Essential tori in spaces of symplectic embeddings

Julian Chaidez, Mihai Munteanu

December 2, 2018

Abstract

Given two n–dimensional symplectic ellipsoids whose symplectic sizes sat-isfy certain inequalities, we show that a map from the n–torus to the spaceof symplectic embeddings from one ellipsoid to the other induces an injectivemap at the level of singular homology with mod 2 coefficients. The proof usesparametrized moduli space of J–holomorphic cylinders in completed symplec-tic cobordisms, provided by bordism maps of full contact homology. This isa pre-arxiv draft, please do not distribute.

Contents

1 Introduction 21.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Proof of the main result 52.1 Moduli spaces in cobordisms . . . . . . . . . . . . . . . . . . . . . . . 62.2 Transversality and compactness . . . . . . . . . . . . . . . . . . . . . 72.3 Counting the curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Proof of main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Contact homology algebra 123.1 Contact dg–algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Computations for ellipsoids . . . . . . . . . . . . . . . . . . . . . . . 16

4 The space of symplectic embeddings 184.1 Frechet manifold structure . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Bordism groups of Frechet manifolds . . . . . . . . . . . . . . . . . . 194.3 Weinstein neighborhood theorem with boundary . . . . . . . . . . . . 22

1

1 Introduction

The study of symplectic embeddings is a major area of focus in symplectic geometry.Remarkably, the space of such embeddings can have a rich and complex structure,even when the domain and range manifolds are relatively simple.

Symplectic embeddings between ellipsoids are a well–studied instance of this phe-nomenon. For a nondecreasing sequence of positive real numbers a = (a1, a2, . . . , an)define the symplectic ellipsoid E(a) as:

E(a) = E(a1, a2, . . . , an) =

(z1, . . . , zn) ∈ Cn

∣∣∣∣ π|z1|2

a1

+ . . .π|zn|2

an≤ 1

. (1.1)

Imbued with the restriction of the standard Liouville form λ = 12

∑ni=1 xidyi− yidxi

on Cn, E(a) is a compact, exact symplectic manifold with boundary. A special caseis the symplectic ball B2n(r), which is simply E(a) for a = (r, . . . , r).

The types of results that one can prove about symplectic embeddings, togetherwith the tools used to do so, are surveyed at length by Schlenk in [18]. Mostresearch has thus far sought to address the existence problem. Let us recall some ofthe more striking progress in this direction. The first nontrivial result was Gromov’seponymous non–squeezing theorem, proven in the seminal paper [7].

Theorem 1.1 ([7]). There exists a symplectic embedding ϕ : B2n(r) → B2(R) ×C2n−2 if and only if r ≤ R.

Theorem 1.1, in particular, demonstrated that there are obstructions to sym-plectic embeddings beyond the volume and initiated the study of quantitative sym-plectic geometry. It can be thought of as a result about ellipsoid embeddings, sinceB2(R)× C2n−2 can be viewed as the degenerate ellipsoid E(R,∞, . . . ,∞).

In dimension 4, the question of when the ellipsoid E(a, b) symplectically embedsinto the ellipsoid E(a′, b′) was answered by McDuff in [10]. Let Nk(a, b)k≥0 de-note the sequence of nonnegative integer linear combinations of a and b, orderednondecreasingly with repetitions.

Theorem 1.2 ([10]). There exists a symplectic embedding int(E(a, b)) → E(a′, b′)if and only if Nk(a, b) ≤ Nk(a

′, b′), for every nonnegative integer k.

Building on techniques introduced in [10], the authors of [11] study the function

c0(a) = infλ∣∣ E(1, a) symplectically embeds into B4(λ)

and show that, in particular, for a ∈ [1, ( 1+

√5

2)4], c0 is given by a piecewise linear

function involving the Fibonacci numbers, which they call the “Fibonacci staircase”.

2

In the more recent work [5] and [6], the authors studied the stabilized ellipsoidembedding problem in higher dimensions, and proved that the functions:

cn(a) = infλ∣∣ E(1, a)× Cn symplectically embeds into B4(λ)× Cn

exhibit a similar behavior involving staircases and Fibonacci numbers.

Beyond problems of existence, one can ask about the algebraic topology ofthe space of symplectic embeddings SympEmb(U, V ) between two symplectic 2n–dimensional manifolds U and V . Very little is known about such questions in di-mensions higher than 4. For instance, in [10] McDuff demonstrated that the spaceof embeddings between 4–dimensional symplectic ellipsoids is connected wheneverit is nonempty. The equivalent question in higher dimensions is open. For otherresults in dimension 4, see [1] and [8].

More recently, in [13], the second author developed methods to show that thecontractibility of certain loops of symplectic embeddings of ellipsoids depends onthe relative sizes of the two ellipsoids.

1.1 Main result

In this paper, we build upon the methods developed in [13] to tackle the question ofdescribing the higher homology groups of spaces of symplectic embeddings betweenellipsoids in any dimension. In particular, we produce certain torus families of em-beddings between ellipsoids whose contractibility depends on the relative symplecticsizes of the ellipsoids involved.

More precisely, we will be studying families of symplectic embeddings that arerestrictions of the following unitary maps. For θ = (θ1, . . . , θn) ∈ T n = (R/2πZ)n,let Uθ denote the unitary transformation:

Uθ(z1, . . . , zn) = (eiθ1z1, . . . , eiθnzn). (1.2)

For any symplectic ellipsoid E(a) and E(b) such that ai < bi, we may define a familyof ellipsoid embeddings by restricting the domain.

Φ : T n → SympEmb(E(a), E(b)) Φ(θ) := Uθ|E(a) (1.3)

The following theorem about the family Φ is the main result of this paper.

Theorem 1.3 (Main theorem). Let E(a) and E(b) be ellipsoids in Cn such thata = (a1, . . . , an) and b = (b1, . . . , bn) satisfy:

ai < bi < ai+1 and bn < 2a1.

Let Φ : T n → SympEmb(E(a), E(b)) be given by Φ(θ) = Uθ |E(a) for Uθ defined asin (1.2). Then the induced map on homology with Z2–coefficients is injective.

Φ∗ : H∗(Tn;Z2) → H∗(SympEmb(E(a), E(b));Z2).

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To appreciate the nontriviality of Theorem 1.3, we note that the torus map U :T n → U(n) is very non–injective on Z2–homology.

Lemma 1.4. Consider the map U : T n → U(n) given by θ 7→ Uθ. Then the inducedmap U∗ : H∗(T

n;Z2)→ H∗(U(n);Z2) on homology with Z2–coefficients is:

(a) surjective if ∗ = 0 or ∗ = 1.

(b) identically 0 if ∗ ≥ 2.

Proof. To show (a), we simply note that T n and U(n) are connected so U0 = Id.Furtheremore, if we consider the loop γ : R/2πZ → T n given by θ 7→ (θ, 0, . . . , 0),we see that the composition:

detC U γ : R/2πZ→ U(1) ' R/2πZ

is the identity. Since detC : U(n) → U(1) induces an isomorphism on H1, theinduced map of U must be surjective on H1.

To show (b) we proceed as so. For each subset L ⊂ 1, . . . , n with size |L| = k,define the k–torus by:

TL = (θ1, . . . , θn) ∈ T n | θj = 0, ∀j /∈ L .

The homology group Hk(Tn;Z2) is generated by the classes [TL], where L runs over

all subsets of size k. It suffices to show that U∗[TL] = 0 for all L with |L| ≥ 2. Wecan factorize TL = TJ×TK for JtK = L and |J | = 2. Then we have a factorization:

TL = TJ × TKιJ×ιK−−−→ U(2)× U(n− 2)

j−→ U(n)

Here j is the inclusions of a product of unitary subgroups, and ιJ and ιK are inclu-sions of the tori into these unitary subgroups. It suffices to show that (ιJ×ιK)∗[TL] =[ιJ ]∗[TJ ]⊗ [ιK ]∗[TK ] = 0, or simply that [ιJ ]∗[TJ ] = 0 ∈ H2(U(2);Z2).

Now we simply note that dim(U(2)) = 4 and H∗(U(2);Z2) ' Z2[c1, c3] where ciis a generator of index i. In particular, H2(U(2);Z2) ' H2(U(2);Z2) = 0.

Lemma 1.4 can be used to prove that the torus map Φ : T n → Symp(E(a), E(b))similarly fails to be injective when E(a) is very small relative to E(b).

Proposition 1.5. Let E(a) and E(b) be ellipsoids in Cn such that a = (a1, . . . , an)and b = (b1, . . . , bn) satisfy

ai < ai+1, bi < bi+1, an < b1.

Let Φ : T n → SympEmb(E(a), E(b)) be given by (1.3). Then the induced map Φ∗on homology with Z2–coefficients has Im(Φ∗) rank 1 in degree ∗ = 0, 1, and rank 0in degree ∗ ≥ 2.

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Proof. Let D : SympEmb(E(a), E(b)) → U(n) denote the map ϕ 7→ r(dϕ|0), givenby taking derivatives dϕ|0 ∈ Sp(2n) at the origin and composing with a retrationr : Sp(2n)→ U(n). Note that, for a and b meeting our assumptions, we can factorthe identity Id : U(n)→ U(n) and Φ : T n → SympEmb(E(a), E(b)) as:

Id : U(n)res−→ Symp(E(a), E(b))

D−→ U(n)

Φ : T nU−→ U(n)

res−→ SympEmb(E(a), E(b))

Here res(ϕ) := ϕ|E(a) denotes restriction of domain. In particular, res : U(n) →Symp(E(a), E(b)) is injective on homology and Im(Φ∗) ' Im(U∗) as graded Z2-vectorspaces. The result thus follows from Lemma 1.4.

Remark 1.6. One can recover the result of Theorem 1.4 in [13] by looking at thefirst order homology H1(T n;Z2) in the conclusion of Theorem 1.3 for n = 2.

Organization. The rest of the paper is organized as so. In Section §2, we givethe proof of Theorem 1.3. The final two section are dedicated to demonstratingsome technical results needed to deduce the steps of the proof. In §3, we recall thedefinition of the contact dg–algebra (constructed in full generality by Pardon in [14])together with the releant computations for symplectic ellipsoids. In §4, we provesome technical results about the topology of all symplectic embeddings spaces.

Acknowledgements. We would like to thank our advisor, Michael Hutchings,for all the helpful discussions. JC was supported by the National Science FoundationGraduate Research Fellowship under Grant No. 1752814.

2 Proof of the main result

In this section, we prove Theorem 1.3, modulo some technical results discussed in§3-4. The strategy is, roughly, the following.

We assume for contradiction that the map Φ∗ induced by the family Φ of (1.3)is not injective in degree k. Using results in §4, we use use this assumption to find acertain family of embeddings, parameterized by a union of k-tori and built from Φ,is null-bordant via a family of embeddings Ψ over some compact space P . We use Ψto construct a moduli space of holomorphic curves MI(J) in cobordisms and someassociated evaluation maps evI to a k–torus T k. We then show that the degree ofrestricted to the boundary is 1 mod 2, which is a contradiction since the evaluationmap extends to the bounding manifold P .

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2.1 Moduli spaces in cobordisms

We now introduce the spaces of holomorphic curves that are relevant to our proof.Throughout §2.1-2.3, we will assume that E(a) and E(b) are ellipsoids with ai/aj 6∈Q and bi/bj 6∈ Q for any i 6= j. In particular, this implies that the natural contactforms on the boundaries ∂E(a) and ∂E(b) are non-degenerate (see Example 3.8).

Recall that, for (Y±, λ±) closed contact manifolds of the same odd dimension,a compact symplectic cobordism from (Y+, λ+) to (Y−, λ−) is a compact symplecticmanifold (W,ω) with boundary ∂W = −Y− t Y+ such that ω|Y± = dλ±. Given acompact symplectic cobordism (W,ω), one can find neighborhoods N− of Y− andN+ of Y+ in W , and symplectomorphisms

(N−, ω) ' ([0, ε)× Y−, d(esλ−)) (N+, ω) ' ((−ε, 0]× Y+, d(esλ+)),

where s denotes the coordinate on [0, ε) and (−ε, 0]. Using these identifications, wecan complete the compact symplectic cobordism (W,ω) by adding cylindrical ends(−∞, 0]× Y− and [0,∞)× Y+ to obtain the completed symplectic cobordism

W = [0,∞)× Y+ ∪Y+ W ∪Y− (−∞, 0]× Y−.

An almost complex structure J on a completed symplectic cobordism W as aboveis called compatible if:

· On [0,∞) × Y+ and (−∞, 0] × Y−, the almost complex structure J is R–invariant, maps ∂s (the R direction) to Rλ± , and maps ξ± to itself compatiblywith dλ±.

· On the compact symplectic cobordism W , the almost complex structure J istamed by ω.

Call J(W ) the set of all such compatible almost complex structures on W .

Notation 2.1. For the remainder of this section, we adopt the following notation.Fix a subset I ⊂ 1, . . . , n. For each i ∈ 1, . . . , n, let Σi denote a copy of the

twice punctured Riemann sphere R × S1 ' CP1 \ 0,∞ with the usual complexstructure. Label the positive and negative punctures of Σi by p+

i and p−i respectively.Denote ΣI = ti∈IΣi.

Moreover, for a symplectic embedding ϕ : E(a) → E(b), let Wϕ denote thecompleted symplectic cobordism obtained by completing the compact symplecticcobordism Wϕ = E(b) \ int(ϕ(E(a))). Finally, let γ+

i = ∂E(b) ∩ zi = 0 and letγ−i = ∂E(a) ∩ zi = 0.

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Definition 2.2 (Unparameterized moduli space). For any symplectic embedding

ϕ ∈ Symp(E(a), E(b)) and any admissible almost–complex structure Jϕ ∈ J(Wϕ),

we denote by MI(Wϕ; Jϕ) the moduli space:

MI(Wϕ; Jϕ) =u : ΣI → Wϕ

∣∣∣ (du)0,1Jϕ

= 0, u→ γ+i at p+

i and u→ ϕ(γ−i ) at p−i

/C×

That is, u : ΣI → Wϕ is a Jϕ–holomorphic curve such that u asymptotes to thetrivial cylinder over γ+

i in R+ × ∂+WΦp ' R+ × ∂E(b) at the puncture p+i and u

asymptotes to the trivial cylinder over ϕ(γ−i ) in R−× ∂−Wϕ ' R−×ϕ(E(a)) at thepuncture p−i , for each i ∈ I. We quotient the space of such maps by the group ofdomain reparameterizations, which is C×.

Definition 2.3 (Parameterized moduli space over P ). Given a compact manifoldwith boundary P and a P–parameterized family of symplectic embeddings Ψ :P ×E(a)→ E(b), let J(Ψ) denote the set of P–parameterized families J = Jpp∈Pof almost complex structures Jp ∈ J(WΨp) for each Ψp. Let MI(J) denote theparameterized moduli space of pairs:

MI(J) =

(p, u)∣∣∣ p ∈ P, u ∈MI(WΨp ; Jp)

.

Given a family of symplectic embeddings Ψ : P ×E(a)→ E(b) where Im(Ψp) isindependent of p for p near ∂P , we let W∂P = E(b) \Ψp(E(a)) for p ∈ ∂P .

2.2 Transversality and compactness

This section is devoted to a proof of generic transversality (Lemma 2.4) and com-pactness (Lemma 2.5) for the moduli space MI(J).

Lemma 2.4 (Transversality). There exists a J∂P ∈ J(W∂P ) and a family of al-most complex structures J ∈ J(Ψ) with J|∂P ≡ J∂P such that the moduli space

MI(W∂P , J∂P ) is a 0–dimensional manifold and the moduli space MI(J) is a (k+1)–

dimensional manifold with boundary ∂MI(J) ' ∂P ×MI(W∂P ; J∂P ).

Proof. This essentially follows from the results of [19] and [20, §7], which we nowdiscuss in some detail.

Observe that all of the Jp–holomorphic curves u appearing in points (p, u) ∈MI(J), for any choice of J, are somewhere injective. By [20, Theorems 7.1–7.2], there

exists a comeager subset Jreg(W∂P ) ⊂ J(W∂P ) such that, for any J∂P ∈ Jreg(W∂P ),

the moduli space MI(W∂P ; J∂P ), which consists entirely of somewhere–injectivecurves, is a manifold with dimension equal to the virtual dimension:

vdim(MI(W∂P ; J∂P )) =∑i∈I

(dim(W∂P )−3)χ(u)+2cτ1(u∗[ΣI ])+µCZ(γ+i )−µCZ(γ−i ).

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Since c1(W∂P ) = 0, χ(u) = 0, and each component of u has two punctures andµCZ(γ+

i ) = µCZ(γ−i ), the virtual dimension is 0.By the parametric version of transversality (see [20, Remark 7.4] and [19, §4.5]),

there exists a family J ∈ J(Ψ), such that J|∂P ≡ J∂P and MI(J) is a manifold with

boundary ∂MI(J) = ∂P ×MI(W∂P ; J∂P ) of dimension:

dim(MI(J)) = dim(P ) + vdim(MI(W∂P ; J∂P )) = k + 1.

Lemma 2.5 (Compactness). Let 0 < a1 < b1 < a2 < b2 < · · · < an < bn < 2a1

and choose J∂P ∈ J(W∂P ) and J ∈ J(Ψ) as in Lemma 2.4. Then the moduli spaces

MI(W∂P ; J∂P ) and MI(J) are compact.

Proof. This is a simple application of SFT compactness, of which we give a simplifieddiscussion below (see Review 2.6).

Thus, let ui be a sequence in MI(W∂P ; J∂P ) or MI(J). We need to show that thelimit building v (under the notation of Review 2.6) has no symplectization levels.By considering components of ΣI and #iΣi individually, we can assume that |I| = 1,i.e. that ΣI has one component and each ui is positively asymptotic to a single γ+

l

for some l ∈ 1, . . . , n.By the action monotonicity, the ends of u+

j , u−j and uW must aymptote to a set

of orbits with total action between al and bl. Due to our assumptions on al and bl,there are no such orbits besides γ−l and γ+

l themselves. Since one symplectizationcomponent must exist in the compactification, uW must be a cylinder from γ+

l toγ−l and there can be no other levels by action considerations.

Review 2.6 (SFT Compactness). Here we give a very simplified review of a versionof SFT compactness as proven in [2, §10].

LetW and P be connected, closed manifolds with boundary, and let (λp, Jp)p∈Pbe a P–parameterized family of Liouville form/compatible almost–complex struc-ture pairs. Fix a closed surface with punctures Σ, and consider a sequence pi ∈ Pand ui : Σ→ (W , Jpi) of Jpi–holomorphic curves satisfying a uniform energy bound.

The SFT compactness theorem states that, after passing to a subsequence, pi →p ∈ P and ui converges to a Jp–holomorphic building :

v = (u+1 , . . . , u

+α , u

W , u−1 , . . . , u−β )

Here α, β ∈ Z≥0 are integers and the elements of the tuple (called levels):

u∗j : Σ∗(j) → Y∗ for ∗ ∈ +,− and uW : ΣW → W∂P

8

denote Jp–holomorphic curves (modulo translation, in the symplectization case).The asymptotics of the u∗i and uW are supposed to be compatible, in the sense thatthe negative ends of u∗i and the positive ends of u∗i+1 must agree (and likewise foru+a and uW , etc.). Note that each symplectization level u∗i can be disconnected, but

it must have at least one component that is not a trivial cylinder R× γ.The surfaces Σ(i) can be glued together along the boundary punctures asymptotic

to matching Reeb orbits, and this glued surface #iΣ(i) is homeomorphic to Σ. Thereis some of additional data, beyond the holomorphic curves themselves, associatedto a holomorphic building. However, we suppress this data since it will play no rolein our argument.

2.3 Counting the curves

This section is devoted to proving the following lemma.

Lemma 2.7 (Odd signed count). The 0–dimensional moduli space MI(W∂P ; J∂P )has an odd number of points.

Proof. First note that we can write the moduli space M(W∂P , J∂P ) in terms of the

moduli spaces M(W∂P ; γ+i , γ

−i , [0]) from Construction 3.4, as so:

MI(W∂P ; J∂P ) '∏i∈I

M(W∂P ; γ+i , γ

−i , [0]).

Here [0] denotes the unique homotopy class of cylinder in π2(W∂P ; γ+i t γ−i ) (by

abuse of notation, we suppress the dependence of this class on i). The argument

in Lemma 2.5 establishes that M(W∂P ; γ+i , γ

−i , [0]) is compact, and in fact that

there can be no genus 0 holomorphic buildings from γ+i to γ−i other than those of

M(W∂P ; γ+i , γ

−i , [0]). Thus we have:

M(W∂P ; γ+i , γ

−i , [0]) = M(W∂P ; γ+

i , γ−i , [0]).

The compactified moduli space is therefore cut out transversely. It suffices to showthat M(W∂P ; γ+

i , γ−i , [0]) has an odd number of points.

To prove this, consider the cobordism W∂P from E(b) to E(a) and the cobordismV∂P := E(c · a) \E(b) from E(c · a) to E(b). Here c 0 is some constant such thatcai > bi for all i. We denote the cobordism maps by:

CH[V∂P ] : CH(∂E(c·a))→ CH(∂E(b)) CH[W∂P ] : CH(∂E(b))→ CH(∂E(a)).

By Lemma 3.9, the map CH[V∂P ] and CH[W∂P ] are isomorphisms. Furthermore,the composition cobordism map CH[W∂P ] CH[V∂P ] agrees with the obvious map:

CH(∂E(c · a)) ' Λ[Pg(∂E(c · a))] ' Λ[Pg(∂E(a))] ' Λ[Pg(∂E(a))].

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This map is induced by the bijection between orbits Pg(∂E(c ·a)) ' Pg(∂E(a)) (see[14, Lemma 1.2] and the proof for a discussion of this fact).

The computation in Example 3.8 implies that on ∂E(c·a), ∂E(a), and ∂E(b) anynon–simple orbit η satisfies |η| ≥ 4n− 2 and any set of 2 or more orbits Γ satisfies|Γ| ≥ 4n− 2. In particular, we have the following isomorphisms for 1 ≤ i ≤ n

CH2(n+i−2)(∂E(c · a)) ' Q〈γcai 〉 CH2(n+i−2)(∂E(a)) ' Q〈γai 〉.

Here γcai and γai are two simple orbits of ∂E(c · a) and ∂E(a) identified under thebijection Pg(∂E(c·a)) ' Pg(∂E(a)). By the discussion above about the compositioncobordism map, we can pick generators [γcai ] and [γai ] of CH2(n+i−2)(∂E(c · a)) andCH2(n+i−2)(∂E(a)) respectively such that:

CH[W∂P ] CH[V∂P ]([γcai ]) = [γai ].

The definition of the cobordism maps on homology classes on the chain level, givenin Construction 3.4, then implies that the following contact homology class must be0. [

#M(W∂P ; γbi , γai , [0])

|Aut(γbi , γai , ; [0])|

· #θM(V∂P ; γcai , γbi , [0])

|Aut(γcai , γbi , ; [0])|

· γai − γai

]= 0 ∈ CH(∂E(a)).

The automorphism groups Aut(γcai , γbi , [0]) and Aut(γbi , γ

ai , [0]) are trivial, since γai ,

γbi , and γcai are all simple, and the above expression can be null–homologous if andonly if it yields a zero multiple of γai . Thus we must have:

#M(W∂P ; γbi , γai , [0]) ·#θM(V∂P ; γcai , γ

bi , [0])− Id = 0.

The above identity is as maps of orientation lines, but identifying the orientationlines with Z yields the analogous identity for a pair of integer–valued point counts.This implies that #M(W∂P ; γbi , γ

ai , [0]) must be ±1 as an integer–valued point count,

which is the desired result.

Remark 2.8 (Moduli space differences). A confusing point, worth mentioning, is

that the moduli spaces MI(W∂P ; J∂P ) and M(W∂P ; γ+i , γ

−i , [0]) consist of slightly dif-

ferent objects. Namely, the moduli space M(W∂P ; γ+i , γ

−i , [0]) consists of reparame-

terization equivalence classes [u,Σ,m] of maps u from any genus 0, twice puncturedRiemann surface Σ (so the domain complex structure can vary), and includes thedata of an asymptotic marker mp which must asymptote to fixed base points bp ofthe limiting orbits γp, for every puncture p.

In the case of a genus 0 surface with two punctures, however, the associated Te-ichmuller space is trivial (see [19]), and a careful reading of the equivalence relationsbetween trees described in [14, §2.1] shows that the asymptotic markers at inputand output vertices are allowed to vary.

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2.4 Proof of main theorem

Given the moduli-space constructions of §2.1-2.3, we can now proceed to the proofof Theorem 1.3.

Proof. As in Lemma 1.4, for each subset L ⊂ 1, . . . , n with size |L| = k we define

TL = (θ1, . . . , θn) ∈ T n | θj = 0, ∀j /∈ L . (2.1)

We again note that the homology group Hk(Tn;Z2) is generated by the classes [TL],

where L runs over all subsets of size k.We will now argue by contradiction to prove Theorem 1.3. To that end, fix an

integer k ∈ 1, . . . , n and assume there is a nonzero homology class [A] =∑

L cL[TL]in Hk(T

n;Z2) such that Φ∗[A] = 0 in Hk(SympEmb(E(a), E(b));Z2). Fix a subsetI = i1, i2, . . . , ik for which cI = 1 in the expansion of the class [A].

Since Φ∗([A]), using Proposition 4.8, we can assume there exists a smooth k+1–dimensional manifold P with boundary ∂P = tcL 6=0TL and a smooth map

Ψ : P → SympEmb(E(a), E(b)),

such that Ψ|TL = Φ|TL for each component TL of ∂P .We may, by passing to sub-ellipsoids, assume that E(a) and E(b) with sub-

ellipsoids satisfying ai/aj 6∈ Q and bi/bj 6∈ Q for all i 6= j. Lemmas 2.4 and 2.5

then imply that there exist J∂P ∈ J(W∂P ) and J ∈ J(Ψ) such that the parametrizedmoduli space MI(J) is a compact (k + 1)–dimensional manifold with boundary

P ×M(W∂P ; J∂P ). We define the evaluation map

evI : MI(J)→ γ−i1 × γ−i2× · · · × γ−ik ' T k,

as follows. For each il ∈ I, fix a point q+il∈ γ+

il. Each (p, u) ∈MI(J) has k cylinder

components u1, . . . , uk such that ul has a positive end at γ+il

and a negative end atγ−il . Let tl ∈ S1 be such that lims→∞ ul(s, tl) = q+

iland let q−il = lims→−∞ ul(s, tl).

Define evI(p, u) = (q−i1 , q−i2, . . . , q−ik).

Let π : MI(J) → P , (p, u) 7→ p, denote the projection to the parameter space.Lemma 2.7 implies that for each TL in the decomposition of A, π−1(TL) is a disjointunion of an odd number of k–tori.

Notice that for each L 6= I, the restriction of evI to each component of π−1(TL)is not surjective and thus it has degree 0. By contrast, the restriction of evI to eachcomponent of π−1(TI) is bijective and has degree 1. Hence, we can conclude that∑

C

deg((evI)|C) ≡ 1 mod 2,

where the sum is taken over the connected components C of the boundary ∂MI(J).Since evI is defined on all of MI(J), the above sum of the degrees must be 0 mod2, hence providing the contradiction.

11

3 Contact homology algebra

In this section, we review the main Floer–theoretic tool in this paper, the full contacthomology CH(Y, ξ) of a closed contact manifold (Y, ξ).

3.1 Contact dg–algebra

In this section, we describe the construction of the contact dg–algebra of a contactmanifold and the cobordism dg–algebra maps induced by exact symplectic cobor-disms. This is largely a review of Pardon’s construction of contact homology in[14].

We begin by fixing notation for the choices of Floer data that are needed todefine the relevant chain groups and cobordism maps.

Setup 3.1 (Contact manifold). Fix the following setup and notation.

(a) (Y, ξ) is a closed contact (2n + 1)–manifold with nondegenerate contact formα and associated Reeb vector-field R.

(b) J is a dα–compatible complex structure on ξ and J is the associated R–

invariant almost complex structure on the symplectization Y := R× Y .

(c) θ ∈ Θ(Y, α, J) is a choice of virtual perturbation data (in the sense of [14] §1.1)

for the compactified moduli spaces of holomorphic curves MI(Y ; γ+,Γ−, β)/Rfor any γ+,Γ− and β (see [14] for elaboration).

We use Data(Y, ξ) to denote the set of choices of data associated to a fixed contactmanifold (Y, ξ).

Setup 3.2 (Exact symplectic cobordism). Fix the following setup and notation.

(a) For each ∗ ∈ +,−, let Y∗, ξ∗ and α∗ be contact data, J∗ and J∗ be complexdata, and θ∗ be virtual perturbation data, all as in Setup 3.1.

Furthermore, fix the following setup and notation for corresponding cobordism data.

(b) (W,λ) is an exact symplectic (2n+2)–cobordism with ∂+W = Y+ and ∂−W =

Y− with completion (W , λ) such that W = (−∞, 0] × Y− ∪W ∪ [0,∞) × Y+

and λ agrees with d(etα∗) and λ on the appropriate pieces.

(c) J is a dλ compatible complex structure on W agreeing with J∗ on symplectic

collar neighborhoods of Y∗, and J is the associated complex structure on theLiouville completion W .

12

(d) θ ∈ Θ(W,λ, J) is a choice of virtual perturbation data (in the sense of [14] §1.3)

for the compactified moduli spaces of holomorphic curves M(W ; γ+,Γ−, β) forall γ+,Γ− and β.

We use Data[W,λ] to denote the set of choices of data as above for a fixed defor-mation class [W,λ] of exact symplectic cobordism. For each ∗ ∈ +,−, there is aprojection map of Floer data:

π∗ : Data[W,λ]→ Data(Y∗, ξ∗) (λ, J, θ) 7→ (α∗, J∗, π∗θ).

Here we use the projection map of VFC data π∗ : Θ(W,λ, J) → Θ(Y∗, α∗, J∗) de-scribed in [14], §1.3. Note that the product map

π+ × π− : Θ(W,λ, J)→ Θ(Y+, α+, J+)×Θ(Y−, α−, J−)

is surjective.

Given the setup discussed above, we give an overview of the construction of thecontact dg–algebra and the cobordism maps.

Construction 3.3. (Contact DG–Algebra) Consider a contact manifold (Y, ξ) alongwith a choice of data D ∈ Data(Y, ξ;Z, [P ]), all as in Setup 3.1, and let m(ξ) denotethe integer:

m(ξ) := 2 · gcd 〈c1(ξ), [Σ]〉 | [Σ] ∈ H2(Y ) .

We now give the construction of the (Z2, H1(Y ;Z))–bigraded differential algebraCC(Y, ξ)D with differential ∂D) of bi–degree (−1, 0), called the contact dg–algebraor the full contact homology algebra. The Z2 part of the grading is called thehomological grading and the H1(Y ;Z) part of the grading is called the loop grading.The subalgebra CC(Y, ξ)D of loop grading 0 elements will have a Zm(ξ) lift of thehomological grading. We denote the homologies of these dg–algebras by:

CH(Y, ξ)D := H(CC(Y, ξ)D) CH(Y, ξ)D := H(CC(Y, ξ)D).

(Algebra) To define the algebra of the total complex CC(Y, ξ)D, consider the setP(Y, α) of unparameterized Reeb orbits of (Y, α). We can divide P(Y, α) into goodorbits Pg(Y, α) and bad orbits Pb(Y, α). An orbit is bad if it is an even cover of anorbit η with CZτ (η) ≡ 1 mod 2 for some (any) trivialization of ξ|η. An orbit is goodif it is not bad.

To each good orbit we can associate an (Z2, H1(Y ;Z))–graded orientation lineoγ supported in bigrading (|γ|, [γ]) where |γ| = CZ(γ, [Σ]) + n − 3 mod 2 (see forinstance [3]) and [γ] ∈ H1(Y ;Z) is simply the homology class of the orbit. For a finiteset of orbits Γ ⊂ Pg(Y, α) of good orbits, we define oΓ := ⊗γ∈Γoγ, |Γ| :=

∑γ∈Γ |γ|

13

and [Γ] =∑

γ∈Γ[γ]. Notationally, we will often oγ or oΓ to also mean oΓ ⊗ Q,suppressing the tensor up to Q. We then define the contact dg–algebra as:

CC(Y, ξ)D :=∧( ⊕

γ∈Pg(Y,α)

oγ

),

that is, CC(Y, ξ)D is the free graded commutative unital Z2-graded Q-algebra gen-erated by oγ for γ ∈ Pg(Y, α).

If we let Pg,(Y, α) denote the space of 0 loop–graded orbits, then we can definethe subalgebra CC(Y, ξ)D by:

CC(Y, ξ)D :=∧( ⊕

γ∈Pg,(Y,α)

oγ

).

The Zm(ξ)–grading on CC(Y, ξ)D is defined as CZ(γ, [Σ]) + n− 3 mod m(ξ).(Differential) The differential ∂D : CC(Y, ξ)D → CC(Y, ξ)D is defined as so. For

any pure element x ∈ oγ+ in the algebra:

∂Dx :=∑

µ(γ+,Γ−,β)=1

#θMI(Y ; γ+,Γ−, β)

|Aut(γ+,Γ−, β)|· x.

On a product xy of elements x, y ∈ CC(Y, α), ∂D satisfies the graded Leibniz rule∂D(xy) = ∂D(x)y + (−1)|x|x∂D(y) for any x, y ∈ CC(Y, ξ)D.

Here MI(Y ; γ+,Γ−, β) is the compactified moduli space of genus 0 curves in

the symplectization Y with one positive puncture asymptotic to γ+ and k negativepunctures asymptotic to Γ− (along with appropriate asymptotic marker behavior atthe punctures). The symbol #θ denotes taking a virtual moduli count in Γ− ⊗ ∨γ+with respect to the VFC data θ comin with the data D. See [14, §1.2, 2.3] for a fulldiscussion.

Construction 3.4 (Cobordism Maps). Consider an exact symplectic cobordism(W,λ) between contact manifolds (Y+, ξ+) and (Y−, ξ−), along with a choice of dataD ∈ Data(W,V, λ) as in Setup 3.2, where A = π+D and B = π−D. Then we canconstruct a morphism of differential Z2–graded algebras:

CC(W,λ)D : CC(Y+, ξ+)A → CC(Y−, ξ−)B.

To define this cobordism map, we declare its value on generators and extend it toan algebra map. On an x ∈ oγ+ for γ+ ∈ P(Y+, α+), it is given by the sum:

CC(W,λ)D(x) =∑

(Γ−,β)|µ(γ+,Γ−,β)=0

#θMII(W ; γ+,Γ−, β)

|Aut(γ+,Γ−, β)|· x.

14

Here MII(W ; γ+,Γ−, β) is the compactified moduli space of genus 0 in the comple-

tion W of W with one positive puncture asymptotic to γ+ and k negative puncturesasymptotic to Γ− (along with appropriate asymptotic marker behavior at the punc-tures). The symbol #θ denotes taking a virtual moduli count in Γ−⊗∨γ+ with respectto the VFC data θ, as with the differential. See [14, §1.3, 2.3] for a full discussion.

If W ' Y × [0, 1] is cylindrical (as a smooth manifold) then CC(W,λ)D respectsthe H1(Y ;Z) grading and provides a map of differential Zm(ξ)–graded algebras:

CC(W,λ)D : CC(Y+, ξ+)A → CC(Y−, ξ−)B.

The salient features of the construction in [14] can be summarized in the followingTheorem, various parts of which are covered in [14, §1.3− 1.8].

Theorem 3.5 ([14] Contact homology). The dg–algebra and maps described inConstructions 3.3 and 3.4 have the following properties.

(a) (Homology) The map ∂D of Construction 3.3 is a differential, i.e. ∂2D = 0.

The homology algebra CH(Y, ξ) ' CH(Y, ξ)D is independent of the choice ofdata D ∈ Data(Y, ξ) up to canonical isomorphism.

(b) (Cobordism) The cobordism map CC(W,λ)D of Construction 3.4 is a map ofdg-algebras, i.e. CC(W,λ)D ∂π+D − ∂π−D CC(W,λ)D = 0. The maps onhomology CH[W,λ]D, given a choice of data D ∈ Data(W,λ), induces a mapCH[W,λ] : CH(Y+, ξ+) → CH(Y−, ξ−) depending only on the deformationclass of the Liouville cobordism [W,λ].

(c) (Nullhomologous subalgebra) The homology algebra CH(Y, ξ) ' CH(Y, ξ)Dis independent of D up to canonical isomorphism, and if W ' Y × [0, 1] thenwe have a map CH[W,λ] : CH(Y+, ξ+) → CH(Y−, ξ−) depending only onthe deformation class [W,λ].

(d) (Contactomorphism) An isotopy class of contactomorphism [Φ] : (Y0, ξ0) →(Y1, ξ1) induces isomorphisms CH[Φ] : CH(Y0, ξ0) ' CH(Y1, ξ1) and CH[Φ] :CH(Y0, ξ0) ' CH(Y1, ξ1).

(e) (Transversality) If the moduli spaces of either Construction 3.3 or 3.4:

MI(Y ; γ+,Γ−, β) MII(W ; γ+,Γ−, β)

are regular (i.e. the linearized ∂J operators on each component (includingbroken components) is surjective, see [14, §2.11]) then:

#θMI(Y ; γ+,Γ−, β) = #MI(Y ; γ+,Γ−, β)

15

#θMII(W ; γ+,Γ−, β) = #MII(W ; γ+,Γ−, β)

Here #S for a oriented zero–dimensional compact manifold S is equivalent toan oriented point count. In particular, if either moduli space is empty then thevirtual count is 0.

3.2 Computations for ellipsoids

We now compute the examples of contact homology and cobordism maps that arerelevant to our construction.

Definition 3.6. A contact form α on a closed contact manifold (Y, ξ) is lacunaryif, for every orbit γ ∈ Pg(Y, ξ), the grading |γ| ∈ Z2 is 0.

The contact homology dg–algebra of a contact manifold with lacunary contactform has vanishing differential for grading reasons. In particular, we have:

Lemma 3.7. Let (Y, ξ) be a contact manifold with lacunary contact form α. Then:

CH(Y, ξ) '∧(

Q[Pg(Y, α)])

CH(Y, ξ) '∧(

Q[Pg,(Y, α)]).

Example 3.8 (Ellipsoids). Let (a1, . . . , an) ∈ (0,∞)n satisfy ai/aj 6∈ Q for eachi 6= j. Recall that the ellipsoid E(a1, . . . , an) ⊂ Cn is given by (1.1).

Consider the contact hypersurface Y = ∂E(a1, . . . , an) ⊂ Cn with contact formα = λ|Y , where λ is the Liouville form λ = 1

2

∑i xidyi−yidxi. We now calculate the

Reeb vector field, orbits, orbit actions and orbit Conley–Zehnder indices of (Y, α).In particular, we show that the contact form α is lacunary, thus computing thecontact homology by Lemma 3.7.

The Reeb vector field Rα is given by Rα = 2π∑

i a−1i

∂∂θi

. Here θi is the angularcoordinate in the ith C factor Ci of Cn. The flow Φα : Y × R → Y by Rα is givenby:

z = (z1, . . . , zn) 7→ Φtα(z) = (e2πt/a1z1, . . . , e

2πt/an). (3.1)

Due to our assumption that ai/aj 6∈ Q for each i 6= j, there are precisely n simpleorbits γi for 1 ≤ i ≤ n. Each γi is a parameterization of the curve of points in Ywith zj = 0 for all j 6= i. The action of γmi is given by Aα(γmi ) = mai by (3.1), sincethe action Aα(γ) =

∫S1 γ

∗α of a Reeb orbit coincides with the period.The Conley–Zehnder index of γmi can be computed as so. Given any contact

manifold (Y, α) with c1(ξ) = 0, a non-degenerate Reeb orbit γ and a homotopy classof symplectic trivialization τ of γ∗ξ, the Conley-Zehnder index CZ(γ) can be writtenas:

CZ(γ) = CZ(γ, τ) + 2c1(γ, τ)

16

Here CZ(γ, τ) is the Conley-Zehnder index of the linearized Reeb flow with respectto the trivialization τ and c1(γ, τ) is the relative first Chern number with respect toτ of the pullback u∗ξ of ξ to a capping disk u of γ.

Returning to the case where (Y, ξ) is the boundary ∂E(a1, . . . , an) as above wenote that, along γmi , the fiber ξγ(t) agrees at each t with the orthogonal complex sub-space to the ith component C⊥i = ⊕j 6=iCj ⊂ Cn ' TCn

γ(t). By 3.1, linearized flow

in this trivialization is a direct sum of loops t 7→ e2πimait/aj for j 6= i for t ∈ [0, 1].Thus we have

CZ(γmi , τ) =∑j 6=i

(2bmai

ajc+ 1

).

The relative Chern number c1(γmi , τ) is given by:

c1(γmi , τ) = m

Thus we have the following formula for the Conley-Zehnder index of γmi :

CZ(γmi ) = n− 1 + 2 |L ∈ Spec(Y, α) | L ≤ mai| .

Finally, note that since H1(Y,Z) = 0 and c1(ξ) = 0 for (Y, ξ), CH(Y, ξ) =CH(Y, ξ) and the algebra is Z–graded. By the above calculation, we have

|γmi | = 2n− 4 + 2|L ∈ Spec(Y, α)|L ≤ mai| ∈ Z.

Therefore the grading is even and the contact form is lacunary. The contact homol-ogy is thus computed by the formula in Lemma 3.7.

Lemma 3.9. Let (Y, ξ) be a closed contact manifold. Let λ be a Liouville form withLiouville vector field Z positively transverse to the fibers of π : Y × I → I. Thenthe deformation class [Y × I, λ] is independent of λ and induces the identity mapCH[Y × I, λ] = Id on CH(Y, ξ).

Proof. First we show that such cobordisms determine a single deformation class.Let Y+ and Y− denote the positive and negative boundary copies of Y , respectively.Let (Y ×R+, λ) denote the partial completion of Y × [0, 1] by attaching Y × [1,∞)with Liouville form etλ|Y+ . We denote the extended Liouville vector field on Y ×R+

by Z. Let T > 0 be a time such that ΦZT (y, 0) ∈ Y × (1,∞) for all y ∈ Y , let

Σ = ΦZT (Y × 0), and let τ : Y → R+ be the smooth function of time such that

ΦZτ(y)(y, 1) ∈ Σ. Then the family λt of Liouville forms:

λs := Ψ∗sλ Ψs : Y → Y defined as Ψs(y, r) = ΦZ(sr)·τ(y)(y, r)

is a deformation of Liouville structures on Y × I with λ0 = λ and λ1 having theproperty that ΦZ1

T (y, 0) = (y, 1).

17

Thus we may assume these properties without altering the deformation class.Pulling back by the diffeomorphism Y × I → Y × I given by (y, r) 7→ ΦZ

Tr(y, 0)yields a form with Liouville vector-field Z = ∂t. Thus the Liouville form can beassumed to be λ = etλ|Y− . Any two such such forms are deformation equivalent,since the space of contact forms is contractible.

Second, we note that CH[Y × I, λ] = Id. In particular, we can choose λ = etαfor some contact form α, and then this is [14, Lemma 1.2].

4 The space of symplectic embeddings

In this section, we discuss some basic results about the Frechet manifold of sym-plectic embeddings SympEmb(U, V ) between symplectic manifolds with boundary.In §4.1, we construct the Frechet manifold structure on SympEmb(U, V ). In §4.2,we discuss the relationship between the bordism groups and homology groups ofa Frechet manifold. Last, we prove a version of the Weinstein neighborhood withboundary as Proposotion 4.13 in §4.3.

4.1 Frechet manifold structure

Let (U, ωU) and (V, ωV ) be n–dimensional compact symplectic manifolds with nonemptycontact boundaries. We now give a proof of the folklore result that the space of sym-plectic embeddings from U to V is a Frechet manifold.

Proposition 4.1. The space SympEmb(U, V ) of symplectic embeddings ϕ : U →int(V ) with the C∞ compact open topology is a metrizable Frechet manifold.

Proof. Let (U×V, ωU×V ), with ωU×V = π∗UωU−π∗V ωV , denote the product symplecticmanifold with corners. Given a symplectic embedding ϕ : U → int(V ), we mayassociate the graph Γ(ϕ) ⊂ U × V given by:

Γ(ϕ) := (u, ϕ(u)) ∈ U × V .

The graph is a Lagrangian submanifold with boundary transverse to the character-istic foliation T (∂U)ω on the contact hypersurface ∂U × int(V ). By the Weinsteinneighborhood theorem with boundary, Proposition 4.13, there is a neighborhoodA of U , a neighborhood B of Γ(ϕ) and a symplectomorphism ψ : A ' B withψ|U : U → Γ(ϕ) given by u 7→ (u, ϕ(u)) and ψ∗ωU×V = ωstd.

Let A(ϕ, ψ) ⊂ ker(d : Ω1(L) → Ω2(L)) and B(ϕ, ψ) ⊂ SympEmb(U, V ) denotethe open subsets given by:

A(ϕ, ψ) := α ∈ Ω1(L)|dα = 0 and Im(α) ⊂ A

18

B(ϕ, ψ) := φ ∈ SympEmb(U, V )| Im(ϕ) ⊂ B.

Then we have maps Φ : A(ϕ, ψ)→ B(ϕ, ψ) and Ψ : A(ϕ, ψ)→ B(ϕ, ψ) given by:

α 7→ Φ[α] := (πV ψ α) (πU ψ α)−1

φ 7→ Ψ[φ] := (ψ−1 (Id×φ)) (πL ψ−1 (Id×φ))−1.

It is a tedious but straight forward calculation to check that ΦΨ = Id and ΨΦ =Id. The fact that Φ and Ψ are continuous in the C∞ compact open topologies onthe domain and images follows from the fact that function composition defines acontinuous map C∞(M,N) × C∞(N,O) → C∞(M,O) for any compact manifoldsM,N and O (in fact, smooth; see [16, Theorem 42.13]).

Since C∞(U, V ) is metrizable under the compact open C∞–topology (see [16,Corollary 41.12]), the subspace SympEmb(U, V ) is also metrizable.

Lemma 4.2. Let L be a compact manifold with boundary and let σ : L→ T ∗L be asection. Then σ(L) is Lagrangian if and only if σ is closed.

Proof. The same as the closed case, see [12, Proposition 3.4.2].

4.2 Bordism groups of Frechet manifolds

We now discuss (unoriented) bordism groups and their structure in the case ofFrechet maifolds. We begin by defining the relevant notions of (continuous andsmooth) bordism.

Definition 4.3 (Bordisms). Let X be a topological space and f : Z → X be amap from a closed manifold. We say that the pair (Z, f) is null–bordant if thereexists a pair (Y, g) of a compact manifold with boundary Y and a continuous mapg : Y → X such that ∂Y = Z and g|∂Y = f . Given a pair of manifold/map pairs(Zi, fi) for i ∈ 0, 1, we say that (Z0, f0) and (Z1, f1) are bordant if (Z0tZ1, f0tf1)is null–bordant.

Definition 4.4 (Smooth bordism). Let X be a Frechet manifold and f : Z → X bea smooth map from a smooth closed manifold. Then (Z, f) is smoothly null–bordantif it is null–bordant via a pair (Y, g) where g : Y → X be a smooth map of Banachmanifolds with boundary. Similarly, a pair (Zi, fi) for i0, 1 is smoothly bordant if(Z0 t Z1, f0 t Z1) is smoothly null-bordant.

The above notions come with accompanying versions of the bordism group.

19

Definition 4.5 (Bordism Group Of X). The n–th bordism group Ωn(X;Z2) of atopological space X is group generated by equivalence classes [Z, f ] of pairs (Z, f),where Z is a closed n–dimensional manifold and f : Z → X is a continuous map,modulo the relation that (Z0, f0) ∼ (Z1, f1) if the pair is bordant. Addition isdefined by disjoint union:

[Z0, f0] + [Z1, f1] := [Z0 t Z1, f0 t f1].

Definition 4.6 (Smooth bordism group of X). The n–th smooth bordism groupΩ∞n (X;Z2) of a Frechet manifold X is group generated by equivalence classes [Z, f ]of pairs (Z, f), where Z is a closed n–dimensional manifold and f : Z → X is asmooth map, modulo the relation that (Z0, f0) ∼ (Z1, f1) if the pair is smoothlybordant. Addition in the group Ω∞∗ (X;Z2) is defined by disjoint union as before.

Lemma 4.7. The natural map Ω∞∗ (X;Z2)→ Ω∗(X;Z2) is an isomorphism.

Proof. The argument uses smooth approximation and is identical to the case whereX is a finite dimensional smooth manifold, which can be found in [4, Section I.9].

Given the above terminology, we can now prove the main result of this subsection,Proposition 4.8. It provides a class of submanifolds for which being null–bordantand being null–homologous are equivalent.

Proposition 4.8. Let X be a metrizable Frechet manifold, and let f : Z → X bea smooth map from a closed manifold Z with Stieffel–Whitney class w(Z) = 1 ∈H∗(Z;Z2). Then f∗[Z] = 0 ∈ H∗(X;Z2) if and only [Z, f ] = 0 ∈ Ω∞∗ (X;Z2).

Proof. Proposition 4.8 will follow immediately from the following results. First, byLemma 4.7, it suffices to show f∗[Z] = 0 ∈ H∗(X;Z2) if and only [Z, f ] = 0 ∈Ω∗(X;Z2). By Proposition 4.9, we can replace X with a CW complex. Lemma 4.10proves the result in this context.

Proposition 4.9 ([15, Theorem 14]). A metrizable Frechet manifold is homotopyequivalent to a CW complex.

Lemma 4.10. Let X homotopy equivalent to a CW complex, and let f : Z → Xbe a continuous map from a closed manifold Z with Stieffel–Whitney class w(Z) =1 ∈ H∗(Z;Z2). Then f∗[Z] = 0 ∈ H∗(X;Z2) if and only [Z, f ] = 0 ∈ Ω∗(X;Z2).

Remark 4.11. Crucially, we make no finiteness assumptions on the CW structure.

Proof. (⇒) Suppose that f∗[Z] = 0 ∈ H2(Z;Z2). Pick a homotopy equivalenceϕ : X ' X ′ with a CW complex X ′. Such an equivalence induces an isomorphismof unoriented bordism groups Ω∗(X;Z2) ' Ω∗(X

′;Z2), so it suffices to show that the

20

pair (Z, ϕ f) is null–bordant, or equivalently to assume that X is a CW complexto begin with.

So assume that X is a CW complex. By Lemma 4.12, we can find a finite sub–complex A ⊂ X such that f(Z) ⊂ A and f∗[Z] = 0 ∈ H∗(A;Z2). By Theorem 17.2of [4], [Z, f ] = 0 ∈ Ω∗(A;Z2) if and only if the Stieffel–Whitney numbers swα,I [Z, f ]are identically 0. Recall that the Stieffel–Whitney number swα,I [Z, f ] associatedto [Z, f ], a cohomology class α ∈ Hk(A;Z2) and a partition I = (i1, . . . , ik) ofdim(Z)− k is defined to be:

swα,I [Z, f ] = 〈wi1(Z)wi2(Z) . . . wik(Z)f ∗α, [Z]〉 ∈ Z2.

Here wj(Z) ∈ Hj(Z;Z2) denotes the j–th Stieffel–Whitney class of Z. By assump-tion, w(Z) = 1 and so wj(Z) = 0 for all j 6= 0. In particular, the only possiblenonzero Stieffel–Whitney numbers have I = (0). But we see that:

swα,(0)[Z, f ] = 〈f ∗α, [Z]〉 = 〈α, f∗[Z]〉 = 0.

Therefore, swα,I [Z, f ] ≡ 0 and [Z, f ] must be null–bordant.(⇐) This direction is completely obvious, since the map Ω∗(X) → H∗(X;Z2)

given by [Z, f ] 7→ f∗[Z] is well defined.

Lemma 4.12. Let X be a CW complex, and let f : Z → X be a map from a closedmanifold Z with f∗[Z] = 0 ∈ H∗(X;Z2). Then there exists a finite sub–complexA ⊂ X with f(Z) ⊂ A and f∗[Z] = 0 ∈ H∗(A;Z2).

Proof. A very convenient tool for this is the stratifold homology theory of [9], whichwe now review briefly.

Given a space M , the n–th stratifold group sHn(M ;Z2) with Z2–coefficients(see Proposition 4.4 in [9]) is generated by equivalence classes of pairs (S, g) of acompact, regular stratifold S and a continuous map g : S → M . Two pairs (Si, gi)for i ∈ 0, 1 are equivalent if they are bordant by a c–stratifold, i.e. if there is apair (T, h) of a compact, regular c–stratifold and a continuous map g : T →M suchthat (∂T, h|∂T ) = (S0 t S1, g0 t g1) (see Chapter 3 and Section 4.4 of [9]). Given amap ϕ : M → N of spaces, the pushforward map ϕ∗ : sH(M ;Z2)→ sH(M ;Z2) onstratifold homology is given (on generators) by [S, g] 7→ [S, ϕ g] = ϕ∗[Σ, g].

Stratifold homology satisfies the Eilenberg–Steenrod axioms (see Chapter 20of [9]), and thus if M is a CW complex then there is a natural isomorphismsH∗(M ;Z2) ' H∗(M ;Z2). If M is a manifold of dimension n, the fundamentalclass [M ] ∈ sHn(M ;Z2) is given by the tautological equivalence class [M ] = [M, Id].

The proof of the lemma is simple with the above machinery in place. Sincef∗[Z] = 0, the pair (Z, f) must be null–bordant via some compact c–stratifold(Y, g). Since Y and its image g(Y ) are both compact, we can choose a sub–complex

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A ⊂ X such that g(T ) ⊂ A ⊂ X. Then the pair (Z, f) are null–bordant by (Y, g) inA as well, so that [Z, f ] = 0 ∈ sH∗(A;Z2) and thus f∗[Z] = 0 ∈ H∗(A;Z2) via theisomorphism sH∗(A;Z2) ' H∗(A;Z2).

4.3 Weinstein neighborhood theorem with boundary

In this section, we prove the analogue of the Weinstein neighborhood theorem for aLagrangian L with boundary, within a symplectic manifold X with boundary. Wecould find no reference for this fact in the literature.

Proposition 4.13 (Weinstein neighborhood theorem with boundary). Let (X,ω)be a symplectic manifold with boundary ∂X and let L ⊂ X be a properly embedded,Lagrangain submanifold with boundary ∂L ⊂ ∂X transverse to T (∂X)ω.

Then there exists a neighborhood U ⊂ T ∗L of L (as the zero section), a neigh-borhood V ⊂ X of L and a diffeomorphism f : U ' V such that ϕ∗(ω|V ) = ωstd|U .

Proof. The proof has two steps. First, we construct neighborhoods U ⊂ T ∗L andV ⊂ X of L, and a diffeomorphism ϕ : U ' V such that:

ϕ|L = Id ϕ∗(ω|V )|L = ωstd|L T (∂U)ωstd = T (∂U)ϕ∗ω (4.1)

Here T (∂U)ωstd ⊂ T (∂U) is the symplectic perpendicular to T (∂U) with respect toωstd (and similarly for T (∂U)ϕ

∗ω. Second, we apply Lemma 4.14 and a Moser typeargument to conclude the result.

(Step 1) Let J be a compatible almost complex structure on X and g be theinduced metric on L. Recall that the normal bundle νgL with respect to g is a bundleover L with Lagrangian fiber, and that J : TL→ νgL gives a natural isomorphism.Let Φg : T ∗L → TL denote the bundle isomorphism induced by the metric g andlet expg denote the exponential map with respect to g.

Since L is compact, we can choose a tubular neighborhood U ′ of νL such thatexpg : U → X is a diffeomorphism onto its image V . We then let U := [J Φg]−1(U ′) ⊂ T ∗L and let:

φg : U ' V (x, v) 7→ expgx(J Φg(v)).

Note that φg|L = Id and [φg]∗ω|L = ωstd|L by the same calculations as in [12,Theorem 3.4.13]. We now must modify U, V and φg to satisfy the last condition of(4.1).

To this end, we apply Lemma 4.15. Taking κ0 = T (∂U)ωstd and κ1 = T (∂U)[φg ]∗ω,we acquire a neighborhood N ⊂ ∂(T ∗L) of ∂L and a family of embeddings ψ :N × I → ∂(T ∗L) with the following four properties.

ψt|∂L = Id d(ψt)u = Id for u ∈ ∂L ψ0 = Id

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[ψ1]∗(T (∂U)ωstd) = T (∂U)[φg ]∗ω.

Note here that we are using the fact that T (∂U)ωstd |L = T (∂U)[φg ]∗ω|L already by theconstruction of φg. By shrinking N and U , we can simply assume that N = ∂U . Lettc : [0, 1) × ∂U ' T ⊂ U be tubular neighborhood coordinates near boundary. Bychoosing the tubular neighborhood coordinates tc : [0, 1) × ∂U ' T appropriately,we can also assume that tc([0, 1)× ∂L) = L∩T . We define a map Φ : U → T ∗L by:

Φ(u) =

(s, ψ1−s(v)) if u = (s, v) ∈ [0, 1)× ∂U via tc

u otherwise.

The map Φ has the following properties which are analogous to those of ψs.

Φ|L = Id d(Φ)u = Id for u ∈ L Φ∗(T (∂L)ωstd) = T (∂L)[φg ]∗ω

Also note that Φ is smooth since ψt is constant for t near 0 and 1. We thus definef as the composition ϕ = φg Φ. It is immediate that f has the properties in (4.1).

(Step 2) We closely follows the Moser type argument of [12, Lemma 3.2.1]. Byshrinking U , we may assume that it is an open disk bundle. Let ωt = (1−t)ωstd+tf ∗ωand τ = d

dt(ωt) = f ∗ω − ωstd. Let κ = T (∂U)ωt (by the previous work, it does not

depend on t). Note that τ satisfies all of the assumptions of Lemma 4.14(4.3). Weprove that κ is invariant under the scaling map φt(x, u) = (x, tu) in Lemma 4.16.We can thus find a σ satisfying the properties listed in (4.2).

Let Zt be the unique family of vector fields satisfying σ = ι(Zt)ωt. Due to theproperties of σ, Zt satisfies the following properties for each t.

Zt|L = 0 Zt|∂U ∈ T (∂U) for all t

The first property is immediate, while the latter is a consequence of the following.

ωt(Zt, ·)|κ = σ|κ = 0 =⇒ Zt ∈ (κ)ωt = T (∂U)

These two properties imply that Zt generates a map Ψ : U ′ × [0, 1] → U for somesmaller tubular neighborhood U ′ ⊂ U with the property that Ψt|L = Id and Ψ∗tωt =ω0 (see [12, §3.2], as the reasoning is identical to the closed case). In particular,we get a map Ψ1 : U ′ → U with Ψ1|L = Id and Ψ∗1f

∗ω. By shrinking U , takingϕ = f Ψ1 and taking V = ϕ(U), we at last acquire the desired result.

The remainder of this section is devoted to proving the various lemmas that weused in the proof above.

Lemma 4.14 (Fiber integration with boundary). Let X be a compact manifold withboundary, π : E → X be a rank k vector bundle with metric and π : U → X be the

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(open) disk bundle of E with closure U . Let κ ⊂ T (∂U) be a distribution on ∂Usuch that dφt(κu) = κφt(u) for all u ∈ U , where φ : U × I → U denote the family ofsmooth maps given by φt(x, u) := (x, tu).

Finally, suppose that τ ∈ Ωk+1(U) is a (k + 1)–form such that:

dτ = 0 τ |X = 0 (ι∗∂Xτ)|κ = 0. (4.2)

Then there exists a k–form σ ∈ Ωk(U) with the following properties:

dσ = τ σ|X = 0 (ι∗∂Xσ)|κ = 0. (4.3)

Proof. We use integration over the fiber, as in [12, p. 109]. Note that the mapsφt : U → φt(U) ⊂ U are diffeomorphisms for each t > 0, φ0 = π, φ1 = Id andφt|X = Id. Therefore we have:

φ∗0τ = 0 φ∗1τ = τ

We may define a vector field Zt for all t > 0 and a k-form σt for all t ≥ 0 as so.

Zt := (d

dtφt) φ−1

t for t > 0 σt := φ∗t (ι(Zt)τ) for t ≥ 0

Although Zt is singular at t = 0, as in [12] one can verify in local coordinatesthat σt is smooth at t = 0. Since Zt|X = 0, the k–form σt satisfies σt|X = 0.Furthermore, for any vector field K ∈ Γ(κ) on ∂X which is parallel to κ, we haveι(K)σt = φ∗t (ι(Zt)ι(dφt(K))τ) = 0 on the boundary, so that ι∗∂X(σt)|κ = 0. Finally,σt satisfies the following equation:

τ = φ∗1τ − φ∗0τ =

∫ 1

0

d

dt(φ∗t τ)dt =

∫ 1

0

φ∗t (LXtτ)dt

=

∫ 1

0

d(φ∗t (ι(Xt)τ))dt =

∫ 1

0

dσtdt = d(

∫ 1

0

σtdt).

Therefore, if we define σ :=∫ 1

0σtdt, it is simple to verify the desired properties using

the corresponding properties for σt.

Lemma 4.15. Let U be a manifold and L ⊂ U be a closed submanifold. Let κ0, κ1

be rank 1 orientable distributions in TU such that κi|L∩TL = 0 and κ0|L = κ1|L.Then there exists a neighborhood U ′ ⊂ U of L and a family of smooth embeddings

ψ : U ′s× I → U with the following four properties.

ψt|∂L = Id d(ψt)u = Id for u ∈ L ψ0 = Id [ψ1]∗(κ0) = κ1.

Furthermore, we can take ψt to be t–independent for t near 0 and 1.

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Proof. Since κ0 and κ1 are orientable, we can pick nonvanishing sections Z0 and Z1

We may assume that Z0 = Z1 along L. We let Zt denote the family of vector fieldsZt := (1− t)Z0 + tZ1. Since Z0 = Z1 along L, we can pick a neighborhood N of Lsuch that Zt is nowhere vanishing for all t. We also select a sub-manifold Σ ⊂ Nwith dim(Σ) = dim(U)− 1 and such that:

Σ t Zt for all t and L ⊂ Σ.

We can find such a Σ by, say, picking a metric and using the exponential map on aneighborhood of L in the sub–bundle νL ∩ κ⊥0 of TL. By shrinking Σ and scalingZt to λZt, 0 < λ < 1, we can define a smooth family of embeddings:

Ψ : (−1, 1)s × Σ× [0, 1]t → N Ψt(s, x) = exp[Zt]s(x).

Here exp[Zt] denotes the flow generated by Zt. We let ψt = Ψt Ψ−10 . To see

the properties of (4.1), note that Ψt(0, l) = l for all l ∈ L and d(Ψt)0,l(s, u) =sZt + u. This implies the first two properties. The third is trivial, while the fourthis immediate from [Φt]∗(∂t) = Zt. We can make ψt constant near 0 and 1 by simplyreparameterizing with respect to t.

Lemma 4.16. Let L be a manifold with boundary and let (T ∗L, ω) be the cotangentbundle with the standard symplectic form. Let κ = T (∂T ∗L)ω denote the charac-teristic foliation of the boundary ∂T ∗L and let φ : T ∗L × (0, 1] → T ∗L denote thefamily of maps φt(x, v) = (x, tv). Then [φt]∗(κ) = κ.

Proof. By passing to a chart, we may assume that L ⊂ R+x1× Rn−1

x and T ∗L ⊂R+x1× Rn−1

x × Rnp . Then κ is simply given on ∂T ∗L ⊂ 0 × ×Rn−1

x × Rnp by:

κ = span(∂p1) = span(∂x1)ω ⊂ T (∂T ∗L).

Under the scaling map, we have [φt]∗(∂p1) = t·∂p1 . This implies that [φt]∗(κ) = κ.

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