Essential tori in spaces of symplectic embeddings
aa r X i v : . [ m a t h . S G ] M a r Essential tori in spaces of symplectic embeddings
Julian Chaidez, Mihai MunteanuMarch 21, 2019
Abstract
Given two 2 n –dimensional symplectic ellipsoids whose symplectic sizessatisfy certain inequalities, we show that a certain map from the n –torus tothe space of symplectic embeddings from one ellipsoid to the other inducesan injective map on singular homology with mod 2 coefficients. The proofuses parametrized moduli spaces of J –holomorphic cylinders in completedsymplectic cobordisms. 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 target 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 = ( a , a , . . . , a n )define the symplectic ellipsoid E ( a ) by E ( a ) = E ( a , a , . . . , a n ) := ( ( z , . . . , z n ) ∈ C n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 π | z i | a i ≤ ) . (1.1)The space E ( a ) carries the structure of an exact symplectic manifold with boundaryendowed with the restriction of the standard Liouville form λ on C n , given by λ = 12 n X i =1 ( x i dy i − y i dx i ) . (1.2)A special case is the symplectic ball B n ( r ), which is simply E ( a ) for a = ( r, . . . , r ).1he types of results that one can prove about symplectic embeddings, togetherwith the tools used to do so, are surveyed at length by Schlenk in [22]. 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 nonsqueezing theorem , proven in the seminal paper [9]. Theorem 1.1 ([9]) . There exists a symplectic embedding B n ( r ) → B ( R ) × C n − if and only if r ≤ R . This result demonstrated that there are obstructions to symplectic embeddingsbeyond the volume and initiated the study of quantitative symplectic geometry.Note that Theorem 1.1 can be seen as a result about ellipsoid embeddings, since B ( R ) × C n − 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 [15]. Let { N k ( a, b ) } k ≥ de-note the sequence of nonnegative integer linear combinations of a and b , orderednondecreasingly with repetitions. Theorem 1.2 ([15]) . There exists a symplectic embedding int( E ( a, b )) → E ( a ′ , b ′ ) if and only if N k ( a, b ) ≤ N k ( a ′ , b ′ ) for every nonnegative integer k . A special case of this embedding problem, where the target ellipsoid is the ball B ( λ ), was studied by McDuff and Schlenk in an earlier paper [17] using methodsdifferent from [15]. In that paper, McDuff and Schlenk give a remarkable calculationof the function c : R + → R + defined by c ( a ) := inf (cid:8) λ (cid:12)(cid:12) E (1 , a ) symplectically embeds into B ( λ ) (cid:9) . In particular, they show that for a ∈ [1 , ( √ ) ], the function c is given by a piece-wise linear function involving the Fibonacci numbers, which they call the Fibonaccistaircase . Some higher dimensional cases of the existence problem for symplectic em-beddings have been studied in a similar manner. For instance, a family of stabilizedanalogues of the function c , which are defined as c n ( a ) := inf (cid:8) λ (cid:12)(cid:12) E (1 , a ) × C n symplectically embeds into B ( λ ) × C n (cid:9) , are studied in the more recent papers [5] and [6].2eyond problems of existence, one can ask about the algebraic topology of thespace of symplectic embeddings SympEmb( U, V ) between two symplectic manifolds U and V , with respect to the C ∞ topology. Again, most results have been provenin dimensions 2 and 4. For instance, in [14], McDuff demonstrated that the spaceof embeddings between 4–dimensional symplectic ellipsoids is connected wheneverit is nonempty. Other results in dimension 4 can be found in [1] and [11].More recently, in [18], 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. In this paper, we build upon the methods developed in [18] to tackle the question ofdescribing the higher homology groups of spaces of symplectic embeddings betweenellipsoids in any dimension.More precisely, we will be studying families of symplectic embeddings that arerestrictions of the following unitary maps. For θ = ( θ , . . . , θ n ) ∈ T n = ( R / π Z ) n ,let U θ denote the unitary transformation U θ ( z , . . . , z n ) := ( e iθ z , . . . , e iθ n z n ) . (1.3)Given symplectic ellipsoids E ( a ) and E ( b ) such that a i < b i for every i ∈ { . . . , n } ,we may define the family of ellipsoid embeddingsΦ : T n → SympEmb( E ( a ) , E ( b )) , Φ( θ ) = U θ | E ( a ) (1.4)by restricting the domain of the maps U θ . The following theorem about the familyΦ is the main result of this paper. Theorem 1.3 (Main theorem) . Let a = ( a , . . . , a n ) and b = ( b , . . . , b n ) be twosequences of real numbers satisfying a i < b i < a i +1 for all i ∈ { , . . . , n − } and a n < b n < a . Furthermore, let
Φ : T n → SympEmb( E ( a ) , E ( b )) be the family of symplectic em-beddings (1.4). Then the induced map Φ ∗ : H ∗ ( T n ; Z / → H ∗ (SympEmb( E ( a ) , E ( b )); Z / on homology with Z / –coefficients is injective. In order to demonstrate the nontriviality of Theorem 1.3, we note that the mapinduced by Φ : T n → Symp( E ( a ) , E ( b )) on Z / E ( a ) is very small relative to E ( b ). More precisely, we have the following.3 roposition 1.4. Let a = ( a , . . . , a n ) and b = ( b , . . . , b n ) be two nondecreas-ing sequences of real numbers satisfying a n < b . Furthermore, let Φ : T n → SympEmb( E ( a ) , E ( b )) be as in (1.4). Then the induced map Φ ∗ on Z / –homologyhas rank in degree ≤ and rank otherwise. Unlike the proof of Theorem 1.3, the proof of Proposition 1.4 is an elementarycalculation in algebraic topology which we defer to § Remark 1.5 (Comparison to [18]) . In dimension 4, the fact that Φ ∗ is injective indegree 1 was proven by the second author in [18]. Specifically, this is equivalent to[18, Theorem 1.4] which states that the loopΨ : S → Symp( E ( a ) , E ( b ))defined by Ψ( t )( z , z ) := (cid:26) ( e πit z , z ) t ∈ (cid:2) , (cid:3) ( e − πit z , z ) t ∈ (cid:0) , (cid:3) is noncontractible. In fact, [18] actually addresses the more general 4–dimensionalcase where E ( a ) and E ( b ) are replaced with convex toric domains in C satisfyingspecific inequalities involving their ECH capacities. We expect Theorem 1.3 to holdat this level of generality, and we hope to address this in future work using somewhatdifferent methods (see Remark 1.7). Remark 1.6 ( Z vs Z / . Our use of Z / Z coef-ficients, allows us to use the methods of § § Z coefficientsas well. We plan to develop the methods needed to work over Z in forthcomingwork. Remark 1.7 (Lagrangian analogues) . In forthcoming work, we hope to demonstrateresults analogous to Theorem 1.3 for families of Lagrangian torus embeddings intoric domains. We anticipate that these results will be useful for demonstrating thevarious generalization of Theorem 1.3 discussed in Remark 1.5.
Organization.
The rest of the paper is organized as follows. In §
2, we establish thegeometric setup and notation. In §
3, we construct the moduli spaces and prove theneeded transversality and compactness properties together with a lemma about thecount of curves in these moduli spaces to build up towards a proof of Theorem 1.3.Lastly, in §
4, we prove some useful technical results about the topology of spaces ofsymplectic embeddings. 4 cknowledgements.
We would like to thank our advisor, Michael Hutchings forall the helpful discussions and for pointing our some significant simplifications toearlier drafts. JC was supported by the NSF Graduate Research Fellowship underGrant No. 1752814. MM was partially supported by NSF Grant No. DMS–1708899.
In this section, we review the concepts from contact geometry and holomorphiccurve theory needed in this paper. For a more comprehensive discussion of thesetopics, see [8], [16], [23] and [24].
We begin by providing a quick overview of basic contact geometry and establishingnotation for §
3. We include a review the Reeb dynamics on the boundary of asymplectic ellipsoid with rationally independent defining parameters.
Review 2.1 (Contact manifolds) . Recall that a contact manifold ( Y, ξ ) is a smooth(2 n − Y together with a rank 2 n − ξ ⊂ T Y that is givenfiberwise by the kernel ξ = ker( α ) of a contact 1–form α ∈ Ω ( Y ). A contact form α is a 1–form on Y satisfying α ∧ dα n − = 0 everywhere.Every contact form α on Y has a naturally associated Reeb vector field R α definedimplicitly from α via the equations ι R α α = 1 , ι R α dα = 0 . (2.1)The Reeb flow Φ α : Y × R → Y is the flow of the vector field R α , i.e. the family ofdiffeomorphisms satisfying d Φ tα ( y ) dt (cid:12)(cid:12)(cid:12)(cid:12) t = s = R α ◦ Φ sα ( y ) . (2.2)A Reeb orbit is a closed orbit of the flow Φ α , i.e. a curve γ : S = R /L Z → Y satisfying dγdt = R α ◦ γ for some positive number L which is called the period . Notethat L coincides with the action A α ( γ ) of γ , which is defined as A α ( γ ) := Z S γ ∗ α. (2.3)A Reeb orbit γ is called nondegenerate if the differential T Φ Lγ (0) of the time L flow satisfies det( T Φ Lγ (0) | ξ − Id ξ ) = 0 . (2.4)5 contact form α is called nondegenerate if every Reeb orbit of α is nondegenerate. Review 2.2 (Conley–Zehnder indices) . Any nondegenerate Reeb orbit γ posseses afundamental numerical invariant called the Conley–Zehnder index
CZ( γ, τ ), whosedefinition and computation we now review.The Conley–Zehnder index CZ( γ, τ ) depends on a choice of symplectic trivializa-tion τ : γ ∗ ξ ≃ S × C n − . The invariant is defined by CZ( γ, τ ) := µ RS ( φ ) were µ RS denotes the Robbin–Salamon index defined in [21] and φ is the path of symplecticmatrices defined as φ : [0 , L ] → Sp(2 n − , φ ( t ) := τ γ ( t ) ◦ T Φ tγ (0) | ξ ◦ τ − γ (0) . In the case where c ( ξ ) = 0 ∈ H ( Y ; Z ) and [ γ ] = 0 ∈ H ( Y ; Z ), a canonicalConley–Zehnder index CZ( γ ) which does not depend on a choice of trivializationcan be associated to γ via the following procedure. Extend γ to a map u : Σ → Y from an oriented surface Σ with boundary ∂ Σ = S satisfying u | ∂ Σ = γ . Pick asymplectic trivialization σ : u ∗ ξ ≃ Σ × C n − and define CZ( γ ) by the formulaCZ( γ ) := CZ( γ, σ | ∂ Σ ) . (2.5)The fact that CZ( γ ) is independent of Σ and σ follows from the vanishing of thefirst Chern class. The index CZ( γ ) can be related to the index CZ( γ, τ ) with respectto a trivialization τ by the formulaCZ( γ ) = CZ( γ, τ ) + 2 c ( γ, τ ) . (2.6)Here c ( γ, τ ) is the relative first Chern number with respect to τ of the pullback u ∗ ξ of ξ to a capping surface u of γ .For the purposes of this paper, we are interested in a specific family of examplesof contact manifolds, namely boundaries ∂E ( a ) of irrational symplectic ellipsoids. Example 2.3 (Ellipsoids) . Let E ( a ) be a symplectic ellipsoid with parameters a = ( a , . . . , a n ) ∈ (0 , ∞ ) n satisfying a i /a j Q for each i = j . The boundary of theellipsoid ∂E ( a ) together with the restriction of the standard Liouville form λ on C n defined by (1.2) is a contact manifold.The discussion in the proof of [10, Lemma 2.1] shows that there are precisely n simple orbits γ i = { ( z , z , . . . , z n ) ∈ ∂E ( a ) | z j = 0 , ∀ j = i } for 1 ≤ i ≤ n . All theorbits γ i are nondegenerate and their action is given by A α ( γ i ) = a i . Moreover, usingthe linearization of the Reeb flow, one can compute the Conley–Zehnder indices ofthe Reeb orbits γ mi to beCZ( γ mi ) = X j = i (cid:18) (cid:22) ma i a j (cid:23) + 1 (cid:19) + 2 m, (2.7)6hich after some smart rewriting becomesCZ( γ mi ) = n − |{ L ∈ Spec(
Y, α ) | L ≤ ma i }| . (2.8)Next, we review the basic terminology of exact symplectic cobordisms and asso-ciated structures. Throughout the discussion for the rest of the section, let ( Y ± , α ± )be closed contact (2 n − α ± . Review 2.4 (Exact symplectic cobordisms) . Recall that an exact symplectic cobor-dism ( W, λ, ι ) from ( Y + , α + ) to ( Y − , α − ) consists of the following data. · A compact, exact symplectic manifold (
W, λ ) with boundary ∂W such thatthe Liouville vector field Z defined by the equation dλ ( Z, · ) = λ is transverseto ∂W everywhere. In this situation, ∂W = ∂ + W ⊔ ∂ − W , where Z pointsoutward along ∂ + W and inward along ∂ − W . · A pair of boundary inclusion maps ι + and ι − , which are strict contactomor-phisms of the form ι + : ( Y + , α + ) ≃ ( ∂ + W, λ | ∂ + W ) , ι − : ( Y − , α − ) ≃ ( ∂ − W, λ | ∂ − W ) . (2.9)We will generally suppress the inclusions in the notation, using ι + and ι − whenneeded. The maps ι + and ι − extend, via flow along Z or − Z , to collar coordinates([0 , ǫ ) × Y − , e s λ − ) ≃ ( N − , λ | N − ) , (( − ǫ, × Y + , e s λ + ) ≃ ( N + , λ | N + ) . (2.10)Here N − and N + are collar neighborhoods of Y − and Y + respectively, the mapspreserve the 1–forms above and s denotes the coordinate on [0 , ǫ ) and ( − ǫ, W, λ, ι ) from ( Y , α ) to ( Y , α ) and ( W ′ , λ ′ , ι ′ )from ( Y , α ) to ( Y , α ), we can form the composition ( W W ′ , λ λ ′ , ι ι ′ ) by glu-ing W and W ′ via the identification ( ι ′ + ) − ◦ ι − of ∂ − W and ∂ + W ′ . The Liouvilleforms and inclusions extend in the obvious way to the glued manifold.Using these identifications (2.10), we can complete the exact symplectic cobor-dism ( W, λ ) by adding cylindrical ends ( −∞ , × Y − and [0 , ∞ ) × Y + to obtain the completed exact symplectic cobordism ( c W , b λ ), given by c W = ( −∞ , × Y − ⊔ ι − W ⊔ ι + [0 , ∞ ) × Y + . (2.11)The Liouville 1–forms λ , e s α − and e s α + glue together to a Liouville form b λ on c W .An important special caase of completed cobordisms is given by the symplectization of a contact manifold ( R × Y, e s α ), which we will denote by b Y .7iven a manifold P (with or without boundary), a P –parametrized family ofexact symplectic cobordisms W from Y + to Y − is a fiber bundle W → P over P withfiber W p at p ∈ P , a 1–form λ on W and a bundle map ι ± : P × Y ± → W such that( W p , λ | W p , ι p ) is an exact symplectic cobordism for each p ∈ P . Review 2.5. (Almost complex structures) Recall that a compatible almost com-plex structure J ξ on the symplectic vector bundle ξ gives rise to an R –invariantcompatible almost complex structure J on the symplectization R × Y , defined by J ( ∂ s ) = R α , J ( R α ) = − ∂ s , J | ξ = J ξ . We denote the set of such translation invariant J on R × Y by J ( Y ).An almost complex structure J on a completed exact symplectic cobordism c W as above is called compatible if it has the following properties. · On the ends [0 , ∞ ) × Y + and ( −∞ , × Y − , J restricts to R –invariant complexstructures arising from J + ∈ J ( Y + ) and J − ∈ J ( Y − ), respectively. · The almost complex structure J is compatible with the symplectic form dλ .We let J ( W ) denote the set of all such compatible almost complex structures ona given exact symplectic cobordism W . More generally, given a P –parametrizedfamily of exact symplectic cobordisms W , we denote by J ( W ) the space of smooth,fiberwise almost complex structures J such that J p ∈ J ( W p ) for each p ∈ P .We note that J ( W ) is contractible for any W (see for instance [16, Proposition4.11]). This implies that the space of families J ( W ) is also contractible, and thatany family J ∂P ∈ J ( W | ∂P ) over ∂P extends to a family J ∈ J ( W ) over all of P .As with contact manifolds, we are interested in a particular family of examplesof exact symplectic cobordisms related to ellipsoid embeddings. Notation 2.6 (Cobordisms of embeddings) . Let E ( a ) and E ( b ) be irrational ellip-soids. Given a symplectic embedding ϕ : E ( a ) → int( E ( b )), we denote by W ϕ theexact symplectic cobordism given by W ϕ := E ( b ) \ int( ϕ ( E ( a ))) , ι + := Id | ∂E ( b ) , ι − := ϕ | ∂E ( a ) . (2.12)More generally, let P be a compact manifold with boundary and Ψ : P × E ( a ) → int( E ( b )) be a P –parametrized family of symplectic embeddings. We then acquirea family of cobordisms W Ψ with fiber ( W Ψ p , λ Ψ p ) given by (2.12).In this context, we label the simple Reeb orbits of ∂E ( b ) by γ + i and the simpleReeb orbits of ∂E ( a ) by γ − i . The simple Reeb orbits of the negative boundary of8 ϕ are, of course, the images ϕ ( γ − i ) and will be denoted as such. Furthermore, ifthe image Im(Ψ p ) of Ψ p is independent of p sufficiently close to ∂P , then we let W ∂P denote E ( b ) \ Ψ p ( E ( a )) for any p ∈ ∂P and we let λ ∂P denote the Liouvilleform. Note that in this case, the cobordisms ( W Ψ p , λ Ψ p , ι Ψ p ) for p ∈ ∂P differ onlyby the boundary inclusion ι Ψ p . In situations where ι Ψ p plays no role, we will oftennot distinguish between ( W Ψ p , λ Ψ p , ι Ψ p ) for different p ∈ ∂P . The proof of Theorem 1.3 is centered around the analysis of certain moduli spacesof holomorphic curves. In this section, we give a quick overview of holomorphiccurves, SFT compactness, and SFT neck stretching.
Definition 2.7 (Holomorphic Curve) . Let (
W, λ ) be an exact symplectic cobordismfrom ( Y + , α + ) to ( Y − , α − ), equipped with an almost complex structure J ∈ J ( W ).Let (Σ , j ) be a Riemann surface acquired by removing a finite set P + of positivepunctures and a finite set P − of negative punctures from a closed Riemann surfaceΣ. Finally, let Γ ∗ = { γ ∗ p | p ∈ P ∗ } be a set of Reeb orbits in Y ∗ for each ∗ ∈ { + , −} .A (parametrized) holomorphic curve u : (Σ , j ) → ( c W , J ) asymptotic to Γ + at P + and Γ − at P − is a smooth map such that · u is ( j, J )–holomorphic, i.e. J u ( p ) ◦ du p = du p ◦ j p for all p ∈ Σ and · for any p ∈ P ∗ for ∗ ∈ { + , −} , there exists a holomorphic chart ϕ : S × R ∗ ≃ Σwith [ u ◦ ϕ ]( S × R ∗ ) ⊂ Y ∗ × R ∗ ⊂ c W andlim r →∗∞ ϕ ( θ, r ) = p, lim r →∗∞ [ π R ◦ u ◦ ϕ ]( θ, r ) = ∗∞ , lim r →∗∞ [ π Y ∗ ◦ u ◦ ϕ ]( · , r ) = γ ∗ p . The left-most limit above is taken in the C –topology. As an alternative to the lasttwo conditions above, we may assert that the limit of u ◦ ϕ ( · , · ∗ R ) converges in C to a parametrization of the trivial cylinder R × γ ∗ p as R → ∞ .Two (parametrized) holomorphic curves u : Σ → c W and u ′ : Σ ′ → c W are equiv-alent if there is a biholomorphism ϕ : Σ → Σ ′ with u = u ′ ◦ ϕ . An (unparametrized)holomorphic curve is a parametrized holomorphic curve up to this equivalence rela-tion. The curves in this paper will be unparametrized, unless otherwise specified.We now provide the reader with brief, very simplified reviews of SFT compact-ness and SFT neck stretching. We refer the reader to [2, §
10] for the original proofsand to [23, § eview 2.8 (SFT Compactness) . Let P be a compact manifold with boundary,and let ( Y ∗ , α ∗ ) for ∗ ∈ { + , −} be closed, nondegenerate contact manifolds. Let W be a P –paramaterized family of exact symplectic cobordisms from Y + to Y − equipped with a P –parametrized family J ∈ J ( W ) such that J p | [0 , ∞ ) × Y + = J + and J p | ( −∞ , × Y − = J − for some fixed almost complex structures J ± ∈ J ( Y ± ). Fix a sur-face Σ, acquired by taking a closed surface Σ and removing a finite set of punctures.Finally, consider a sequence p i ∈ P and u i : Σ → ( c W p i , J p i ) of J p i –holomorphiccurves asymptoting to collections of Reeb orbits Γ + (at the positive end of c W p i ) andΓ − (at the negative end of c W p i ) independent of i .The SFT compactness theorem states that, after passing to a subsequence, p i → p ∈ P and u i converges (in the SFT Gromov topology, see [2, § J p –holomorphic building , which is a tuple of the form v = ( u +1 , . . . , u + M , u W , u − , . . . , u − N ) . (2.13)Here M, N ∈ Z ≥ are integers and the elements of the tuple (called levels ) areholomorphic maps from punctured surfaces of the form u ∗ j : S ∗ j → ( R × Y ∗ , J ∗ ) for ∗ ∈ { + , −} and u W : S W → ( c W p , J p ) . The maps u ∗ j and the map u W are considered modulo domain reparametrization, andmodulo translation when the target manifold is a symplectization. The surfaces S j can be glued together along the boundary punctures asymptotic to matching Reeborbits, and this glued surface j S j is homeomorphic to Σ.All of the curves u ∗ j and u W must be asymptotic to a Reeb orbit at each positiveand negative puncture. We denote the collections of positive and negative limit Reeborbits of u W (with multiplicity) by Γ + ( u W ) and Γ − ( u W ), respectively, and we adoptsimilar notation for u ∗ j . The asymptotics of the u ∗ j and u W must be compatible,in the sense that the negative ends of u ∗ j and the positive ends of u ∗ j +1 must agree(and likewise for u + M and u W , etc.). Furthermore, we must have Γ + ( u +1 ) = Γ + and Γ − ( u − N ) = Γ − . Finally, every symplectization level u ∗ j must have at least onecomponent that is not a trivial cylinder R × γ .Since ( W p , λ p ) is an exact symplectic cobordism, one may apply Stoke’s theoremto derive the following expression for the energies of the levels of v : E ( u W ) := Z S W [ u W ] ∗ dλ p = X η + ∈ Γ + ( u W ) A ( η + ) − X η − ∈ Γ − ( u W ) A ( η − ) , (2.14) E ( u ± j ) := Z S j [ u ± j ] ∗ d ( e t α ± ) = X η + ∈ Γ + ( u ± j ) A ( η + ) − X η − ∈ Γ − ( u ± j ) A ( η − ) . (2.15)10he positivity of the energy of any holomorphic curve implies that the right handsides of (2.14) and (2.15) are nonnegative. More generally, if we let A [Γ] denote thetotal action of a collection of Reeb orbits, then we have the string of inequalities A [Γ − ] = A [Γ( u − N )] ≤ · · · ≤ A [Γ( u − )] ≤ A [Γ( u W )] ≤≤ A [Γ( u + M )] ≤ · · · ≤ A [Γ( u +1 )] = A [Γ + ] . (2.16)There is some additional data, beyond the holomorphic curves themselves, asso-ciated to a holomorphic building. However, we suppress this data since it will playno role in any of our arguments below. Review 2.9 (SFT Neck Stretching) . Let ( Y ∗ , α ∗ ) for ∗ ∈ { , , } be closed, non-degenerate contact manifolds and let ( U, λ U , J U ) and ( V, λ V , J V ) be a pair of exactsymplectic cobordisms from Y to Y and Y to Y , respectively, equipped withcompatible almost complex structures J R i on their completions. We denote theboundary inclusions of the contact manifolds Y ∗ into U and V by ι U ∗ for ∗ ∈ { , } and ι V ∗ for ∗ ∈ { , } .The neck stretching domain W R = U R V for parameter R ∈ [0 , ∞ ) is the exactsymplectic cobordism from ( Y , e R α ) to ( Y , e − R α ) given by U R V := U ⊔ ι U [ − R, R ] × Y ⊔ ι V V. (2.17)The Liouville forms and complex structures glue to give complex structure J R = J U R J V and λ R = λ U R λ V on the neck stretching domain for each parameter R .As in Review 2.8, fix a punctured surface Σ, and consider a sequence R i ∈ [0 , ∞ )and u i : Σ → ( c W R i , J R i ) of J R i –holomorphic curves asymptoting to collections ofReeb orbits Γ + on Y and Γ − on Y , independent of i . We remark that the contactforms on the contact boundaries of ( W R , λ R ) are equivalent up to multiplication bya scalar, so the Reeb dynamics are independent of R and it is sensible to refer tofixed asymptotics for the curves u i .The SFT neck stretching theorem provides a topology in which any such sequence( R i , u i ) converges (after passing to a subsequence) to a holomorphic building v ofthe form v = ( u , . . . , u A , u U , u , . . . , u B , u V , u , . . . , u C ) . (2.18)Here A, B, C ∈ Z ≥ are integers and the elements of the tuple (called levels ) areholomorphic maps from punctured surfaces of the form u ∗ j : S ∗ j → ( Y ∗ × R , J ∗ ) for ∗ ∈ { , , } ,u U : S U → ( b U , J U ) , and u V : S V → ( b V , J V ) . S j can be gluedtogether along the boundary punctures asymptotic to matching Reeb orbits, andthis glued surface j S j is homeomorphic to Σ.The analogous remarks from Review 2.8, regarding orbit asymptotics and actionmonotonicity, hold for the building v in (2.18). In this section, we prove Theorem 1.3 assuming a of technical result, Lemma 3.10,which is proven in §
4. Here is a brief overview of the proof to help guide the reader.We assume by contradiction that the map Φ ∗ induced by the family Φ of (1.4)is not injective in degree k . Using this assumption and the results in §
4, wefind a certain family of symplectic embeddings, parametrized by a union of anodd number of k –tori ⊔ m T k and built from Φ, which is null–bordant in the spaceSympEmp( E ( a ) , E ( b )). This means that the family extends to a smooth ( k + 1)–dimensional family of symplectic embeddings Ψ : P → SympEmp( E ( a ) , E ( b )) where P is a smooth, compact, ( k + 1)–dimensional manifold with boundary ∂P ≃ ⊔ m T k .Using Ψ, we construct a moduli space of holomorphic curves M I ( J ) in completedsymplectic cobordisms parametrized by P . Moreover, we construct an associatedevaluation map ev I : M I ( J ) → T k to a k –torus T k . We then show that the degree ofthis evaluation map is 1 mod 2 when restricted to any of the torus components of ∂ M I ( J ). This is the contradiction, since the evaluation map extends to the boundingmanifold M I ( J ) and so must have degree 0. We now introduce the spaces of holomorphic curves that are relevant to our proof,and derive the salient properties of these spaces. These are generic transversality(Lemma 3.4), compactness (Lemma 3.5), and a point count result (Lemma 3.6).
Notation 3.1 (Curve domains) . Fix a subset I ⊂ { , . . . , n } and denote by | I | thesize of I . For the remainder of §
3, we adopt the following notation.For each i ∈ I , let Σ i denote a copy of the twice punctured Riemann sphere R × S ≃ CP \ { , ∞} with the usual complex structure j C P and let Σ i denote thecorresponding copy of C P itself. Let p + i and p − i denote the points ∞ and 0 in thecopy Σ i of C P . We refer to p + i and p − i as the positive and negative punctures ofΣ i , respectively. Denote by Σ I the disjoint union ⊔ i ∈ I Σ i .12 efinition 3.2 (Unparametrized moduli space) . Let E ( a ) and E ( b ) be irrationalellipsoids, W be an exact symplectic cobordism from ∂E ( b ) to ∂E ( a ) and J ∈ J ( W )be an admissible almost complex structure on W .We define the moduli space M I ( J ) by M I ( J ) := ( u : Σ I → c W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( du ) , J = 0 u → γ ± i at p ± i ) (cid:14) ( C × ) | I | . (3.1)That is, u : Σ I → c W is a J –holomorphic curve such that u is asymptotic to thetrivial cylinder over ι + ( γ + i ) in [0 , ∞ ) × ∂ + W ϕ ≃ [0 , ∞ ) × ι + ( ∂E ( b )) at the puncture p + i and u is asymptotic to the trivial cylinder over ι − ( γ − i ) in ( −∞ , × ∂ − W ϕ ≃ ( −∞ , × ι − ( ∂E ( a )) at the puncture p − i , for each i ∈ I . We quotient the space ofsuch maps by the group of domain reparametrizations, which is the product ( C × ) | I | of the biholomorphism groups C × of each component cylinder Σ i ≃ R × S . Definition 3.3 (Parametrized moduli space over P ) . Let P be a manifold withboundary, W be a P –parametrized family of exact symplectic cobordisms ∂E ( b ) → ∂E ( a ), and J ∈ J ( W ) be a P –parametrized family of complex structure.We define the parametric moduli space M I ( J ) to be the space of pairs M I ( J ) := { ( p, u ) | p ∈ P, u ∈ M I ( J p ) } . (3.2)Our first order of business is establishing transversality for these moduli spaces. Lemma 3.4 (Transversality) . Let E ( a ) and E ( b ) be irrational symplectic ellipsoidswith parameters a = ( a , . . . , a n ) and b = ( b , . . . , b n ) satisfying a i < b i , a i < a , and b i < b for all i with ≤ i ≤ n . (3.3) Let W be a P –parametrized family of exact cobordisms from ∂E ( b ) to ∂E ( a ) . Then:(a) There exists a generic subset J reg ( W | ∂P ) ⊂ J ( W | ∂P ) such that for any J ∂P ∈ J reg ( W | ∂P ) all u ∈ M I ( J ∂P ) is parametrically Fredholm regular (see [24, Re-mark 7.4] and [23, Definition 4.5.5]). The moduli space M I ( J ∂P ) is then a (dim( P ) − –dimensional manifold.(b) Given any J ∂P as in (a), there exists a J ∈ J reg ( W ) such that J | ∂P = J ∂P andsuch that every ( p, u ) ∈ M I ( J ) is parametrically Fredholm regular (see [24,Remark 7.4] and [23, Definition 4.5.5]). In particular, M I ( J ) is a dim( P ) –dimensional manifold with boundary ∂ M I ( J ) ≃ M I ( J ∂P ) . roof. This essentially follows from the general transversality results of [23] and [24, § u ∈ M I ( J ∂P ) must be somewhere injective (see[24, p. 123]) for any choice of J ∂P . Indeed, note that all of the orbits γ − i and γ + i are simple. This means that none of them can be factored as η ◦ ϕ where η is aclosed Reeb orbit and ϕ : S → S is a k –fold cover with k ≥
2. This implies that u is simple as well, i.e. that u cannot factor as v ◦ φ where v : Σ ′ → c W ∂P is a J ∂P –holomorphic curve and φ : Σ I → Σ ′ is a holomorphic branched cover. Simple curvesare somewhere injective. In fact, these conditions are equivalent in our setting, see[24, Theorem 6.19]. The same reasoning shows that any curve u appearing as afactor in a point ( p, u ) ∈ M I ( J ) is somewhere injective for any choice of J .(Part (a)) we now apply the appropriate parametric version of transversality(see [24, Theorems 7.1–7.2], [24, Remark 7.4] and [23, § J reg ( W | ∂P ) ⊂ J ( W | ∂P ) with thefollowing property: for any J ∂P ∈ J reg ( W | ∂P ), any point ( p, u ) ∈ M I ( J ∂P ) where u is somewhere injective is (parametrically) Fredholm regular. In particular, due tothe discussion above, every ( p, u ) ∈ M I ( J ∂P ) is Fredholm regular for such a choiceof J ∂P . The dimension of M I ( J ∂P ) at a point ( p, u ) is given by the formuladim ( p,u ) ( M I ( J ∂P )) = dim( ∂P ) + ind( u ) , (3.4)where ind( u ) denotes the Fredholm index of u , given by the following index formula.ind( u ) = X i ∈ I (cid:0) ( n − χ (Σ i ) + 2 c ( u | Σ i , τ ) + CZ( γ + i , τ ) − CZ( γ − i , τ ) (cid:1) . (3.5)The Conley–Zehnder indices CZ( γ ± , τ ) and relative Chern numbers c ( u | Σ i , τ ) areas in Review 2.2, and τ denotes a trivialization of ξ over ⊔ i ( γ + i ⊔ γ − i ).To simplify the dimension formula, note that ∂E ( a ) and ∂E ( b ) are simply-connected and we have assumed that H ( W p ; Z ) = 0. Thus we may choose τ bytaking capping disks D i for γ − i , thus inducing trivializations of ξ along γ − i , andthen extending τ to a trivialization along Σ i to induce trivializations of ξ along γ + i . The resulting trivialization has c ( u | Σ i , τ ) = 0, CZ( γ + i , τ ) = CZ( γ + i ), andCZ( γ − i , τ ) = CZ( γ − i ). Here CZ( γ + i ) and CZ( γ − i ) denote the canonical indices de-scribed in Review 2.2. Thus, using this special choice of τ and noting that χ (Σ i ) = 0,the formulas (3.4-3.5) simplify todim( M I ( J ∂P )) = dim( P ) − X i ∈ I (cid:0) CZ( γ + i ) − CZ( γ − i ) (cid:1) . (3.6)Finally, we observe that the hypotheses (3.3) and the Conley–Zehnder index formula(2.8) imply that CZ( γ + i ) = CZ( γ − i ) = n − i . Therefore, the moduli space M I ( J ∂P ) is (dim( P ) − § J ∈ J ( W ) to agree with a fixed parametrically transverse J ∂P onthe boundary. This concludes our discussion for this lemma.Next we discuss compactness. For the next lemma, we will refer extensively tothe review of SFT compactess (Review 2.8) provided in § Lemma 3.5 (Compactness) . Let E ( a ) and E ( b ) be irrational symplectic ellipsoidswith parameters a = ( a , . . . , a n ) and b = ( b , . . . , b n ) . Let P is a compact manifoldwith boundary, let W be a P –family of exact symplectic cobordisms with H ( W p ) = H ( W p ) = 0 and let J ∈ J reg ( W ) be a family of regular almost complex structures.Then the moduli space M I ( J ) has the following compactness properties:(a) If a n < a and b n < b , and P has dimension less than or equal to 1, thenthe moduli space M I ( J ) is compact.(b) If moreover a i < b i < a i +1 for all i ∈ { , . . . , n − } and b n < a , then for P of any dimension M I ( J ) is compact.Proof. We prove (a) and (b) by showing that any broken building u arising as alimit of a sequence in M I ( J ) must consist of a single cobordism level, and thus mustbe an element of M I ( J ). Note that the hypotheses of (b) imply those of (a).Thus let ( p i , u i ) be a sequence in M I ( J ). Since P is compact, we may passto a subsequence so that p i → p ∈ P . By SFT compactness, after passing to asubsequence u i converges to a limit building v . We use the notation of Review 2.8for this building. By considering components of Σ I and j S j , we can assume that | I | = 1, i.e. that Σ I has one component and each u i is asymptotic to i –independentends γ + l and γ − l where 1 ≤ l ≤ n .(Part (a)) First, consider a positive symplectization level u + k . By action mono-tonicity, we know that A [Γ ± ( u + k )] ≤ A ( γ + l ). This implies, due to the hypothesis of(a), that each Γ ± ( u + k ) is a singleton consisting of an embedded orbit of E ( b ), i.e.that there is a sequence { α k } M such that α = l andΓ − ( u + k ) = { γ + α k } for all k ∈ { , . . . , M } . Since the genus of the building v must be equal to that of the u i , each u + k musttherefore be a cylinder from γ + α k − to γ + α k .Next, consider the cobordism level u W . Since Γ + ( u W ) = Γ − ( u + M ) = { γ + α M } , weknow by the same discussion as in Lemma 3.4 that u W is somewhere injective and the15arametric moduli space M ( p, u W ) containing ( p, u W ) is parametrically Fredholmregular. Therefore we know M ( p, u W ) is a manifold of dimensiondim( M ( p, u W )) = dim( P ) + ind( u W ) ≥ . (3.7)Here the index is given by the following formula, similar to (3.5):ind( u ) = ( n − χ ( S W ) + 2 c ( u W , τ ) + X η + ∈ Γ + ( u W ) CZ( η + , τ ) − X η − ∈ Γ − ( u W ) CZ( η − , τ ) . As in Lemma 3.4, the Conley–Zehnder indices CZ( η ± , τ ) and relative Chern numbers c ( u W , τ ) are as in Review 2.2, and τ is a trivialization of ξ over Γ + ( u W ) ⊔ Γ − ( u W ).Now we simplify this dimension formula. Due to the assumption that H ( W p ) = H ( W p ) for each fiber W p of W , we can choose a trivialization τ extending over u W such that CZ( η ± , τ ) = CZ( η ± ) and c ( u W , τ ) = 0. Furthermore, we must have g ( S W ) = 0 since otherwise the total genus of v would be greater than 0. This impliesthat χ ( S W ) = 1 − | Γ − ( u W ) | . Thus the index formula simplifies to the following.dim( M ( p, u W )) = dim( P )+( n − γ + α M ) − X η − ∈ Γ − ( u W ) (cid:0) ( n −
3) + CZ( η − ) (cid:1) (3.8)Applying the hypothesis (a) and the CZ index formula (2.8) in Example 2.3, wenote that( n −
3) + CZ( γ ± i ) = 2 n − i and ( n −
3) + CZ(( γ ± i ) m ) ≥ n − γ ± i and any m ≥ P ) ≤
1, this implies that (3.8) is negative if either | Γ − ( u W ) | ≥
2, or η − = ( γ − i ) m for m ≥ i , or if η − = γ − m for m > α M (where γ + α M is thepositive end of u W )). Thus we must have Γ − ( u W ) = { γ − β } for some β ≤ α M .Finally, we can argue analogously to the positive symplectization case to showthat there is a sequence { β k } N such that β N = l andΓ + ( u − k ) = { γ − β k } for all k ∈ { , . . . , N } . We have thus shown that every level of v is Fredholm regular and cylindrical. There-fore we have the following inequalities of the Conley–Zehnder indices.CZ( γ − l ) = CZ( γ − β N ) ≤ · · · ≤ CZ( γ − β ) ≤ CZ( γ + α M ) ≤ · · · ≤ CZ( γ + α ) = CZ( γ + l ) . Since CZ( γ − l ) = CZ( γ + l ), we thus conclude that every symplectization level u ± k isindex 0. This implies that they are somewhere injective branched covers of trivial16ylinders, which must in fact be trivial cylinders. This is only possible if M = N = 0and thus v only has a cobordism level.(Part (b)) Begin by considering a positive symplectization level u + j of v . Dueto action monotonicity (2.16), the collections Γ + ( u + j ) and Γ − ( u + j ) of positive andnegative limit Reeb orbits of ( Y + , α + ) ≃ ( ∂E ( b ) , λ | ∂E ( b ) ) must satisfy a l i = A ( γ − l i ) ≤ A [Γ − ( u + j )] ≤ A [Γ + ( u + j )] ≤ A ( γ + l i ) = b l i . Consider Γ + ( u + j ) only. Due to the hypotheses of (b), Γ + ( u + j ) cannot contain eithera copy of γ + r for r > l i or a copy of an iterate ( γ + r ) m for any m ≥ r .Otherwise, we would have A [Γ + ( u + j )] > b l i . This implies that Γ + ( u + j ) can onlycontain Reeb orbits γ + r for r ≤ l i . Moreover, since A [Γ + ( u + j )] ≥ a l i and A ( γ + r ) = b r < a l i for r < l i (again by (b)), we must have Γ + ( u + j ) = { γ + l i } . The same reasoningshows that Γ − ( u + j ) = { γ + l i } .This demonstrates that the energy E ( u + j ) of the level u + j is 0 by (2.15) and thelevel u + j must be a branched cover of a trivial cylinder (see [23, Lemma 9.9]). Sincethe ends are embedded, u + j must be simple and thus a trivial cylinder. This isdisallowed by the SFT compactness statement, so u + j cannot exist.The same reasoning implies that negative levels u − j of v cannot exist. Thus thebuilding v consists of only a cobordism level u W .Finally, we state and prove the following curve count result, Lemma 3.6. Forthis proof, we will use SFT neck stretching as discussed in Review 2.9. Lemma 3.6 (Curve count) . Let E ( a ) and E ( b ) be irrational symplectic ellipsoidswith parameters a = ( a , . . . , a n ) and b = ( b , . . . , b n ) satisfying a i < b i < a i +1 for all i ∈ { , . . . , n − } and b n < a . (3.9) Let ϕ : E ( a ) → E ( b ) be a symplectic embedding which is isotopic to the inclusion ι : E ( a ) → E ( b ) . Finally, let W ϕ be the symplectic cobordism associated to ϕ and let J ∈ J reg ( W ϕ ) be a regular almost complex structure (provided by Lemma 3.4). Thenthe number of points M I ( J ) in the moduli space M I ( J ) is odd. Remark 3.7 (Floer theoretic proof) . Morally, one may view the holomorphic cylin-ders in M I ( J ) as contributing to the cobordism map CH ( W ϕ ) : CH ( ∂E ( b )) → CH ( ∂E ( a )) (where one can take CH to be either the full contact homology or per-haps cylindrical contact homology). The invariance of the signed or mod 2 pointcount of M I ( J ) may be viewed essentially as a consequence of the deformation in-variance of this cobordism map. A proof of Lemma 3.6 in this spirit is possible usingthe foundations from e.g. [20]. Here we provide a simpler argument which does notuse Floer theory directly. 17 roof. We first address the case where ϕ = ι and then tackle the general case.(Case of ϕ = ι ) Pick an ǫ > : E ( ǫ · b ) → E ( a ), and let W ι , W and W ι ◦ be the cobordisms associated to ι, and ι ◦ respectively. Pick regular J ι ∈ J reg ( W ι ) and J ∈ J reg ( W ) so that(by Lemmas 3.4(b) and 3.5(a)), the spaces M I ( J ι ) and M I ( J ) are compact 0–dimensional manifolds with Fredholm regular points.Note that c W ι ◦ is equivalent (as a completed cobordism) to the symplectizationof ∂E ( b ). Thus we can choose J ι ◦ ∈ J reg ( W ι ◦ ) to be translation invariant and suchthat the moduli spaces M I ( J ι ◦ ) are all transverse (due to the same arguments asin Lemma 3.5, but using the transversality theorem [23, Theorem 8.1] for symplec-tizations). Any J ι ◦ –holomorphic cylinder u from γ + i to γ + i with index 0 must betranslation invariant (since otherwise the dimension of the moduli space would bepositive) and embedded, thus a trivial cylinder. Thus M I ( J ι ◦ ) consists of a singlepoint { u } , which is a product of trivial cylinders.Now note that we may write W ι ◦ = E ( b ) \ E ( ǫ · b ) as a composition of cobordisms W ι ◦ = W ι W and consider the neck stretching domain W Rι ◦ = W ι R W for each R .This yields a [0 , ∞ )–parametrized family of cobordisms W whose fiber at R ∈ [0 , ∞ )is W Rι ◦ . Let J ∈ J ( W ) be an almost complex structure which agrees with the gluedstructure J R := J ι R J for R near ∞ , and consider the moduli space M I ( J ) := { ( R, u ) | R ∈ [0 , ∞ ) and u ∈ M I ( J R ) } . (3.10)For sufficiently large R , there exists a gluing map of the formglue : M I ( J ι ) × M I ( J ) × ( R , ∞ ) → M I ( J ) , ( u ι , u , R ) ( R, glue R ( u ι , u )) . This map is constructed in (for instance) [20, § R sufficiently large. Using Lemma 3.4 and the references therein, we maychoose J to be parametrically regular over the region [0 , R ] of P and to agree with J and J ι R J at 0 and R ∈ [ R , ∞ ). We may form a compactified moduli space M I ( J ) := M I ( J ) ⊔ glue M I ( J ι ) × M I ( J ) × ( R , ∞ ] . (3.11)This moduli space is a compact 1–manifold with boundary ∂ M I ( J ) = M I ( J ι ◦ ) ⊔ M I ( J ι ) × M I ( J ) . (3.12)The number of boundary points of a compact 1–manifold is even. In particular M I ( J ι ◦ ) + M I ( J ι ) · M I ( J ) = 0 mod 2 . Since M I ( J ι ◦ ) has an odd number of points, it follows that M I ( J ι ) and M I ( J ) doto. 18General Case) Let J be as chosen in the lemma statement, and let Φ : E ( a ) × [0 , → E ( b ) be an isotopy of symplectic embeddings with Φ = ι and Φ = ϕ .Let W denote the [0 , ∂E ( b ) to ∂E ( a ) with fiber W Φ t at t ∈ [0 , J ι ∈ J reg ( W ι ) as permittedby Lemma 3.4(b). Then by Lemma 3.4(b) and 3.5(a), we may choose a regular J = { J t } t ∈ [0 , ∈ J reg ( W ) such the parametric moduli space M I ( J ) over [0 ,
1] isFredholm regular, and such that J = J ι and J = J . It follows that ∂ M I ( J ) = M I ( J ) ⊔ M I ( J ι ) and M I ( J ) = M I ( J ι ) mod 2 . Thus the general case follows from the case where ϕ = ι . Lemma 3.8.
The compactification M I ( J ) (see (3.11)) of the moduli space M I ( J ) (see (3.10)) is compact in the neck stretching topology discussed in Review 2.9.Proof. It suffices to take a sequence ( R i , u i ) ∈ M I ( J ) and show that it has a conver-gent subsequence in M I ( J ). If R i ≤ C for some fixed upper bound C , then ( R i , u i )is a sequence of holomorphic curves in cobordisms parametrized over a compact,1–dimensional space [0 , C ] and we can apply Lemma 3.5. Thus we may assume that R i → ∞ as i → ∞ . Let v denote the limit building provided by Review 2.9.Let γ bi , γ ai , and γ ǫbi denote the simple orbits on ∂E ( b ) , ∂E ( a ), and ∂E ( ǫb ) respec-tively, ordered by increasing action as usual. We denote the levels of v by v = ( u b , . . . , u bA , u ι , u a , . . . , u aB , u , u ǫb , . . . , u ǫbC ) . By considering components, we may assume that | I | = 1 and u i is asymptotic to γ bl at the positive end and γ ǫbl on the negative end for some l . An identical argument tothat in Lemma 3.5(a) shows that every level of v is transverse, embedded, genus 0and asymptotic to a single embedded orbit at the positive end and a single embeddedorbit at the negative end.The remainder of the proof is also like Lemma 3.5(a). Since the index of eachlevel must be nonnegative by transversality, the index of the limit orbits must benondecreasing. Then CZ( γ bi ) = CZ( γ ǫbi ) implies that every orbit has the same index,and that all of the levels are index 0. In particular, any symplectization level mustbe a trivial cylinder, and thus can’t exist. This implies the result. In this section, we use the moduli spaces constructed in § otation 3.9. For any n ∈ Z + and any I ⊂ { , . . . , n } , define the | I | –torus by T I = { ( θ , . . . , θ n ) ∈ T n | θ j = 0 , ∀ j / ∈ I } . (3.13)Note that the k th homology group H k ( T n ; Z /
2) of the n –torus T n is generated bythe fundamental classes [ T I ], where I runs over all subsets of size | I | = k .For Theorem 1.3, we also require the following result, which is proven in § Lemma 3.10.
Let U and V be compact symplectic manifolds with boundary. Let Z be a closed manifold with total Stieffel–Whitney class w ( Z ) = 1 ∈ H ∗ ( Z ; Z / andlet Φ be a smooth family of symplectic embeddings Φ : Z → SympEmb(
U, V ) with Φ ∗ [ Z ] = 0 . Then there exists a compact manifold P with boundary Z and an extension of Φ toa smooth family Ψ of symplectic embeddings Ψ : P → SympEmb(
U, V ) with Ψ | ∂P = Φ . Given the above preparation, we are now ready for the proof of Theorem 1.3.
Proof. (Theorem 1.3) We pursue the argument by contradiction outlined at thebegining of §
3. Fix an integer k with 1 ≤ k ≤ n and suppose that there were anonzero Z / n –torus of the form[ A ] = X L c L [ T L ] with Φ ∗ [ A ] = 0 ∈ H k (SympEmb( E ( a ) , E ( b )); Z / . (3.14)Let Z = ⊔ c L =0 T L . Then Lemma 3.10 states that there exists a smooth ( k + 1)–dimensional manifold P with boundary ∂P = Z and a smooth family of embeddingsΨ : P → SympEmb( E ( a ) , E ( b )) with Ψ | T L = Φ | T L for each T L ⊂ Z. (3.15)By passing to subellipsoids, we may assume that E ( a ) and E ( b ) are irrational.In this setting, Lemmas 3.4 and 3.5 state that there exist choices of J ∂P ∈ J ( c W ∂P )and J ∈ J (Ψ) such that the parametrized moduli space M I ( J ) is a compact ( k +1)–dimensional manifold with boundary ∂ M ( J ) ≃ Z × M ( c W ∂P ; J ∂P ). On theparametrized moduli space M I ( J ), we can define an evaluation mapEv I : Y i ∈ I γ + i × M I ( J ) → Y i ∈ I γ − i ≃ T k (3.16)20ia the following procedure. Let q = ( q i ) i ∈ I be a point in × i ∈ I γ + i and let ( p, u ) ∈ M I ( J ). According to (3.2), ( p, u ) is a pair of a point p ∈ P and an equivalence classof holomorphic maps u : Σ I → c W Ψ p up to reparametrization. Pick a representativeholomorphic curve e u of u , which consists of k maps e u i : Σ i → c W Ψ p for each i ∈ I .We have limit parametrizations of γ + i and γ − i induced by e u i , defined bylim + e u i : S → γ + i ⊂ ∂E ( b ) , lim + e u i ( t ) := lim s → + ∞ π ∂ + W Ψ p e u i ( s, t ) , lim − e u i : S → Ψ p ( γ − i ) ⊂ Ψ p ( ∂E ( a )) , lim − e u i ( t ) := lim s →−∞ π ∂ − W Ψ p e u i ( s, t ) . Here π ∂ + W Ψ p and π ∂ − W Ψ p denote projection to the positive and negative boundariesof W Ψ p . Note that these projections are only defined in the limit as s → ±∞ . Interms of these parametrizations, we define the evaluation map Ev I by the formulaEv I ( q ; p, u ) := (cid:18) [Ψ − p ◦ lim − e u i ◦ (lim + e u i ) − ]( q i ) (cid:19) i ∈ I ∈ Y i ∈ I γ − i . (3.17)This definition is independent of the choice of representative e u . Finally, fix anarbitrary q in the product Q i ∈ I γ + i and defineev I : M I ( J ) → Y i ∈ I γ − i , ev I ( p, u ) := Ev I ( q ; p, u ) . (3.18)Now consider the restriction of ev I to each component T L × { u } of the boundary Z × M ( c W ∂P , J ∂P ) of M ( J ). Since the equivalence class of curve u is independent of θ ∈ T L ⊂ T n , we can use (3.17) and (3.18) to writeev I ( θ, u ) = (Ψ − θ ( r i )) i ∈ I with r := (cid:18) [lim − e u i ◦ (lim + e u i ) − ]( q i ) (cid:19) i ∈ I ∈ Y i ∈ I γ − i . Here r is independent of θ . Using the fact that Ψ − | Z = Φ − and the formula (1.4)for the family of embeddings Φ, we have the formulaev I ( θ, u ) = (Φ − θ ( r i )) i ∈ I = ( e − πiθ i · r i ) i ∈ I . (3.19)In the right–most expression of (3.19), we identify r i with an element of C n via theinclusion γ − i ⊂ E ( a ) ⊂ C n .The expression (3.19) allows us to compute the degree of ev I on each component T L × { u } . There are two cases. If L = I , then (3.19) shows that ev I | T L ×{ u } is degree1. If I = L , then for any j ∈ L \ ( L ∩ I ), θ j is constant for every θ ∈ T L and it follows21rom (3.19) that the degree of ev I | T L ×{ u } is 0. To derive our final contradictiction,we now observe that the total degree mod 2 of ev I restricted to the boundary isdeg(ev I | ∂ M I ( J ) ) = X T L ×{ u }⊂ ∂ M I ( J ) deg(ev I | T L ×{ u } ) == | M I ( J ∂P ) | ≡ . (3.20)The right–most equality in (3.20) crucially uses the point count of Lemma 3.6. Theequality (3.20) also provides the contradiction, since the degree of the restricton ofa map to a boundary must be 0 mod 2. This concludes the proof.Having concluded the proof of Theorem 1.3, we now move on to Proposition 1.4.The proof is much less involved than that of Theorem 1.3, and does not use any ofthe machinery from § U ( n ). Lemma 3.11.
Consider the map U : T n → U ( n ) given by θ U θ . Then theinduced map U ∗ : H ∗ ( T n ; Z / → H ∗ ( U ( n ); Z / on Z / –homology is:(a) surjective if ∗ = 0 or ∗ = 1 .(b) identically if ∗ ≥ .Proof. To show (a), we first note that T n and U ( n ) are connected so U ∗ | H = Id.Furthermore, if we consider the loop γ : R / π Z → T n given by θ ( θ, , . . . , C ◦ U ◦ γ : R / π Z → U (1) ≃ R / π Z is the identity. Since det C : U ( n ) → U (1) induces an isomorphism on H , theinduced map of U must be surjective on H .To show (b) we proceed as follows. It suffices to show that U ∗ [ T L ] = 0 for all L with | L | ≥
2. We can factorize T L = T J × T K for J ⊔ K = L and | J | = 2, and T L = T J × T K ι J × ι K −−−→ U (2) × U ( n − j −→ U ( n ) . Here j is the inclusion of a product of unitary subgroups, and ι J and ι K are inclusionsof the tori into these unitary subgroups. It suffices to show that ( ι J × ι K ) ∗ [ T L ] =[ ι J ] ∗ [ T J ] ⊗ [ ι K ] ∗ [ T K ] = 0, or simply that [ ι J ] ∗ [ T J ] = 0 ∈ H ( U (2); Z / U (2)) = 4 and H ∗ ( U (2); Z / ≃ Z / c , c ] where c i is a generator of index i . In particular, H ( U (2); Z / ≃ H ( U (2); Z /
2) = 0.22sing Lemma 3.11, we can now prove Propositon 1.4. The point is that the entireunitary group U ( n ) embeds into SympEmb( E ( a ) , E ( b )) via domain restriction when a i < b j for all i and j (which is equivalent to a n < b by our ordering convention). Proof. (Proposition 1.4) Let D : SympEmb( E ( a ) , E ( b )) → U ( n ) denote the map ϕ r ( dϕ | ), given by taking derivatives dϕ | ∈ Sp(2 n ) at the origin and composingwith a retraction r : Sp(2 n ) → U ( n ). Under the hypotheses on a and b , we canfactor the identity Id : U ( n ) → U ( n ) and Φ : T n → SympEmb( E ( a ) , E ( b )) asId : U ( n ) res −→ Symp( E ( a ) , E ( b )) D −→ U ( n ) , Φ : T n U −→ 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 Z –graded Z / In this section, we discuss some basic results about the Fr´echet manifold of sym-plectic embeddings SympEmb(
U, V ) between symplectic manifolds with boundary.In § U, V ). In § § Let (
U, ω U ) and ( V, ω V ) be 2 n –dimensional compact symplectic manifolds withnonempty contact boundaries. We now give a proof of the folklore result thatthe space of symplectic embeddings from U to V is a Fr´echet 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 Fr´echet manifold.Proof. Let ( U × V, ω U × V ), with ω U × V = π ∗ U ω U − π ∗ V ω V , denote the product symplecticmanifold with corners. Given a symplectic embedding ϕ : U → int( V ), we mayassociate the graph Γ( ϕ ) ⊂ U × V given byΓ( ϕ ) := { ( u, ϕ ( u )) ∈ U × V } . T ( ∂U ) ω on the contact hypersurface ∂U × int( V ). By the Weinsteinneighborhood theorem with boundary, Proposition 4.13, there is a neighborhood A of U , a neighborhood B of Γ( ϕ ) and a symplectomorphism ψ : A ≃ B with ψ | U : U → Γ( ϕ ) given by u ( u, ϕ ( u )) and ψ ∗ ω U × V = ω std .Let A ( ϕ, ψ ) ⊂ ker( d : Ω ( L ) → Ω ( L )) and B ( ϕ, ψ ) ⊂ SympEmb(
U, V ) denotethe open subsets given by A ( ϕ, ψ ) := (cid:8) α ∈ Ω ( L ) (cid:12)(cid:12) dα = 0 and Im( α ) ⊂ A (cid:9) , 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 Φ[ α ] := ( π V ◦ ψ ◦ α ) ◦ ( π U ◦ ψ ◦ α ) − ,φ Ψ[ φ ] := ( ψ − ◦ (Id × φ )) ◦ ( π L ◦ ψ − ◦ (Id × φ )) − . It is a tedious but straightforward calculation to check that Φ ◦ Ψ = Id and Ψ ◦ Φ = 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 manifolds M , N , and O (in fact, smooth; see [13, Theorem 42.13]).Since C ∞ ( U, V ) is metrizable under the compact open C ∞ –topology (see [13,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 [16, Proposition 3.4.2].
We now discuss (unoriented) bordism groups and their structure in the case ofFr´echet 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 map g : Y → X such that ∂Y = Z and g | ∂Y = f . Given a pair of manifold/map pairs( Z i , f i ) for i ∈ { , } , we say that ( Z , f ) and ( Z , f ) are bordant if ( Z ⊔ Z , f ⊔ f )is null–bordant. 24 efinition 4.4 (Smooth bordism) . Let X be a Fr´echet manifold and f : Z → X bea smooth map from a smooth closed manifold. Then ( Z, f ) is smoothly null–bordant if it is null–bordant via a pair (
Y, g ) where g : Y → X be a smooth map of Banachmanifolds with boundary. Similarly, a pair ( Z i , f i ) for i ∈ { , } is smoothly bordant if ( Z ⊔ Z , f ⊔ Z ) is smoothly null–bordant.The above notions come with accompanying versions of the bordism group. Definition 4.5 (Bordism group of X ) . The n –th bordism group Ω n ( X ; Z /
2) 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 ( Z , f ) ∼ ( Z , f ) if the pair is bordant. Addition isdefined by disjoint union[ Z , f ] + [ Z , f ] := [ Z ⊔ Z , f ⊔ f ] . Definition 4.6 (Smooth bordism group of X ) . The n –th smooth bordism group Ω ∞ n ( X ; Z /
2) of a Fr´echet 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 isa smooth map, modulo the relation that ( Z , f ) ∼ ( Z , f ) if the pair is smoothlybordant. Addition in the group Ω ∞∗ ( X ; Z /
2) is defined by disjoint union as before.
Lemma 4.7.
The natural map Ω ∞∗ ( X ; Z ) → Ω ∗ ( X ; Z ) is an isomorphism.Proof. The argument uses smooth approximation and is identical to the case where X 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 Fr´echet manifold, and let f : Z → X bea smooth map from a closed manifold Z with Stieffel–Whitney class w ( Z ) = 1 ∈ H ∗ ( Z ; Z / . Then f ∗ [ Z ] = 0 ∈ H ∗ ( X ; Z / if and only [ Z, f ] = 0 ∈ Ω ∞∗ ( X ; Z ) .Proof. Proposition 4.8 will follow immediately from the following results. First, byLemma 4.7, it suffices to show f ∗ [ Z ] = 0 ∈ H ∗ ( X ; Z /
2) if and only [
Z, f ] = 0 ∈ Ω ∗ ( X ; Z ). By Proposition 4.9, we can replace X with a CW complex. Lemma 4.10proves the result in this context. Proposition 4.9 ([19, Theorem 14]) . A metrizable Fr´echet manifold is homotopyequivalent to a CW complex. emma 4.10. Let X homotopy equivalent to a CW complex, and let f : Z → X be a continuous map from a closed manifold Z with Stieffel–Whitney class w ( Z ) =1 ∈ H ∗ ( Z ; Z / . Then f ∗ [ Z ] = 0 ∈ H ∗ ( X ; Z / if and only [ Z, f ] = 0 ∈ Ω ∗ ( X ; Z ) . Remark 4.11.
Crucially, we make no finiteness assumptions on the CW structure.
Proof. ( ⇒ ) Suppose that f ∗ [ Z ] = 0 ∈ H ( Z ; Z / ϕ : X ≃ X ′ with a CW complex X ′ . Such an equivalence induces an isomorphism ofunoriented bordism groups Ω ∗ ( X ; Z ) ≃ Ω ∗ ( X ′ ; Z ), so it suffices to show that thepair ( 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 ; Z / Z, f ] = 0 ∈ Ω ∗ ( A ; Z ) if and only if the Stieffel–Whitney numbers sw α,I [ Z, f ]are identically 0. Recall that the Stieffel–Whitney number sw α,I [ Z, f ] associatedto [
Z, f ], a cohomology class α ∈ H k ( A ; Z ) and a partition I = ( i , . . . , i k ) ofdim( Z ) − k is defined to besw α,I [ Z, f ] = h w i ( Z ) w i ( Z ) . . . w i k ( Z ) f ∗ α, [ Z ] i ∈ Z . Here w j ( Z ) ∈ H j ( Z ; Z ) denotes the j –th Stieffel–Whitney class of Z . By assump-tion, w ( Z ) = 1 and so w j ( Z ) = 0 for all j = 0. In particular, the only possiblenonzero Stieffel–Whitney numbers have I = (0). But we see thatsw α, (0) [ Z, f ] = h f ∗ α, [ Z ] i = h α, f ∗ [ Z ] i = 0 . Therefore, sw α,I [ Z, f ] ≡ Z, f ] must be null–bordant.( ⇐ ) This direction is completely obvious, since the map Ω ∗ ( X ) → H ∗ ( X ; Z / Z, f ] 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 ; Z / . Then there exists a finite sub–complex A ⊂ X with f ( Z ) ⊂ A and f ∗ [ Z ] = 0 ∈ H ∗ ( A ; Z / .Proof. A very convenient tool for this is the stratifold homology theory of [12], whichwe now review briefly.Given a space M , the n –th stratifold group sH n ( M ; Z /
2) with Z / S, g ) of acompact, regular stratifold S and a continuous map g : S → M . Two pairs ( S i , g i )for i ∈ { , } 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 ) = ( S ⊔ S , g ⊔ g ) (see Chapter 3 and Section 4.4 of [12]). Given a26ap ϕ : M → N of spaces, the pushforward map ϕ ∗ : sH ( M ; Z ) → sH ( M ; Z ) onstratifold homology is given (on generators) by [ S, g ] [ S, ϕ ◦ g ] = ϕ ∗ [Σ , g ].Stratifold homology satisfies the Eilenberg–Steenrod axioms (see Chapter 20of [12]), and thus if M is a CW complex then there is a natural isomorphism sH ∗ ( M ; Z ) ≃ H ∗ ( M ; Z ). If M is a manifold of dimension n , the fundamentalclass [ M ] ∈ sH n ( M ; Z ) is given by the tautological equivalence class [ M ] = [ M, Id].The proof of the lemma is simple with the above machinery in place. Since f ∗ [ 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 A ⊂ X such that g ( T ) ⊂ A ⊂ X . Then the pair ( Z, f ) are null–bordant by (
Y, g ) in A as well, so that [ Z, f ] = 0 ∈ sH ∗ ( A ; Z ) and thus f ∗ [ Z ] = 0 ∈ H ∗ ( A ; Z ) via theisomorphism sH ∗ ( A ; Z ) ≃ H ∗ ( A ; Z ). 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 ) = ω std | U .Proof. The proof has two steps. First, we construct neighborhoods U ⊂ T ∗ L and V ⊂ X of L , and a diffeomorphism ϕ : U ≃ V such that ϕ | L = Id , ϕ ∗ ( ω | V ) | L = ω std | L , T ( ∂U ) ω std = T ( ∂U ) ϕ ∗ ω . (4.1)Here T ( ∂U ) ω std ⊂ T ( ∂U ) is the symplectic perpendicular to T ( ∂U ) with respect to ω std (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 ν g L with respect to g is a bundleover L with Lagrangian fiber, and that J : T L → ν g L gives a natural isomorphism.Let Φ g : T ∗ L → T L denote the bundle isomorphism induced by the metric g andlet exp g denote the exponential map with respect to g .Since L is compact, we can choose a tubular neighborhood U ′ of νL such thatexp g : U → X is a diffeomorphism onto its image V . We then let U := [ J ◦ Φ g ] − ( U ′ ) ⊂ T ∗ L φ g : U ≃ V, ( x, v ) exp gx ( J ◦ Φ g ( v )) . Note that φ g | L = Id and [ φ g ] ∗ ω | L = ω std | L by the same calculations as in [16,Theorem 3.4.13]. We now must modify U , V , and φ g to satisfy the last condition of(4.1).To this end, we apply Lemma 4.15. Taking κ = T ( ∂U ) ω std and κ = T ( ∂U ) [ φ g ] ∗ 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: ψ t | ∂L = Id , d ( ψ t ) u = Id for u ∈ ∂L, ψ = Id , [ ψ ] ∗ ( T ( ∂U ) ω std ) = T ( ∂U ) [ φ g ] ∗ ω . Note here that we are using the fact that T ( ∂U ) ω std | L = T ( ∂U ) [ φ g ] ∗ ω | L already by theconstruction of φ g . By shrinking N and U , we can simply assume that N = ∂U . Lettc : [0 , × ∂U ≃ T ⊂ U be tubular neighborhood coordinates near boundary. Bychoosing the tubular neighborhood coordinates tc : [0 , × ∂U ≃ T appropriately,we can also assume that tc([0 , × ∂L ) = L ∩ T . We define a map Φ : U → T ∗ L byΦ( u ) = (cid:26) ( s, ψ − s ( v )) if u = ( s, v ) ∈ [0 , × ∂U via tc ,u otherwise . The map Φ has the following properties which are analogous to those of ψ s :Φ | L = Id , d (Φ) u = Id for u ∈ L, Φ ∗ ( T ( ∂L ) ω std ) = T ( ∂L ) [ φ g ] ∗ ω . Also note that Φ is smooth since ψ t is constant for t near 0 and 1. We thus define f as the composition ϕ = φ g ◦ Φ. It is immediate that f has the properties in (4.1).(Step 2) We closely follows the Moser type argument of [16, Lemma 3.2.1]. Byshrinking U , we may assume that it is an open disk bundle. Let ω t = (1 − t ) ω std + tf ∗ ω and τ = ddt ( ω t ) = f ∗ ω − ω std . Let κ = T ( ∂U ) ω t (by the previous work, it does notdepend 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 φ t ( x, u ) = ( x, tu ) in Lemma 4.16.We can thus find a σ satisfying the properties listed in (4.2).Let Z t be the unique family of vector fields satisfying σ = ι ( Z t ) ω t . Due to theproperties of σ , Z t satisfies the following properties for each t . Z t | L = 0 , Z t | ∂U ∈ T ( ∂U ) for all t. The first property is immediate, while the latter is a consequence of the fact that ω t ( Z t , · ) | κ = σ | κ = 028mplies Z t ∈ ( κ ) ω t = T ( ∂U ) . These two properties imply that Z t generates a map Ψ : U ′ × [0 , → U for somesmaller tubular neighborhood U ′ ⊂ U with the property that Ψ t | L = Id and Ψ ∗ t ω t = ω (see [16, § : U ′ → U with Ψ | L = Id and Ψ ∗ f ∗ ω . By shrinking U , taking ϕ = f ◦ Ψ 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(open) disk bundle of E with closure U . Let κ ⊂ T ( ∂U ) be a distribution on ∂U such that dφ t ( κ u ) = κ φ t ( u ) for all u ∈ U , where φ : U × I → U denote the family ofsmooth maps given by φ t ( x, u ) := ( x, tu ) .Finally, suppose that τ ∈ Ω k +1 ( U ) is a ( k + 1) –form such that dτ = 0 , τ | X = 0 , ( ι ∗ ∂X τ ) | κ = 0 . (4.2) Then there exists a k –form σ ∈ Ω k ( U ) with dσ = τ, σ | X = 0 , ( ι ∗ ∂X σ ) | κ = 0 . (4.3) Proof.
We use integration over the fiber, as in [16, p. 109]. Note that the maps φ t : U → φ t ( U ) ⊂ U are diffeomorphisms for each t > φ = π , φ = Id and φ t | X = Id. Therefore we have φ ∗ τ = 0 , φ ∗ τ = τ. We may define a vector field Z t for all t > k –form σ t for all t ≥ Z t := ( ddt φ t ) ◦ φ − t for t > , σ t := φ ∗ t ( ι ( Z t ) τ ) for t ≥ . Although Z t is singular at t = 0, as in [16] one can verify in local coordinatesthat σ t is smooth at t = 0. Since Z t | X = 0, the k –form σ t satisfies σ t | X = 0.Furthermore, for any vector field K ∈ Γ( κ ) on ∂X which is parallel to κ , we have ι ( K ) σ t = φ ∗ t ( ι ( Z t ) ι ( dφ t ( K )) τ ) = 0 on the boundary, so that ι ∗ ∂X ( σ t ) | κ = 0. Finally, σ t satisfies the equation τ = φ ∗ τ − φ ∗ τ = Z ddt ( φ ∗ t τ ) dt = Z φ ∗ t ( L X t τ ) dt = Z d ( φ ∗ t ( ι ( X t ) τ )) dt = Z dσ t dt = d ( Z σ t dt ) . σ := R σ t dt , it is simple to verify the desired properties usingthe corresponding properties for σ t . Lemma 4.15.
Let U be a manifold and L ⊂ U be a closed submanifold. Let κ , κ be rank orientable distributions in T U such that κ i | L ∩ T L = { } and κ | L = κ | 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: ψ t | ∂L = Id , d ( ψ t ) u = Id for u ∈ L, ψ = Id , [ ψ ] ∗ ( κ ) = κ . Furthermore, we can take ψ t to be t –independent for t near and .Proof. Since κ and κ are orientable, we can pick nonvanishing sections Z and Z We may assume that Z = Z along L . We let Z t denote the family of vector fields Z t := (1 − t ) Z + tZ . Since Z = Z along L , we can pick a neighborhood N of L such that Z t is nowhere vanishing for all t . We also select a submanifold Σ ⊂ N with dim(Σ) = dim( U ) − ⋔ Z t 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 ⊥ of T L . By shrinking Σ and scaling Z t to λZ t , 0 < λ <
1, we can define a smooth family of embeddingsΨ : ( − , s × Σ × [0 , t → N, Ψ t ( s, x ) = exp[ Z t ] s ( x ) . Here exp[ Z t ] denotes the flow generated by Z t . We let ψ t = Ψ t ◦ Ψ − . To seethe properties of (4.1), note that Ψ t (0 , l ) = l for all l ∈ L and d (Ψ t ) ,l ( s, u ) = sZ t + u . This implies the first two properties. The third is trivial, while the fourthis immediate from [Φ t ] ∗ ( ∂ t ) = Z t . We can make ψ t constant near 0 and 1 by simplyreparametrizing 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 , → T ∗ L denote thefamily of maps φ t ( x, v ) = ( x, tv ) . Then [ φ t ] ∗ ( κ ) = κ .Proof. By passing to a chart, we may assume that L ⊂ R + x × R n − x and T ∗ L ⊂ R + x × R n − x × R np . Then κ is simply given on ∂T ∗ L ⊂ { } × × R n − x × R np by κ = span( ∂ p ) = span( ∂ p , . . . , ∂ p n , ∂ x , . . . , ∂ x n ) ω ⊂ T ( ∂T ∗ L ) . Under the scaling map, we have [ φ t ] ∗ ( ∂ p ) = t · ∂ p . This implies that [ φ t ] ∗ ( κ ) = κ .30 eferences [1] S. Anjos, F. Lalonde, M. Pinsonnault, The homotopy type of the space of sym-plectic balls in rational ruled –manifolds , Geometry and Topology 13 (2009):1177-1227.[2] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactnessresults in symplectic field theory , Geometry and Topology 7.2 (2003): 799-888.[3] F. Bourgeois, K. Mohnke,
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