅, the
annulus A(γ)⊃ γ and the isotopyϕ
t
: S
2
→ S
2
chosen as above, we say that the contact
structureξ
Γ
S
2
×{0}
,Φ
on S
2
× [0,1] is induced by a pure braid of the dividing set, whereΦ :
S
2
×[0,1]→ S
2
×[0,1] is induced byϕ
t
as before. We denote such contact structures by
ξ
Γ,Φ(
˜
Γ,D
2
ϵ
(y),γ)
. Whenthereisnoconfusion,wealsoabbreviateitbyξ˜
Γ,D
2
ϵ
,γ
.
45
Remark 5.6.6. For any simply connected region D⊂ S
2
×{0} containing
˜
Γ
S
2
×{0}
, one can
isotopsothatDbecomesarounddiskwithsmallradiusasrequiredinDefinition5.6.5. The
isotopyclass of the contactstructure on S
2
×[0,1] induced by a pure braid of the dividing
setonlydependsonthechoiceof D⊃
˜
Γ
S
2
×{0}
andtheisotopyclassofγ.
Remark 5.6.7. If ξ is a contact structure on S
2
× [0,1] induced by a pure braid of the
dividingset,thenΓ
S
2
×{0}
=Γ
S
2
×{1}
.
Before we give a complete classification of contact structures on S
2
× [0,1] induced
by pure braids of the dividing set, we make a digression into the study of its homotopy
classes using the generalized version of the Pontryagin-Thom construction for manifolds
withboundarydiscussedinSection3.2.
We can always assume that the isotopy ϕ
t
(
˜
Γ,D
2
ϵ
(y),γ) : S
2
→ S
2
, t ∈ [0,1], dis-
cussed in Definition 5.6.5 is supported in a disk D
2
⊂ S
2
. Trivialize the tangent bundle
of D
2
× [0,1] by embedding it intoR
3
so that D
2
is contained in the xy-plane. Consider
the Gauss map G : (D
2
× [0,1],ξ˜
Γ,D
2
ϵ
,γ
)→ S
2
. By Lemma 5.6.2, we can assume with-
out loss of generality that the dividing set is a disjoint union of round circles in D
2
×{t}
for all 0≤ t≤ 1, and p = (1,0,0)∈ S
2
⊂ R
3
is a regular value. Suppose the number
of connected components #Γ
D
2
×{0}
= m, then the Pontryagin submanifoldB = G
−1
(p) is
an oriented framed monotone braid in the sense thatB transversely intersects D
2
×{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 fram-
ing, and consequently the self-linking number ofB is exactly writhe(B). It follows from
the generalized Pontryagin-Thom construction that the homotopy class of a contact struc-
ture on D
2
× [0,1] relative to the boundary is uniquely determined by the relative framed
cobordism class of its Pontryagin submanifoldB, and hence is uniquely determined by
writhe(B) since H
1
(D
2
× [0,1],∂(D
2
× [0,1]);Z) = 0. One may think of writhe(B) as a
46
relative version of the Hopf invariant associated with boundary relative homotopy classes
ofmaps D
2
×[0,1]≃ B
3
→ S
2
.
Example 5.6.8. IfΓ
D
2
×{0}
is the disjoint union of two isolated circles, and
˜
Γ
D
2
×{0}
= S
1
⊂
D
2
ϵ
(y) is the circle on the left as depicted in Figure 5.11. The isotopyϕ
t
parallel transports
D
2
ϵ
(y) along the oriented loopγ. We compute the homotopy class of the contact structure
ξ˜
Γ,D
2
ϵ
,γ
.
(a) (b)
p
1
p
2
p
1
p
2
p
1
p
2
p
1
p
2
+
+
− −
− −
D
2
×[0,1]
γ
Figure 5.11: (a) The contact structure on S
2
×[0,1] induced by a full twist of the dividing
circles, where{p
1
,p
2
} are pre-images of the regular value p = (1,0,0) ∈ S
2
. (b) The
orientedbraidwiththeblackboardframingBasthePontryaginsubmanifold.
According to the Pontryagin-Thom construction, since writhe(B) =−2, the homotopy
class ofξ˜
Γ,D
2
ϵ
,γ
is in general different from the I-invariant contact structure, and the differ-
ence is measuredby decreasingthe 3-dimensional obstruction class o
3
by 2.
3
Example5.6.9. IfΓ
D
2
×{0}
is the disjoint union of three circles, and
˜
Γ
D
2
×{0}
= S
1
⊂ D
2
ϵ
(y) is
the circle on the left as depicted in Figure 5.12. The isotopy ϕ
t
parallel transports D
2
ϵ
(y)
along the oriented loopγ. Wecompute the homotopy class of the contact structureξ˜
Γ,D
2
ϵ
,γ
.
In this case, one computes that writhe(B) = 0, hence ξ˜
Γ,D
2
ϵ
,γ
is homotopic to the I-
invariantcontact structure.
3
However,ifthedivisibilityoftheEulerclassis2,thenϕ
t
givesacontactstructurewhichishomotopicto
the I-invariantcontactstructure. WewilldiscussthedivisibilityoftheEulerclassindetailinSection5.7.
47
p
1
p
2
p
3
p
1
p
2
p
3
(a) (b)
D
2
×[0,1]
p
1
p
2
p
3
p
1
p
2
p
3
+ − + −
+ − + −
γ
Figure 5.12: (a) A braiding by a full twist of the left-hand side dividing circle along γ,
where{p
1
,p
2
,p
3
}= G
−1
(p)isthepre-imageoftheregularvalue p= (1,0,0)∈ S
2
. (b)The
orientedframedbraidBasthePontryaginsubmanifold.
Now we are ready to classify the contact structures induced by pure braids of the di-
viding set up to stable isotopy in the sense of Definition 5.6.5. One goal is to establish an
isotopy equivalence relation between a pure braid of the dividing set and the bypass trian-
gle attachment. To start with, we consider the contact structures induced by two special
pure braids of the dividing set as depicted in Figure 5.13. In Figure 5.13(a), the dividing
set
˜
Γ⊂ D
2
ϵ
(y) is a single circle, and the dividing set contained in the disk bounded by γ
and disjoint from
˜
Γ is also a single circle. In Figure 5.13(b), the dividing set
˜
Γ⊂ D
2
ϵ
(y)
consists of m isolated circles nested in another circle, and the dividing set contained in the
diskboundedbyγanddisjointfrom
˜
Γconsistsofnisolatedcirclesnestedinanothercircle.
Wealsoassumethateithermornisnotzero. Fortechnicalreasons,itisconvenienttohave
thefollowingdefinitions.
Definition 5.6.10. Given two disjoint embedded circlesγ,γ
′
⊂ D
2
, we sayγ < γ
′
if and
onlyifγ iscontainedinthediskboundedbyγ
′
.
Definition5.6.11. LetΓ⊂ D
2
be a finite disjoint union of embedded circles. The depth of
Γ is the maximum length of chainsγ
1
< γ
2
<··· < γ
r
, whereγ
i
⊂ Γ is a single circle for
anyi∈{1,2,··· ,r}.
Observe that the depth of the dividing set in Figure 5.13(a) is 1, and the depth of the
48
dividing set in Figure 5.13(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
thesetwospecialcases.
(a) (b)
γ γ
Γ
′
Γ
′ |{z} |{z}
m n
Figure5.13: Twospecialisotopiesofdividingsets.
Lemma5.6.12. If(S
2
×[0,1],ξ˜
Γ,D
2
ϵ
,γ
)isacontactmanifoldwithcontactstructureinduced
byapurebraidofthedividingsetwhere
˜
Γ⊂ D
2
ϵ
andγarechosenasinFigure5.13(a),then
(S
2
×[0,1],ξ˜
Γ,D
2
ϵ
,γ
)isisotopicrelativetotheboundaryto(S
2
×[0,1],△
2
),where△
2
denotes
thecontactstructureobtainedbyattachingtwobypasstriangleson(S
2
×{0},ξ˜
Γ,D
2
ϵ
,γ
|
S
2
×{0}
).
Proof. Letα be an admissible arc as depicted in Figure 5.14(b). Suppose that both bypass
trianglesareattachedalongα.
(a) (b)
γ
α
α
ξ˜
Γ,D
2
ϵ
,γ
△
α
△
α
Figure 5.14: (a) The contact structure is induced by parallel transporting
˜
Γ⊂ D
2
ϵ
alongγ.
(b)Attachingtwobypasstrianglesalongtheadmissiblearcα.
Observe that△
α
= σ
α
∗σ
α
′∗σ
α
′′, whereσ
α
,σ
α
′ andσ
α
′′ are all trivial bypass attach-
ments. Hence the contact manifold (S
2
×[0,1],△
2
α
) can be foliated by convex surfaces by
Lemma 5.3.2. In other words, it is induced by an isotopy. By Theorem 4.2.1, we know
49
that attaching two bypass triangles△
2
α
decreases o
3
by 2, i.e., o
3
(ξ,ξ∗△
2
α
) =−2. In Ex-
ample5.6.8,wecheckedbyPontryagin-Thomconstructionthatξ˜
Γ,D
2
ϵ
,γ
alsodecreaseso
3
by
2. Observe that the isotopy class 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△
2
α
is isotopicΦ˜
Γ,D
2
ϵ
,γ
in the region where both opera-
tionsaresupported. ByextendingtheisotopybyidentitytotherestofS
2
,weconcludethat
(S
2
×[0,1],ξ˜
Γ,D
2
ϵ
,γ
)isisotopicrelativetotheboundaryto(S
2
×[0,1],△
2
).
Lemma5.6.13. If(S
2
×[0,1],ξ˜
Γ,D
2
ϵ
,γ
)isacontactmanifoldwithcontactstructureinduced
by a pure braid of the dividing set where
˜
Γ⊂ D
2
ϵ
and γ are chosen as in Figure 5.13(b),
then(S
2
×[0,1],ξ˜
Γ,D
2
ϵ
,γ
)isstablyisotopicto(S
2
×[0,1],△
2(m−1)(n−1)
).
Proof. Let α ⊂ S
2
×{1} be an admissible arc as depicted in the left-hand side of Fig-
ure5.15(a). ByLemma5.6.3,if ˜ αisthepush-downofα,thenξ
Γ,Φ(
˜
Γ,D
2
ϵ
,γ)
∗σ
α
≃σ
˜ α
∗ξ
Γ
′
,Φ
,
whereΓ
′
is obtained fromΓ by attaching a bypass alongα. We remark here thatξ
Γ,Φ(
˜
Γ,D
2
ϵ
,γ)
andξ
Γ
′
,Φ
are contact structures induced by the same isotopy, but are push-forwards of dif-
ferent contact structures on S
2
× [0,1]. Choose
˜
Γ
′
⊂ D
2
′
ϵ
to be the m isolated circles on
the left andγ
′
be an oriented loop as depicted in the right-hand side of Figure 5.15(a). Let
ξ
˜
Γ
′
,D
2
′
ϵ
,γ
′ be the contact structure induced by an isotopy which parallel transports
˜
Γ
′
⊂ D
2
′
ϵ
along γ
′
. Then Lemma 5.6.2 implies that ξ
Γ
′
,Φ
is isotopic, relative to the boundary, to
ξ˜
Φ
∗ξ
˜
Γ
′
,D
2
′
ϵ
,γ
′,where
˜
Φisinducedbyanisotopythatroundstheoutmostdividingcircle. An
iteratedapplicationofLemma5.6.12impliesthatξ
˜
Γ
′
,D
2
′
ϵ
,γ
′≃△
2m(n−1)
.
We next isotop the contact structure σ
˜ α
∗ξ˜
Φ
. Consider the n isolated circles nested
in a larger circle. Let
˜
Γ
′′
⊂ D
2
′′
ϵ
be the leftmost circle among the n circles and γ
′′
be an
orientedloopasdepictedintheright-handsideofFigure5.15(b). Wepullup ˜ αthroughan
isotopy which parallel transports
˜
Γ
′′
⊂ D
2
′′
ϵ
alongγ
′′
, and observe that the pull-up of ˜ α is
isotopictoα. ByusingLemma5.6.3onemoretime,wegettheisotopyofcontactstructures
50
σ
˜ α
∗ξ˜
Φ
≃ ξ
˜
Γ
′′
,D
2
′′
ϵ
,γ
′′∗σ
α
. It is left to determine the isotopy class of the contact structure
ξ
˜
Γ
′′
,D
2
′′
ϵ
,γ
′′. Sinceγ
′′
isorientedcounterclockwise,byapplyingLemma5.6.12(n−1)times,
wegetastableisotopyξ
˜
Γ
′′
,D
2
′′
ϵ
,γ
′′∼△
2(1−n)
,i.e.,ξ
˜
Γ
′′
,D
2
′′
ϵ
,γ
′′∗△
2(n−1)
isisotopictotheI-invariant
contactstructure.
(a)
(b)
≃
≃
σ
˜ α
ξ
˜
Γ
′′
,D
2
′′
ϵ
,δ
′′
σ
α
˜ α
γ
′′
α
ξ˜
Γ,D
2
ϵ
,γ
△
α
γ
α
σ
˜ α
ξ
˜
Γ
′
,D
2
′
ϵ
,γ
′
σ
α
′∗σ
α
′′
˜ α
γ
′
˜
Γ
˜
Γ
′
˜
Γ
′′
Figure5.15: (a)Pushingdownthebypassattachmentσ
α
. (b)Pullingupthebypassattach-
mentσ
˜ α
.
To summarize what we have done so far, we have the following (stable) isotopies of
51
contactstructures:
ξ˜
Γ,D
2
ϵ
,γ
∗△
α
=ξ
Γ,Φ(
˜
Γ,D
2
ϵ
,γ)
∗σ
α
∗σ
α
′∗σ
α
′′
≃σ
˜ α
∗ξ
Γ
′
,Φ
∗σ
α
′∗σ
α
′′
≃σ
˜ α
∗ξ˜
Φ
∗ξ
˜
Γ
′
,D
2
′
ϵ
,γ
′∗σ
α
′∗σ
α
′′
≃σ
˜ α
∗ξ˜
Φ
∗△
2m(n−1)
∗σ
α
′∗σ
α
′′
≃ξ
˜
Γ
′′
,D
2
′′
ϵ
,γ
′′∗σ
α
∗△
2m(n−1)
∗σ
α
′∗σ
α
′′
∼△
2(1−n)
∗σ
α
∗△
2m(n−1)
∗σ
α
′∗σ
α
′′
≃△
2(m−1)(n−1)
∗σ
α
∗σ
α
′∗σ
α
′′
=△
2(m−1)(n−1)
∗△
α
.
Note that the third equation from the bottom is only a stable isotopy so that the (possi-
bly) negative power of the bypass triangle attachment makes sense. See Definition 5.5.9.
We will use the same trick in the proof of the following Proposition 5.6.14 without further
mentioning. Hencebydefinition,ξ˜
Γ,D
2
ϵ
,γ
isstablyisotopicto△
2(m−1)(n−1)
asdesired.
Wenowcompletelyclassifycontactstructureson S
2
×[0,1]inducedbypurebraidsof
thedividingset.
Proposition 5.6.14. If (S
2
×[0,1],ξ˜
Γ,D
2
ϵ
,γ
) is a contact manifold with contact structure in-
duced by a pure braid of the dividing set, thenξ˜
Γ,D
2
ϵ
,γ
is stably isotopic to (S
2
× [0,1],△
l
)
forsome l∈N.
Proof. Recall that
˜
Γ⊂ D
2
ϵ
is a codimension 0 submanifold ofΓ
S
2
×{0}
, andγ is an oriented
loopinthecomplementofΓ
S
2
×{0}
asinDefinition5.6.5. Let
˜
Γ
′
betheunionofcomponents
ofΓ
S
2
×{0}
contained in a disk bounded byγ and outside of A(γ). We may choose the disk
so that−γ is the oriented boundary. Since the contact structureξ˜
Γ,D
2
ϵ
,γ
is induced by a pure
52
braidofthedividingset,wehaveΓ
S
2
×{0}
=Γ
S
2
×{1}
. Hencewealsoview
˜
Γand
˜
Γ
′
asdividing
sets on S
2
×{1}. Choose pairwise disjoint admissible arcs α
1
,α
2
,··· ,α
r
,α
r+1
,··· ,α
k
on
S
2
×{1}suchthatthefollowingconditionshold:
1. α
1
,α
2
,··· ,α
r−1
are admissible arcs contained in D
2
ϵ
such that by attaching bypasses
alongthesearcs,thedepthof
˜
Γbecomesatmost2.
2. α
r
,α
r+1
,··· ,α
k
are admissible arcs contained in the disk bounded byγ and outside
of A(γ) such that by attaching bypasses along these arcs, the depth of
˜
Γ
′
becomes at
most2.
Observethatwechooseα
1
,α
2
,··· ,α
k
suchthattheisotopyclassofeachα
i
isinvariant
under thetime-1mapϕ
1
whichissupported in A(γ)\ D
2
ϵ
. Hence, by abuseof notation, we
do not distinguish α
i
and its push-down through ϕ
t
(
˜
Γ,D
2
ϵ
,γ). By Lemma 5.6.3, we have
the isotopy of contact structuresξ˜
Γ,D
2
ϵ
,γ
∗σ
α
1
∗···∗σ
α
k
≃σ
α
1
∗···∗σ
α
k
∗ξ
Φ
, whereξ
Φ
is
the contact structure induced by a finite composition of special pure braids of the dividing
set considered in Lemma 5.6.12 and Lemma 5.6.13, Therefore ξ
Φ
is stable isotopic to a
power of the bypass triangle attachment, say△
l
for some l∈ N. To summarize, we have
thefollowing(stable)isotopiesofcontactstructures,relativetotheboundary.
53
ξ˜
Γ,D
2
ϵ
,γ
∗△
k
≃ξ˜
Γ,D
2
ϵ
,γ
∗△
α
1
∗···∗△
α
k
=ξ˜
Γ,D
2
ϵ
,γ
∗(σ
α
1
∗σ
α
′
1
∗σ
α
′′
1
)∗···∗(σ
α
k
∗σ
α
′
k
∗σ
α
′′
k
)
≃ (ξ˜
Γ,D
2
ϵ
,γ
∗σ
α
1
∗···∗σ
α
k
)∗(σ
α
′
1
∗σ
α
′′
1
)∗···∗(σ
α
′
k
∗σ
α
′′
k
)
≃ (σ
α
1
∗···∗σ
α
k
∗ξ
Φ
)∗(σ
α
′
1
∗σ
α
′′
1
)∗···∗(σ
α
′
k
∗σ
α
′′
k
)
∼ (σ
α
1
∗···∗σ
α
k
∗△
l
)∗(σ
α
′
1
∗σ
α
′′
1
)∗···∗(σ
α
′
k
∗σ
α
′′
k
)
≃△
l
∗(σ
α
1
∗σ
α
′
1
∗σ
α
′′
1
)∗···∗(σ
α
k
∗σ
α
′
k
∗σ
α
′′
k
)
=△
l
∗△
k
.
Henceξ˜
Γ,D
2
ϵ
,γ
isstablyisotopicto△
l
bydefinition.
To conclude this section, we prove the following technical result which asserts that
undercertainassumptionsanduptopossiblebypasstriangleattachments,onecanseparate
twobypasses.
Proposition5.6.15. Let(S
2
,Γ)beaconvexspherewithdividingsetΓandα⊂ (S
2
,Γ)bean
admissiblearcsuchthatthebypassattachmentσ
α
increases#Γby2. Supposethat(S
2
,Γ
′
)
is the new convex sphere obtained by attachingσ
α
to (S
2
,Γ) and supposeβ⊂ (S
2
,Γ
′
) is
another admissible arc such that the bypass attachmentσ
β
decreases #Γ
′
by 2. Then there
exists an admissible arc
˜
β⊂ (S
2
,Γ) disjoint fromα, a mapΦ : S
2
× [0,1]→ S
2
× [0,1]
induced by an isotopy, and an integer l∈N such thatσ
α
∗σ
β
∼ σ
α
∗σ˜
β
∗△
l
∗ξ
Φ
relative
totheboundary.
Proof. Letδbethearcofanti-bypassattachmenttoσ
α
containedin(S
2
,Γ
′
)asdiscussedin
Remark2.3.3. ThenδintersectsΓ
′
inthreepoints{p
1
,p
2
,p
3
}asdepictedinFigure5.16(b).
Letδ
1
andδ
2
be subarcs ofδ from p
1
to p
2
and from p
2
to p
3
respectively. Observe that,
54
in order to find an admissible arc
˜
β⊂ (S
2
,Γ) which is disjoint from α and satisfy all the
conditions in the lemma, it suffices to find an admissible arc on (S
2
,Γ
′
), which we still
denoteby
˜
β,andwhichisdisjointfromδandalsosatisfiestheconditionsinthelemma. In
fact, by symmetry, we only need
˜
β to be disjoint fromδ
1
. Without loss of generality, we
canassumethatβintersectsδtransverselyandtheintersectionpointsaredifferentfrom p
1
,
p
2
and p
3
.
α
δ
p
1
p
2
p
3
(a) (b)
Figure 5.16: (a) The convex sphere (S
2
,Γ) with an admissible arc α. (b) The convex
sphere(S
2
,Γ
′
)obtainedbyattachingabypassalongα,whereδisthearcoftheanti-bypass
attachment.
Claim: Uptoisotopyandpossiblyafinitenumberofbypasstriangleattachments,onecan
arrangeso thatβ andδ
1
do not cobound a bigon Bon S
2
as depicted in Figure 5.17(a).
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 isotopβ to eliminate B. If otherwise, we
consider a minimal bigon bounded by β and δ
1
in the sense that the interior of the bigon
does not intersect with β. Take a disk D
2
ϵ
⊂ B containing all components of the dividing
set
˜
Γ in B, namely, Γ
′
∩ D
2
ϵ
=
˜
Γ andΓ
′
∩ (B\ D
2
ϵ
) =∅. By our assumption, the bypass
attachmentσ
β
decreases#Γ
′
by2,soβmustintersectΓ
′
inthreepointswhicharecontained
in three different connected components ofΓ
′
respectively. One can find an oriented loop
55
δ
1
β
γ
˜
β
(a) (b) (c)
Figure 5.17: (a) The admissible arcβ together withδ
1
bound a minimal bigon, which con-
tains other components of the dividing set in the interior. (b) Choose a disk D
2
ϵ
containing
all the dividing sets
˜
Γ in the bigon and an oriented loopγ so that it intersectsβ in exactly
one point. (c) The pull-up ofβ through the contact structureξ˜
Γ,D
2
ϵ
,γ
bounds a trivial bigon
withδ
1
.
γ : [0,1]→ S
2
\Γ
′
withγ(0)=γ(1)∈ D
2
ϵ
suchthatγ intersectsβinonepoint. Orientγ in
suchawaythatitgoesfromγ∩βtoγ(1)intheinteriorof BasdepictedinFigure5.17(b).
Suppose that Φ : S
2
× [0,1] → S
2
× [0,1] is induced by an isotopy ϕ
t
which parallel
transports D
2
ϵ
alongγ. Bypullingupthethebypassattachmentσ
β
throughξ
Γ
′
,Φ
,wegetthe
followingisotopyofcontactstructures(cf. proofofLemma5.6.13):
σ
β
∗ξ
Γ
′′
,Φ(D
2
ϵ
,γ)
≃ξ
Γ
′
,Φ(
˜
Γ,D
2
ϵ
,γ)
∗σ˜
β
whereΓ
′′
isobtainedfromΓ
′
byattachingabypassalongβ,and
˜
βisthepull-upofβwhich
isisotopictotheonedepictedinFigure5.17(c).
Since
˜
βandδ
1
coboundatrivialbigon,afurtherisotopyof
˜
βwilleliminatethebigonso
thatβ
′
doesnotintersectδ
1
inthislocalpicture. ByProposition5.6.14,thecontactstructure
ξ
Γ
′
,Φ(
˜
Γ,D
2
ϵ
,γ)
isstablyisotopicto△
n
forsomen∈N. DefineΦ
−1
: S
2
×[0,1]→ S
2
×[0,1]by
(x,t)7→ (ϕ
−1
t
(x),t), then it is easy to see thatξ
Γ
′′
,Φ(D
2
ϵ
,γ)
∗ξ
Γ
′′
,Φ
−1
(D
2
ϵ
,γ)
is isotopic, relative to
the boundary, to an I-invariant contact structure. Since we will use this trick many times,
56
wesimplywriteξ
Φ
−1 forξ
Γ
′′
,Φ
−1
(D
2
ϵ
,γ)
whenthereisnoconfusion. Tosummarize,wehave
σ
β
≃ξ
Γ
′
,Φ(
˜
Γ,D
2
ϵ
,γ)
∗σ˜
β
∗ξ
Γ
′′
,Φ
−1
(D
2
ϵ
,γ)
∼△
n
∗σ˜
β
∗ξ
Γ
′′
,Φ
−1
(D
2
ϵ
,γ)
≃σ˜
β
∗△
n
∗ξ
Γ
′′
,Φ
−1
(D
2
ϵ
,γ)
Byapplyingtheaboveargumentfinitely manytimes, we caneliminate allbigons bounded
byβandδ
1
. Hencetheclaimisproved.
Let us assume thatβ intersectsδ
1
nontrivially, andβ andδ
1
do not cobound any bigon
on S
2
. Weconsiderthefollowingtwocasesseparately.
Case 1. Suppose β does not intersect any of the three components of the dividing set
generated by the bypass attachment σ
α
. Let Γ
1
, Γ
2
and Γ
3
be the three dividing circles
whichintersectwithβ.
Γ
1
Γ
2
Γ
3
β
γ
˜
β
(a) (b) (c)
Figure 5.18: (a) The convex sphere (S
2
,Γ
′
) with an admissible arc β intersecting δ
1
in
exactlyonepoint. (b)Chooseadisk D
2
ϵ
containingΓ
1
andanorientedloopγ,alongwhich
we apply the isotopy. (c) The pull-up of β through the contact structure ξ
Γ
1
,D
2
ϵ
,γ
bounds a
trivialbigonwithδ
1
.
Ifβ intersectsδ
1
in exactly one point as depicted in Figure 5.18(a), then we choose a
disk D
2
ϵ
⊃ Γ
1
and an oriented loopγ in the complement of the dividing set as depicted in
57
Figure5.18(b)suchthatσ
β
≃ξ
Γ
′
,Φ(Γ
1
,D
2
ϵ
,γ)
∗σ˜
β
∗ξ
Φ
−1∼△
m
∗σ˜
β
∗ξ
Φ
−1 byargumentsasbefore
for some m∈N, where
˜
β intersectsδ
1
in exactly two points and cobound a trivial bigon as
depictedinFigure5.18(c). Henceanobviousfurtherisotopyof
˜
βmakesitdisjointfromδ
1
asdesired.
If β intersects δ
1
in more than one point, we orient β so that it starts from the point
q=β∩Γ
1
asdepictedinFigure5.19(a). Letq
1
andq
2
bethefirstandthesecondintersection
points ofβ withδ
1
respectively. Note that since we assumeβ andδ
1
do not cobound any
bigon, there is no more intersection point β∩δ
1
along δ
1
between q
1
and q
2
. Let
−−→
qq
1
,
−−→
q
1
q and
−−−→
q
1
q
2
be oriented subarcs of β and
−−−→
q
2
q
1
be an oriented subarc of δ
1
. We obtain a
closed, oriented (but not embedded) loopγ =
−−→
qq
1
∪
−−−→
q
1
q
2
∪
−−−→
q
2
q
1
∪
−−→
q
1
q by gluing the arcs
together. To apply Proposition 5.6.14 in this case, we take an embedded loop close toγ as
depicted in Figure 5.19(b), which we still denote byγ. Let D
2
ϵ
be a small disk containing
Γ
1
as usual. Again by pulling up the bypass attachment σ
β
through ξ
Γ
′
,Φ(Γ
1
,D
2
ϵ
,γ)
, we have
(stable)isotopiesofcontactstructuresσ
β
≃ξ
Γ
′
,Φ(Γ
1
,D
2
ϵ
,γ)
∗σ˜
β
∗ξ
Φ
−1∼△
r
∗σ˜
β
∗ξ
Φ
−1 forsome
r∈ N, where
˜
β andδ
1
bound a trivial bigon. Hence an obvious further isotopy eliminates
the trivial bigon and decreases #(β∩δ
1
) by 2. By applying the above argument finitely
many times, we can reduce to the case whereβ intersectsδ
1
in exactly one point, but we
havealreadysolvedtheproblemin this case. Weconclude that under the hypothesisat the
beginningofthiscase,thereexistsa
˜
βdisjointwithδ
1
suchthatσ
α
∗σ
β
∼σ
α
∗σ˜
β
∗△
l
∗ξ
Φ
forsomeisotopyΦandaninteger l∈N.
Case2. Supposeβnontriviallyintersectstheunionofthethreecomponentsofthedividing
setgeneratedbythebypassattachmentσ
α
. Withoutlossofgenerality,wepickanintersec-
tionpointr asdepictedinFigure5.20(a). Orientβsothatitstartsfromr. Letr
1
bethefirst
intersection point ofβ andδ
1
. Thenβ,δ
1
andΓ
′
bound a triangle△rr
1
p
1
. By the assump-
tionthatthereexistsnobigonboundedbyβandδ
1
,theinteriorofthetriangle△rr
1
p
1
does
not intersect with β. If the interior of the triangle△rr
1
p
1
contains no components of the
58
Γ
1
q
q
1
q
2
β
γ
β
′
(a) (b) (c)
Figure 5.19: (a) The convex sphere (S
2
,Γ
′
) with an admissible arcβ intersectingδ
1
in at
least two points, say, q
1
and q
2
. (b) The embedded, oriented loop γ approximating the
broken loop ⃗ qq
1
∪ ⃗ q
1
q
2
∪ ⃗ q
2
q
1
∪ ⃗ q
1
q. (c) The pull-up of β through the contact structure
ξ
Γ
1
,D
2
ϵ
,γ
boundsatrivialbigonwithδ
1
.
r
p
1
r
1
γ β
˜
β
(a) (b) (c)
Figure 5.20: (a) The admissible arc β, the dividing set Γ
′
and δ
1
cobound a topological
triangle△rr
1
p
1
, which may contain other components of the dividing set in the interior.
(b) Choose the disk D
2
ϵ
to contain all the components of the dividing set in the topological
triangle△rr
1
p
1
, and an oriented loopγ which intersectsβ in exactly one point. (c) By ap-
plyingtheisotopyalongγ,theadmissiblearcβbecomesβ
′
whichboundsatrivialtriangle
withthedividingsetandδ
1
.
dividing set, then it is easy to isotopβ so that #(β∩δ
1
) decreases by 1. If otherwise, take
a small disk D
2
ϵ
⊂△rr
1
p
1
containing all components of the dividing set
˜
Γ in△rr
1
p
1
, i.e.,
△rr
1
p
1
\D
2
ϵ
doesnotintersectwiththedividingsetΓ
′
. Letγ beanorientedloopbasedata
pointinD
2
ϵ
whichdoesnotintersectwiththedividingset,andintersectsβexactlyonce. By
pullingupthebypassattachmentσ
β
throughξ
Φ(
˜
Γ,D
2
ϵ
,γ)
,wehave(stable)isotopiesofcontact
structuresσ
β
≃ ξ
Γ
′
,Φ(
˜
Γ,D
2
ϵ
,γ)
∗σ˜
β
∗ξ
Φ
−1∼ σ˜
β
∗△
n
∗ξ
Φ
−1 so that
˜
β,δ
1
andΓ
′
bound a trivial
triangleinthesensethattheinteriorofthetriangledoesnotintersectwiththedividingset.
Hencewecanfurtherisotop
˜
βtoeliminatethetrivialtriangleandhencedecrease#(
˜
β∩δ
1
)
59
by 1. By applying such isotopies finitely many times, we get an admissible arc
˜
β such that
#(
˜
β∩δ
1
)= 0andsatisfyalltheconditionsoftheproposition.
5.7 Classificationofovertwistedcontactstructureson S
2
×[0,1]
We have established enough techniques to classify overtwisted contact structures on S
2
×
[0,1].
Proposition 5.7.1. Letξ be an overtwisted contact structure on S
2
×[0,1] such that S
2
×
{0,1}isconvexwithΓ
S
2
×{0}
=Γ
S
2
×{1}
= S
1
. Thenξ∼△
n
forsomen∈N,where△
n
denotes
thecontactstructureonS
2
×[0,1]obtainedbyattachingnbypasstrianglestoS
2
×{0}with
thestandardtightneighborhood.
Proof. By Giroux’s criterion of tightness, both S
2
×{0} and S
2
×{1} have neighborhoods
which are tight. Take an increasing sequence 0 = t
0
< t
1
<··· < t
n
= 1 such that ξ is
isotopic to a sequence of bypass attachmentsσ
α
0
∗σ
α
1
∗···∗σ
α
n−1
, whereα
i
⊂ S
2
×{t
i
}
are admissible arcs along which a bypass is attached. Define the complexity of a bypass
sequencetobe c= max
0≤i≤n
#Γ
S
2
×{t
i
}
. The idea isto showthat if c> 3, then we can always
decrease cby2byisotopingthebypasssequenceandsuitablyattachingbypasstriangles.
To achieve this goal, we divide the admissible arcs on (S
2
,Γ) into four types (I), (II),
(III) and (IV), according to the number of components ofΓ intersecting the admissible arc
asdepictedinFigure5.21,whereweonlydrawthedividingsetwhichintersectstheadmis-
sible arc. Observe that bypass attachment of type (I) increases #Γ by 2, bypass attachment
oftype(II)and(III)donotchange#Γ,andbypassattachmentoftype(IV)decreases#Γby
2. Hence the complexity of a sequence of bypass attachments changes only if the types of
bypassesinthesequencechange. ByrepeatedapplicationofLemma5.6.3,wemayassume
that contact structures induced by isotopies are contained in a neighborhood of S
2
×{1}.
60
By assumption, S
2
×{1} has a tight neighborhood. Hence according to Remark 5.5.1, we
shallonlyconsidersequencesofbypassattachmentsmodulocontactstructuresinducedby
isotopies.
α
α
α α
(I) (II) (III) (IV)
Figure5.21: Fourtypesofadmissiblearcsαon(S
2
,Γ).
Claim1: Wecanisotopthesequenceofbypassattachmentssuchthatonlybypassesoftype
(I)and(IV)appear.
To prove the claim, we first show that a bypass attachment of type (III) can be elimi-
nated. Take an admissible arc α of type (III). If the bypass attachment along α is trivial,
then by Lemma 5.3.2, the bypass attachmentσ
α
is induced by an isotopy. Otherwise there
exists an admissible arc β disjoint from α as depicted in Figure 5.22(a)
4
such that if one
attaches a bypass along α, followed by a bypass attached along β, then the later bypass
attachmentistrivial.
α
β
α
σ
β
α
β
α
σ
β
(a) (b)
Figure5.22: Addingtrivialbypasses.
4
Inliterature,wesayβisobtainedfromαby left rotation.
61
Bythedisjointnessofadmissiblearcsαandβ,wegetthefollowingisotopiesofcontact
structures,
σ
α
≃σ
α
∗σ
β
≃σ
β
∗σ
α
.
Observethatσ
β
∗σ
α
isacompositionoftype(I)andtype(IV)bypassattachments. Hence
a finite number of such isotopies will eliminate all bypass attachments of type (III) in a
sequence.
Similarly suppose that σ
α
is the bypass attachment of type (II) in a sequence and is
nontrivial. Then there must exist other components of the dividing set as shown in Fig-
ure5.22(b). ChooseanadmissiblearcβdisjointfromαasdepictedinFigure5.22(b)such
that if one attaches a bypass alongα, followed by a bypass attached alongβ, then the later
bypass attachment is trivial. By the disjointness of α and β again, we get the following
isotopiesofcontactstructures:
σ
α
≃σ
α
∗σ
β
≃σ
β
∗σ
α
.
Observe thatσ
β
∗σ
α
isa composition of bypass attachments both of type (III), hence by a
further isotopy will turnσ
α
into a composition of bypass attachments of type (I) and (IV).
Afinitenumberofsuchisotopieswilleliminatebypassesoftype(II).Theclaimfollows.
From now on, we assume that any bypass attachment in σ
α
0
∗σ
α
1
∗···∗σ
α
n−1
either
increasesordecreases#Γby2.
Assume that the complexity of the bypass sequence is achieved at level S
2
×{t
r
} for
62
some r∈{0,1,··· ,n} and is at least 5, i.e., #Γ
S
2
×{t
r
}
= c≥ 5. Then it is easy to see that
σ
α
r−1
is type (I) andσ
α
r
is type (IV). By Proposition 5.6.15, we can always assume thatα
r
isdisjointfromα
r−1
modulofinitelymanybypasstriangleattachments. Hencewecanview
bothα
r−1
andα
r
as admissible arcs on S
2
×{t
r−1
}. To finish the proof of the proposition, it
sufficestoprovethefollowingclaim.
Claim 2: We can isotop the composition of bypass attachments σ
α
r−1
∗σ
α
r
such that the
localmaximumof#Γat S
2
×{t
r
}decreasesbyatleast2.
To prove the claim, letγ⊂Γ
S
2
×{t
r−1
}
be the dividing circle which nontrivially intersects
α
r−1
. Wedoacase-by-caseanalysisdependingonthenumberofpointsα
r
intersectingwith
γ.
Case 1: Ifα
r
intersectsγ in at most one point, then one easily check that by applying iso-
topyσ
α
r−1
∗σ
α
r
≃σ
α
r
∗σ
α
r−1
tothesequenceofbypassattachments,#Γ
S
2
×{t
r
}
decreasesby4.
Case 2: If α
r
intersects γ in exactly two points, then once again we apply the isotopy
σ
α
r−1
∗σ
α
r
≃σ
α
r
∗σ
α
r−1
tothesequenceofbypassattachments. Nowobservethatσ
α
r
∗σ
α
r−1
isacompositionofbypassattachmentsoftype(III).Intheproofoftheclaimabove,wesee
thatanybypassattachmentoftype(III)isisotopictoacompositionofabypassattachment
of type (IV) followed by a bypass attachment of type (I). Such an isotopy also decreases
thelocalmaximumof#Γby4.
Case 3: Ifα
r
also intersectsγ inthree points, we consider a disk D bounded byγ andα
r−1
asdepictedinFigure5.23(a). IfDcontainsnocomponentofthedividingsetintheinterior,
then σ
α
r−1
∗σ
α
r
is isotopic to a bypass triangle attachment, more precisely, there exists a
trivial bypass along an admissible arcδ on S
2
×{t
r
} such thatσ
α
r−1
∗σ
α
r
∗σ
δ
is a bypass
63
triangle attachment alongα
r−1
. Suppose D contains at least one connected component of
the dividing set. Letβ be an admissible arc on S
2
×{t
r−1
} disjoint fromα
r−1
andα
r
such
thatitintersectsγintwopointsandthedividingsetcontainedin Dinonepointasdepicted
inFigure5.23(b).
α
r−1
α
r−1
αr αr
D
γ γ
β
(a) (b)
Figure5.23: Addingatrivialbypasstodecreasethecomplexity.
We have the following isotopies of contact structures due to Lemma 5.5.6 and the dis-
jointnessofadmissiblearcs:
σ
α
r−1
∗σ
α
r
∗△≃σ
α
r−1
∗σ
α
r
∗△
β
=σ
α
r−1
∗σ
α
r
∗σ
β
∗σ
β
′∗σ
β
′′
≃σ
β
∗σ
α
r−1
∗σ
α
r
∗σ
β
′∗σ
β
′′
One can check that the last five bypass attachments above are all of type (III). Hence
we can further isotop as before to eliminate type (III) bypass attachments to decrease the
localmaximumof#Γby2.
To summarize, we have proved that any sequence of bypass attachments σ
α
0
∗σ
α
1
∗
···∗σ
α
n−1
onS
2
×[0,1]isstablyisotopictoanothersequenceofbypassattachmentswhose
complexityisatmost3,whichisclearlyisotopictoapowerofbypasstriangleattachments.
Thusthepropositionisproved.
64
5.8 ProofofTheorem1.0.2
NowwearereadytofinishtheproofofTheorem1.0.2.
Proof of Theorem 1.0.2. By Proposition 5.4.3, we can isotopξ andξ
′
so that they agree in
a neighborhood of the 2-skeleton. Without loss of generality, we can furthermore assume
thatthereexistsanembeddedclosedball B
3
⊂ M suchthat
1. ∂B
3
isconvexandhasatightneighborhoodin M withrespecttobothξ andξ
′
.
2. ξ=ξ
′
in M\ B
3
.
3. Therestrictionofξ andξ
′
to M\ B
3
andto B
3
areallovertwisted.
Take a small ball B
3
ϵ
⊂ B
3
in a Darboux chart so that bothξ|
B
3
ϵ
andξ
′
|
B
3
ϵ
are tight. We
identify B
3
\ B
3
ϵ
with S
2
×[0,1] and represent the contact structuresξ|
B
3
\B
3
ϵ
andξ
′
|
B
3
\B
3
ϵ
by
two sequences of bypass attachments. By Proposition 5.7.1, both ξ|
B
3
\B
3
ϵ
and ξ
′
|
B
3
\B
3
ϵ
are
stably isotopic to some power of the bypass triangle attachment, in other words, there are
isotopiesofcontactstructuresξ|
B
3
\B
3
ϵ
∗△
r
≃△
n+r
andξ
′
|
B
3
\B
3
ϵ
∗△
s
≃△
m+s
forsomen,m,r,s∈
N. By assumption, the restriction of ξ and ξ
′
to M\ B
3
are overtwisted, so there exist
bypass triangle attachments along any admissible arc on ∂B
3
according to Lemma 5.3.1.
By simultaneously attaching sufficiently many bypass triangles to ξ|
B
3
\B
3
ϵ
and ξ
′
|
B
3
\B
3
ϵ
, we
canfurtherassumethatξ|
B
3
\B
3
ϵ
≃△
n
,ξ
′
|
B
3
\B
3
ϵ
≃△
m
andξ=ξ
′
on M\ B
3
.
Letd bethelargestintegersuchthattheEulerclasse(ξ)= e(ξ
′
)∈ H
2
(M;Z)dividedby
disstillanintegralclass. SuchadisknownasthedivisibilityoftheEulerclass. Combining
some elementary obstruction theory with Theorem 4.2.1, we have d|(m−n). To complete
the proof of the theorem, we need to show thatξ|
M\B
3 is isotopic toξ|
M\B
3∗△
d
relative to
the boundary. Since d = g.c.d.{e(Σ)|Σ∈ H
2
(M)}, it suffices to prove the following more
generalfact.
65
Lemma5.8.1. LetΣbeaclosedsurfaceofgenusgandηbeanI-invariantcontactstructure
onΣ×[0,1]. Thenη∗△
l
isstablyisotopictoηrelativetotheboundary,wherel= e(η)(Σ).
Proof. Sinceweonlyconsiderstableisotopiesofcontactstructures,onecanprescribeany
dividingsetΓ
Σ
onΣsuchthattheEulerclassevaluatesonΣtol. Inparticular,weconsider
thedividingsetonΣasdepictedinFigure5.24,namely,thereareg+1circlesγ
1
∪···∪γ
g+1
dividing Σ 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 circlesΓ
0
andΓ
1
respectively. We also choose admissible arcs{α
1
,α
2
,··· ,α
p−1
} and{β
1
,β
2
,··· ,β
q−1
}, and
orientγ
i
,1≤ i≤ g+1,inawayasdepictedinFigure5.24.
γ
1
γ
2 γ
g+1
α
1
α
2
α
p−1
...
β
1
β
q−1
...
...
−
+
+
+ + +
− − −
Σ
Γ
0
Γ
1
Figure5.24: ThedividingsetonΣ.
Aneasycalculationshowsthatl= 2(p−q). ChoosesmalldisksD
2
ϵ,0
, D
2
ϵ,1
inΣsuchthat
D
2
ϵ,0
∩Γ
Σ
=Γ
0
and D
2
ϵ,1
∩Γ
Σ
=Γ
1
. Observe that the bypass triangle attachment along any
α
i
andβ
j
consistsofthreetrivialbypassattachments,henceisisotopictocontactstructures
inducedbyapurebraidofthedividingset. Moreprecisely,letγ
−
i
,i= 1,2,··· ,g+1,bean
orientedloopinthenegativeregionwhichisparalleltoγ
i
. Wehavethefollowingisotopies
ofcontactstructures△
2
α
1
∗···∗△
2
α
p−1
≃η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
1
∪···∪γ
−
g+1
)
≃η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
1
)
∗···∗η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
g+1
)
,
where we think of γ
−
1
∪···∪ γ
−
g+1
as an oriented loop homologous to the union of the
γ
i
’s. Similarly one can study the bypass triangle attachments along the β
j
’s, but with an
opposite orientation. Let γ
+
i
be an oriented loop in the positive region which is parallel
to γ
i
for 1≤ i≤ g+ 1. We have the following (stable) isotopies of contact structures
△
−2
β
1
∗···∗△
−2
β
q−1
∼ η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
1
∪···∪γ
+
g+1
)
≃ η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
1
)
∗···∗ η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
g+1
)
. Here we only
66
have a stable isotopy because of our choice of the orientation of γ
i
. To summarize the
computationsabove,wegetthefollowing(stable)isotopiesofcontactstructures:
η∗△
l
≃η∗(△
2
α
1
∗···∗△
2
α
p−1
)∗(△
−2
β
1
∗···∗△
−2
β
q−1
)
≃η∗(η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
1
)
∗···∗η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
g+1
)
)∗(η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
1
)
∗···∗η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
g+1
)
)
≃η∗(η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
1
)
∗η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
1
)
)∗···∗(η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
g+1
)
∗η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
g+1
)
)
where the last step follows from the fact that isotopies that parallel transport D
2
ϵ,0
and D
2
ϵ,1
aredisjoint.
Nowitsufficestoprovethatη
Φ(Γ
0
,D
2
ϵ,0
,γ
−
i
)
∗η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
i
)
isstablyisotopictoan I-invariant
contact structure for 1 ≤ i ≤ g + 1. To see this, take an annular neighborhood A
i
of
γ
i
containing D
2
ϵ,0
and D
2
ϵ,1
and an admissible arc δ
i
which intersects Γ
0
, Γ
1
, and γ
i
as
depictedinFigure5.25. WecanassumethattheisotopiesΦ(Γ
0
,D
2
ϵ,0
,γ
−
i
)andΦ(Γ
1
,D
2
ϵ,1
,γ
+
i
)
are supported in A
i
. For simplicity of notation, we denote the composition η
Φ(Γ
0
,D
2
ϵ,0
,γ
−
i
)
∗
η
Φ(Γ
1
,D
2
ϵ,1
,γ
+
i
)
byη
γ
i
.
Γ
0
Γ
1
γ
i
δ
i
+
−
+ −
Figure5.25: Anannulusneighborhood A
i
ofγ
i
containingΓ
0
andΓ
1
.
Bypushingdownthebypassattachmentσ
δ
i
throughη
γ
i
,wehavethefollowingisotopies
67
ofcontactstructures:
η
γ
i
∗△
δ
i
=η
γ
i
∗σ
δ
i
∗σ
δ
′
i
∗σ
δ
′′
i
≃σ˜
δ
i
∗η
Φ(γ
i
)
∗σ
δ
′
i
∗σ
δ
′′
i
≃σ
δ
i
∗σ
δ
′
i
∗σ
δ
′′
i
=△
δ
i
where
˜
δ
i
is the push-down of δ
i
which is isotopic to δ
i
, and the η
Φ(γ
i
)
is easily seen to be
isotopictoan I-invariantcontactstructure. Theargumentworksforalli∈{1,2,··· ,g+1},
henceweestablishthestableisotopyasdesired.
68
References
[1] Daniel Bennequin. Entrelacements et équations de Pfaff. In Third Schnepfenried
geometry conference, Vol. 1 (Schnepfenried, 1982), volume 107 of Astérisque, pages
87–161.Soc.Math.France,Paris,1983.
[2] Yakov Eliashberg. Classification of overtwisted contact structures on 3-manifolds.
Invent.Math.,98(3):623–637,1989.
[3] Yakov Eliashberg and Mikhael Gromov. Convex symplectic manifolds. In Several
complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), volume52
of Proc. Sympos. Pure Math., pages 135–162. Amer. Math. Soc., Providence, RI,
1991.
[4] JohnB.EtnyreandKoHonda. OnconnectedsumsandLegendrianknots. Adv.Math.,
179(1):59–74,2003.
[5] Hansjörg Geiges. An introduction to contact topology, volume 109 of Cambridge
Studies in Advanced Mathematics. CambridgeUniversityPress,Cambridge,2008.
[6] Emmanuel Giroux. Convexité en topologie de contact. Comment. Math. Helv.,
66(4):637–677,1991.
[7] Emmanuel Giroux. Sur les transformations de contact au-dessus des surfaces. In
Essays on geometry and related topics, Vol. 1, 2, volume 38 of Monogr. Enseign.
Math.,pages329–350.EnseignementMath.,Geneva,2001.
[8] Robert E. Gompf. Handlebody construction of Stein surfaces. Ann. of Math. (2),
148(2):619–693,1998.
[9] Ko Honda. Contact structures, heegaard floer homology and triangulated categories.
In preparation.
[10] Ko Honda. On the classification of tight contact structures. I. Geom. Topol., 4:309–
368,2000.
[11] KoHonda. Gluingtightcontactstructures. Duke Math. J.,115(3):435–478,2002.
[12] Heinz Hopf. Über die Abbildungen der dreidimensionalen Sphäre auf die
Kugelfläche. Math. Ann.,104(1):637–665,1931.
69
[13] JohnW.Milnor. Topologyfromthedifferentiableviewpoint. BasedonnotesbyDavid
W.Weaver.TheUniversityPressofVirginia,Charlottesville,Va.,1965.
[14] Ichiro Torisu. On the additivity of the Thurston-Bennequin invariant of Legendrian
knots. PacificJ.Math.,210(2):359–365,2003.
70
Abstract (if available)
Abstract
The goal of this thesis is to study homotopy classes of oriented 2-plane fields in 3-manifolds and apply them to 3-dimensional contact topology. Using a generalized version of the Pontryagin-Thom construction, we compute the homotopy class of a bypass attachment. Based on this computation, we give a new proof to the classification theorem of overtwisted contact structures, originally due to Eliashberg, using convex surface theory.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Huang, Yang
(author)
Core Title
On the homotopy class of 2-plane fields and its applications in contact topology
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Mathematics
Publication Date
06/20/2012
Defense Date
05/14/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
2-plane field,bypass,contact topology,homotopy,OAI-PMH Harvest,overtwisted contact structure
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Honda, Ko (
committee chair
), Bonahon, Francis (
committee member
), Kocer, Yilmaz (
committee member
)
Creator Email
huangyan@usc.edu,hymath@gmail.com
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Tags
2-plane field
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contact topology
homotopy
overtwisted contact structure