Sutured ECH is a Natural Invariant

148 Pages·2022·1.559 MB·English

by  agatay Kutluhan, Steven Sivek, C. H. Taubes

Number 1350
Sutured ECH is a Natural Invariant
C¸ ag˘atay Kutluhan
Steven Sivek
C. H. Taubes
January 2022 • Volume 275 • Number 1350 (fourth of 6 numbers)
Number 1350
Sutured ECH is a Natural Invariant
C¸ ag˘atay Kutluhan
Steven Sivek
C. H. Taubes
January 2022 • Volume 275 • Number 1350 (fourth of 6 numbers)
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Contents
Chapter 1. Introduction 1
1.1. Organization 4
1.2. Acknowledgments 4
Chapter 2. Sutured ECH and some related constructions 5
2.1. Sutured ECH 5
2.2. Contact 1-handles 7
2.3. Liouville forms on the boundary 10
2.4. Closed manifolds and continuation maps 14
Chapter 3. Independence of the almost complex structure 17
Chapter 4. Independence of the contact form 29
4.1. An isomorphism for isotopic contact forms 29
4.2. Isomorphisms and embedding data 35
4.3. A natural version of Theorem 4.5 38
Chapter 5. Some properties of the contact class 43
Chapter 6. Invariance under gluing 1-handles 47
Chapter 7. Stabilization and a canonical version of sutured ECH 53
Appendix A. Appendix by C. H. Taubes 67
A.1. Setting the stage 67
A.2. Monopoles 69
A.3. Instantons 85
A.4. A priori bounds for instantons 97
A.5. Local convergence to pseudo-holomorphic curves 105
A.6. Convergence of instantons as T →∞ 121
A.7. Comparing instantons on Y∞ and YT 128
Bibliography 135
iii

Abstract
We show that sutured embedded contact homology is a natural invariant of
sutured contact 3-manifolds which can potentially detect some of the topology of
the space of contact structures on a 3-manifold with boundary. The appendix, by
C. H. Taubes, proves a compactness result for the completion of a sutured contact
3-manifold in the context of Seiberg–Witten Floer homology, which enables us to
complete the proof of naturality.
ReceivedbytheeditorJanuary13,2015,and,inrevisedform,February21,2016,November
27,2017,January24,2018,andJune14,2018.
ArticleelectronicallypublishedonDecember22,2021.
DOI:https://doi.org/10.1090/memo/1350
2020 MathematicsSubjectClassification. Primary53D40,57M27.
C¸a˘gatayKutluhanwassupportedbyNSFgrantDMS-1360293. StevenSivekwassupported
byNSFpostdoctoralfellowshipDMS-1204387. C.H.TaubeswassupportedbyNSFgrantDMS-
1401192.
(cid:2)c2022 C¸a˘gatay Kutluhan, Steven Sivek, and C. H. Taubes
v

CHAPTER 1
Introduction
Embedded contacthomology, definedby Hutchings, isaninvariant of closed3-
manifoldsY equippedwithnondegeneratecontactformsλ. Givenahomologyclass
Γ∈H (Y)andagenericsymplectization-admissiblealmostcomplexstructureJ on
1
R×Y,oneconsidersthefreeAbeliangroupECC(Y,λ,Γ,J)generatedbyadmissible
orbit sets Θ={(Θ ,m )} with the following properties:
i i
• {Θ } is a finite collection of distinct embedded Reeb orbits in (Y,λ), and
i
the m are positive integers;
i
• mi =1(cid:2)if Θi is a hyperbolic Reeb orbit;
• [Θ]:= m [Θ ] is equal to Γ.
i i i
This group admits a differential ∂ defined by its action on the set of generators
via a suitable count of J-holomorphic curves in R×Y with ECH index, denoted
I, equal to 1. To be more explicit, given two generators Θ = {(Θ ,m )} and
i i
Θ(cid:3) = {(Θ(cid:3),m(cid:3))}, the coefficient (cid:6)∂Θ,Θ(cid:3)(cid:7) is a signed count, modulo R-translation,
i i
of I = 1 J-holomorphic curves asymptotic to R×Θ with multiplicity m at +∞
i i
and to R×Θ(cid:3) with multiplicity m(cid:3) at −∞. The resulting homology is denoted
i i
ECH(Y,λ,Γ,J), while the direct sum over all Γ is denoted ECH(Y,λ,J). Embed-
dedcontacthomologyisequippedwithanaturalZ[U]-modulestructure, andthere
exists a contact class c(λ) = [∅] ∈ ECH(Y,λ,0,J) corresponding to the empty set
of Reeb orbits. For more information, see the detailed expositions by Hutchings
[11,12].
It was conjectured that ECH(Y,λ,J) should in fact be a topological invariant
of Y, independent of J and of the contact form λ, but direct proofs of this have
beenelusive. Theindependence wasfinally establishedby Taubes, whoshowed the
following in a long series of papers [28–32]:
Theorem 1.1 (Taubes). There is a canonical isomorphism of relatively graded
Z[U]-modules
∼ (cid:3)−∗
ECH∗(Y,λ,Γ,J)=HM (Y,sξ+PD(Γ)),
(cid:3)∗
where HM denotes a particular version of Seiberg–Witten Floer cohomology [19]
and s is the canonical Spinc structure associated to ξ =ker(λ). This isomorphism
ξ
sends the ECH contact class c(λ) to the contact invariant [18] in Seiberg–Witten
Floer cohomology.
More recently, following an extension of Heegaard Floer homology to balanced
sutured manifolds by Juh´asz [15] and subsequent analogues in Seiberg–Witten
and instanton Floer homologies by Kronheimer and Mrowka [20], Colin, Ghiggini,
Honda, and Hutchings [4] defined a generalization of embedded contact homology
to sutured contact manifolds. To any sutured contact 3-manifold (M,Γ) with con-
tact form α and generic almost complex structure J tailored to (M,α), they define
1
2 1. INTRODUCTION
the sutured embedded contact homology group
ECH(M,Γ,α,J),
with a direct sum decomposition
(cid:4)
ECH(M,Γ,α,J)= ECH(M,Γ,α,J,h),
h∈H1(M)
just as in the case of closed manifolds, and conjecture [4, Conjecture 1.2] that this
is a topological invariant of (M,Γ).
In this paper we prove that sutured ECH doesnot depend onthe contact form
α or the almost complex structure J.
Theorem 1.2. Let (M,Γ) be a sutured manifold, and let α ,α be adapted,
0 1
nondegenerate contact forms on (M,Γ) such that ker(α ) and ker(α ) are isotopic
0 1
rel a neighborhood of Γ. For any two generic almost complex structures J and J
0 1
tailored to (M,α ) and (M,α ), respectively, there is an isomorphism
0 1
∼
ECH(M,Γ,α ,J )=ECH(M,Γ,α ,J )
0 0 1 1
which respects direct sum decompositions over H (M) and sends the contact class
1
c(α ) to the contact class c(α ).
0 1
We hope to address the question of topological invariance in future work.
Theorem1.2wasprovedsimultaneouslybyColin,Ghiggini,andHonda[3,The-
orem10.2.2], usingaconstructionwhichisvirtuallyidenticaltoours. Ourstrategy
of proof is to embed sutured contact manifolds (M,Γ,α) inside certain families
(Y ,α ) of closed contact manifolds whose construction is reminiscent of an open
n n
book decomposition. The contact forms α are such that given a constant L > 0,
n
the ECH generators with symplectic action less than L associated to (M,Γ,α,J)
for a generic tailored almost complex structure J and to (Y ,α ,J ) for an appro-
n n n
priate extension J of J coincide canonically when n is sufficiently large, as well as
n
the moduli spaces of pseudo-holomorphic curves which define the respective differ-
entials. Hutchings and Taubes [14] constructed canonical isomorphisms between
filteredECHgroupsfortheclosedcontactmanifolds(Y ,α )withdifferentchoices
n n
of J , and we can transfer these back to the filtered ECH groups for (M,Γ,α). We
n
then turn isotopies of contact forms on (M,Γ) into exact symplectic cobordisms
between the Y and use the cobordism maps defined in [14] for filtered ECH to
n
relate the ECH groups for pairs of isotopic contact forms.
By construction, the isomorphism of Theorem 1.2 potentially depends on a
choice of what we call embedding data (see Definition 3.3) for (M,Γ,α), appearing
in both our work and in that of Colin–Ghiggini–Honda [3]. However, one might
expect that the embedding data (essentially the triple (Y ,α ,J )) should play a
n n n
fairlyminorrole,sincethefilteredECHgroupwhichcoincideswiththeappropriate
filtered ECH group for (M,Γ,α,J) is concentrated near the submanifold M ⊂
Y . (See Section 2.1 for the precise definition of filtered ECH.) We can make
n
this precise using the correspondence results proved in the appendix to this paper
concerning Seiberg–Witten theory for completions of (M,Γ,α,J), allowing us to
prove an analogue of [14, Theorem 1.3] for sutured ECH.
1. INTRODUCTION 3
Theorem 1.3. Let (M,Γ,α) be a sutured contact manifold.
(1) If α is an L-nondegenerate contact form, then there is a canonically de-
fined group ECHL(M,Γ,α) which is isomorphic to ECHL(M,Γ,α,J) for
any J.
(2) If L<L(cid:3) and α is L(cid:3)-nondegenerate, then there is a canonical map
iL,L(cid:2) :ECHL(M,Γ,α)→ECHL(cid:2)(M,Γ,α)
induced by the maps iL,L(cid:2) of (2.1) and satisfying iL,L(cid:2)(cid:2) =iL(cid:2),L(cid:2)(cid:2)◦iL,L(cid:2) for
J
L<L(cid:3) <L(cid:3)(cid:3).
(3) The direct limit ECH(M,Γ,α) of the system ({ECHL(M,Γ,α)} ,
L
{iL,L(cid:2)}L,L(cid:2)) is canonically isomorphic to ECH(M,Γ,α,J) for any J.
The same results also produce a stronger version of Theorem 1.2:
Theorem 1.4. Fix a contact form α which is adapted to the sutured manifold
(M,Γ), and let Ξ(M,Γ,α) denote the space of cooriented contact structures on
(M,Γ) which agree with ker(α) on a neighborhood of ∂M. Then sutured ECH
canonically defines a local system on Ξ(M,Γ,α), i.e. a functor
ECH :Π (Ξ(M,Γ,α))→AbGroup,
1
from the fundamental groupoid of Ξ(M,Γ,α) to the category of Abelian groups. It
canonically assigns a group ECH(M,Γ,ξ,α| ) to any ξ ∈ Ξ(M,Γ,α), depending
∂M
only on ξ and on the restriction of α to ∂M, and an isomorphism
F :ECH(M,Γ,ξ ,α| )−∼→ECH(M,Γ,ξ ,α| )
ξs 0 ∂M 1 ∂M
to any homotopy class of paths ξ ⊂ Ξ(M,Γ,α). These groups and isomorphisms
s
decompose naturally with respect to H (M), and there is a contact class c(ξ) ∈
1
ECH(M,Γ,ξ,α| ) satisfying F (c(ξ ))=c(ξ ) for all paths ξ .
∂M ξs 0 1 s
Remark 1.5. All of the theorems stated above are true with integer coeffi-
cients. In Theorems 1.2 and 1.4 this requires some care, because we use the ECH
cobordismmapsdefinedbyHutchingsandTaubes[14,Theorem1.9](mostnotably
throughout Section 4), and these are only proved to work over Z/2Z. The issue
is that one must choose a homology orientation to avoid a sign ambiguity in the
corresponding Seiberg–Witten Floer cobordism map. In our case, we work with
topologicallyproduct cobordisms, andthesehave canonical homologyorientations,
so the cobordism maps on ECH exist and have the desired properties over Z.
We do not know whether the local system of Theorem 1.4 has nontrivial mon-
odromy: inotherwords,whetherthereexistsaclosedloopξ :(S1,∗)→Ξ(M,Γ,α)
s
for some (M,Γ,α) which induces a nontrivial automorphism of ECH(M,Γ,
ξ∗,α|∂M). This could potentially be used to detect interesting topology in the
space of contactstructures on(M,Γ). Ingeneral little is knownabout such spaces,
though some cases are understood due to work of Eliashberg [6,7], Ding–Geiges
[5], and Geiges–Klukas [8], among others.
The fact that the isomorphisms of Theorem 1.4 may depend on the paths ξ
s
rather thanjust their endpoints is not terribly surprising, however, because similar
phenomena occur in other sutured homology theories. In these other homology
theories, one associates a natural invariant to a sutured manifold and a canonical
isomorphismtoanydiffeomorphismofsuturedmanifolds;thisnaturalitywasproved
forsuturedFloerhomologybyJuha´szandThurston[16]andforsuturedmonopole