Embedded contact homology of prequantization bundles
EEmbedded contact homology of prequantization bundles
Jo Nelson ∗ and Morgan Weiler Abstract
The 2011 PhD thesis of Farris [Fa] demonstrated that the ECH of a prequantizationbundle over a Riemann surface is isomorphic as a Z -graded group to the exterioralgebra of the homology of its base. We extend this result by computing the Z -gradingon the chain complex, permitting a finer understanding of this isomorphism. We fill insome technical details, including the Morse-Bott direct limit argument and some writhebounds. The former requires the isomorphism between filtered Seiberg-Witten Floercohomology and filtered ECH as established by Hutchings–Taubes [HT13]. The latterrequires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner–Hutchings–Zhang [CGHZ]. In a subsequent paper, we plan to extend thesemethods for Seifert fiber spaces and relate the ECH U -map to orbifold Gromov-Witteninvariants of the base. Contents ∗ Partially supported by NSF grant DMS-1810692. a r X i v : . [ m a t h . S G ] J u l The many flavors of J S -invariant domain dependent J . . . . . . . . . . . . 47 J to domain independent J ECH References 82
Embedded contact homology (ECH) is an invariant of three dimensional contact manifolds,due to Hutchings [Hu14], with powerful applications to dynamics and symplectic embeddingproblems. Most computations of ECH rely on enumerative toric methods, and a generalMorse-Bott program for ECH does not presently exist. Following the framework given byFarris [Fa] and providing additional details in preparation for our future work, we show thatfor prequantization bundles, there is an appropriately filtered ECH differential which onlycounts cylinders corresponding to unions of fibers over Morse flow lines of a perfect Morsefunction on the base. We then make use of direct limits for filtered ECH, as established in[HT13], to provide a Morse-Bott means of computing ECH for prequantization bundles overclosed Riemann surfaces. This permits us to conclude that the ECH of a prequantizationbundle over a Riemann surface is isomorphic as a Z -graded group to the exterior algebraof the homology of this base. In a subsequent paper we will extend our work to Seifert fiberspaces and consider the U-map on the ECH of prequantization bundles and Seifert fiberspaces, enabling us to compute their ECH capacities. Let Y be a closed three-manifold with a contact form λ . Let ξ “ ker p λ q denote the associatedcontact structure, and let R denote the associated Reeb vector field, which is uniquelydetermined by λ p R q “ , dλ p R, ¨q “ . Reeb orbit is a map γ : R { T Z Ñ Y for some T ą γ p t q “ R p γ p t qq , moduloreparametrization. A Reeb orbit is said to be embedded whenever this map is injective. Fora Reeb orbit as above, the linearized Reeb flow for time T defines a symplectic linear map P γ : p ξ γ p q , dλ q ÝÑ p ξ γ p q , dλ q . (1.1)The Reeb orbit γ is nondegenerate if P γ does not have 1 as an eigenvalue. The contact form λ is called nondegenerate if all Reeb orbits are nondegenerate; generic contact forms havethis property. Fix a nondegenerate contact form below. A nondegenerate Reeb orbit γ is elliptic if P γ has eigenvalues on the unit circle and hyperbolic if P γ has real eigenvalues. If τ is a homotopy class of trivializations of ξ | γ , then the Conley-Zehnder index CZ τ p γ q P Z isdefined; see the review in § τ , and is even when γ is positive hyperbolic and odd otherwise. Wesay that an almost complex structure J on R ˆ Y is λ -compatible if J p ξ q “ ξ ; dλ p v, J v q ą v P ξ ; J is invariant under translation of the R factor; and J pB s q “ R , where s denotes the R coordinate. We denote the set of all λ -compatible J by J p Y, λ q .ECH is defined roughly defined as follows, with a complete description given in § Y equipped with a nondegenerate contact form λ and generic λ -compatible J , the ECH chain complex (with respect to a fixed homology class Γ P H p Y q )is the Z vector space freely generated by finite sets of pairs α “ tp α i , m i qu where the α i are distinct embedded Reeb orbits, the m i are positive integers, the total homology class ř i m i r α i s “ Γ, and m i “ α i is hyperbolic. Let M J p α, β q denote the set of J -holomorphic currents from α to β . The ECH differential is a mod 2 count of ECH index1 currents. The definition of the ECH index is the key nontrivial part of ECH [Hu02b],and under the assumption that J is generic, guarantees that the curves are embedded,except possibly for multiple covers of trivial cylinders R ˆ γ where γ is a Reeb orbit. Let ECH ˚ p Y, λ, Γ , J q denote the homology of the ECH chain complex. It turns out that thishomology does not depend on the choice of J or on the contact form λ for ξ , and so definesa well-defined Z { ECH ˚ p Y, ξ, Γ q . The proof of invariance goes through Taubes’isomorphism with Seiberg-Witten Floer cohomology [T10I].There is a filtration on ECH which enables us to compute it via successive approximations,as explained in § § symplectic action or length of an orbit set α “ tp α i , m i qu is A p α q : “ ÿ i m i ż α i λ If J is λ -compatible and there is a J -holomorphic current from α to β , then A p α q ě A p β q by Stokes’ theorem, since dλ is an area form on such J -holomorphic curves. Since B counts J -holomorphic currents, it decreases symplectic action, i.e., xB α, β y ‰ ñ A p α q ě A p β q Let
ECC L ˚ p Y, λ, Γ; J q denote the subgroup of ECC ˚ p Y, λ, Γ; J q generated by orbit sets ofsymplectic action less than L . Since B decreases action, it is a subcomplex; we denote therestriction of B to ECC L ˚ by B L . It is shown in [HT13, Theorem 1.3] that the homology of ECC ˚ p Y, λ, Γ; J q is independent of J , therefore we denote its homology by ECH L ˚ p Y, λ, Γ q ,which we call filtered ECH . 3 .2 Main theorem and outline of the proof Given a closed Riemann surface p Σ g , ω q of genus g ě
0, consider the the principal S -bundle p : Y Ñ Σ g with Euler class e “ ´ π r ω s . A connection on Y is an imaginary-valued 1-form A P Ω p P, i R q that is S -invariant and satisfies A p X q “ i , where the vector field X p p q : “ ddθ ˇˇˇˇ θ “ pe iθ , θ P R { π Z , is the derivative of the S -action of the bundle. Since the Euler class of Y is ´ π r ω s , theconnection can be chosen such that its curvature is F A “ dA “ i p ˚ ω ;see [MS3rd, Thm. 2.7.4]. The 1-form λ : “ ´ iA is a contact form on Y and satisfies dλ “ p ˚ ω . Since λ is S -invariant and λ p X q “ λ is R : “ X , and all Reeb orbits are degenerate andhave period 2 π . This contact manifold p Y, λ q is called the prequantization bundle of p Σ g , ω q ,and the construction can be generalized to a symplectic base of arbitrary dimension. TheHopf fibration is an example of a prequantization bundle of Euler class -1 over the sphere,while the lens space L p| e | , q is an example of a prequantization bundle of arbitrary negativeEuler class e over the sphere. Our main result is the following computation of the ECH ofprequantization bundles. Theorem 1.1.
Let p Y, ξ “ ker λ q be a prequantization bundle over p Σ g , ω q of negative Eulerclass e . Then as Z -graded Z -modules, à Γ P H p Y ; Z q ECH ˚ p Y, ξ, Γ q – Λ ˚ H ˚ p Σ g ; Z q . Moreover, each Γ P H p Y ; Z q satisfying ECH ˚ p Y, ξ, Γ q ‰ corresponds to a number in t , . . . , ´ e ´ u , and under this correspondence ECH ˚ p Y, ξ, Γ q – à d P Z ě Λ Γ `p´ e q d H ˚ p Σ g ; Z q (1.2) as Z -graded Z -modules. When Γ “ , the Z -graded isomorphism (1.2) can be upgraded toan isomorphism of Z -graded Z -modules, where if ˚ denotes the grading on ECH and ‚ thegrading on the right hand side of (1.2), then ˚ “ ´ ed ` p χ p Σ g q ` e q d ` ‚ . (1.3) When Γ ‰ , we have a relatively Z -graded isomorphism. Let α P Λ Γ `p´ e q d α H ˚ p Σ g ; Z q and β P Λ Γ `p´ e q d β H ˚ p Σ g ; Z q , and let α and β also denote the corresponding homogeneous ECHelements under (1.2). Let | ¨ | ˚ denote the ECH grading and | ¨ | ‚ the grading on the righthand side of (1.2). Then | α | ˚ ´ | β | ˚ “ ´ e p d α ´ d β q ` p χ p Σ g q ` ` e qp d α ´ d β q ` | α | ‚ ´ | β | ‚ . xample 1.2. As a refinement of Theorem 1.1, we compute the Z -graded ECH of the lensspaces L p| e | , q , obtaining the ECH version of [KMOS, Corollary 3.4]: ECH ˚ p Y, ξ, Γ q “ Z if ˚ P Z ě § §
2, we provide a shortprimer on the ingredients that go into ECH. Then, in § H . Our computationof the ECH index uses the blueprint given in [Fa, §
3] and fills in a number of details whichwere not previously available. Before we can give the computation, we need to perturb thedegenerate canonical contact form λ .As in [Ne20] we use the following perturbation in § H : Σ Ñ R with | H | C ă λ , λ ε : “ p ` ε p ˚ H q λ. The perturbed Reeb vector field is R ε “ R ` ε p ˚ H ` ε ˜ X H p ` ε p ˚ H q where ˜ X H is the horizontal lift of the Hamiltonian vector field X H to ξ . Note that if p P Crit p H q then X H p p q “
0. Fix L ą
0; there exists ε ą γ of R ε satisfiesthe following dichotomy: • if A p γ q ă L then γ is nondegenerate and projects to p P Crit p H q ; • if A p γ q ą L then γ loops around the tori above the orbits of X H , or is a larger iterateof a fiber above p P Crit p H q .We denote the k -fold cover projecting to p P Crit p H q by γ kp . We have the following expressionfor the Conley-Zehnder index (see [vKnotes], [Ne20, § CZ τ p γ kp q “ RS τ p fiber k q ´ dim p Σ g q ` ind p p H q . Using the constant trivialization τ of ξ “ p ˚ T Σ g , we have that the Robbin-Salamon indexof the degenerate fiber is RS τ p fiber k q “
0. Thus CZ τ p γ kp q “ ind p p H q ´ . If ind p p H q “ γ p is positive hyperbolic. Since p is a bundle, all linearized return mapsare close to the identity, thus there are no negative hyperbolic orbits. If ind p p H q “ , γ p is elliptic. Our convention for defining the Hamiltonian vector field is X H “ ω p¨ , dH q . H to be perfect, we denote the index zero elliptic orbit by e ´ , the indextwo elliptic orbit by e ` , and the hyperbolic orbits by h , . . . , h g . The critical points of aperfect H form a basis for H ˚ p Σ g ; Z q . The generators of CC L ˚ p Y, λ ε p L q , Γ q for some L , where λ ε p L q are of the form e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` where m i “ ,
1, and hence form a basis for theexterior algebra Λ ˚ H ˚ p Σ g ; Z q when both are given a Z grading, under the map generatedby sending each critical point to its Morse homology class.For homologous orbit sets α and β , let d denote the difference in the number of embeddedorbits, counted with multiplicity, appearing in α and β , divided by ´ e : d “ M ´ N | e | , (1.4)where M : “ m ´ ` m ` ¨ ¨ ¨ ` m g ` m ` and similarly N : “ n ´ ` n ` ¨ ¨ ¨ ` n g ` n ` .We obtain the following formula for the ECH index, cf. Proposition 3.5: Proposition 1.3.
Let p Y, λ q be a prequantization bundle over a surface Σ g with Euler class e P Z ă . The ECH index in ECH L ˚ p Y, λ ε p L q , Γ q satisfies the following formula for any Γ , L ,and orbit sets α and β , where L and ε p L q are as in Lemma 3.1, I p α, β q “ χ p Σ g q d ´ d e ` dN ` m ` ´ m ´ ´ n ` ` n ´ . (1.5) Remark 1.4.
The above formula recovers the ECH index for perturbations of the standard S . The index formula in [Hu14], when applied to the contact forms on the ellipsoids E p ` δ n , ´ η n q where δ n , η n Ñ ´ δ n ´ η n P R ´ Q , converges to our formula with g “ e “ ´ B only counts cylinders corresponding toMorse flows on Σ g , therefore Bp e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` q is a sum over all ways to apply B Morse to h i or e ` . In § B is zero because H is perfect.Before sketching the correspondence between Morse flows and the filtered ECH differ-ential (the content of § § L there exists ε p L q ą ECC L ˚ p Y, λ ε , J q consist solely of orbits which are fibers over critical points. We have: Proposition 1.5.
With
Y, λ , and ε p L q as in Lemma 3.1, for any Γ P H p Y ; Z q , there is a di-rect system consisting of all the ECH L ˚ p Y, λ ε p L q , Γ q . The direct limit lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q is the homology of the chain complex generated by orbit sets tp α i , m i qu where the α i are fibersabove critical points of H and ř i m i r α i s “ Γ . Then in §
7, rather than considering degenerations of moduli spaces directly, we insteadpass to filtered ECH and take direct limits by appealing to the isomorphism with filteredSeiberg-Witten theory [HT13]. We prove:
Theorem 1.6.
With
Y, λ, ε p L q as in Lemma 3.1, and for any Γ P H p Y ; Z q , lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q “ ECH ˚ p Y, ξ, Γ q . B L only counts cylinders whichare the union of fibers over Morse flow lines in Σ g . Cylinder counts associated to the aboveperturbation permit the use of fiberwise S -invariant J “ p ˚ j Σ g , even for multiply coveredcurves, by automatic transversality, cf. [Ne20]. However for somewhere injective curves withgenus, we cannot obtain transversality using a S -invariant J , and following [Fa, § S -invariant domain dependent almost complex structures J P J S ˙Σ in § Proposition 1.7.
Let α and β be nondegenerate orbit sets and J P J S ˙Σ be generic. If deg p α, β q ą and I p α, β q “ then M J p α, β q “ H . In § g and show that deg p α, β q “ d , where d is as in (1.4), c.f. Definition 4.5. By Lemma 4.6,a curve C which contributes nontrivially to the ECH differential has degree zero if and onlyif it is a cylinder. We note that Proposition 1.7 and its proof previously appeared in [Fa,Corollary 6.2.3], and we repeat his arguments from which this corollary follows, includingthe regularity result for S -invariant domain dependent almost complex structures Theorem4.18; cf. Corollary 4.20.However, the definition of ECH relies on the choice of a generic λ -compatible domainindependent almost complex structure J , rather than a domain dependent J . For highergenus curves, we cannot achieve transversality with S -invariant domain independent J . Asobserved by Farris in [Fa, § t J t u t Pr , s interpolating between J : “ J P J S ˙Σ and J : “ J P J p Y, λ q . We obtain the following resultin § Proposition 1.8.
Let α and β be admissible orbit sets with I p α, β q “ and deg p α, β q ą . For generic paths t J t u t Pr , s , the moduli space M t : “ M J t p α, β q is cut out transversely savefor a discrete number of times t , ..., t (cid:96) P p , q . For each such t i , the ECH differential canchange either by: (a) The creation or destruction of a pair of oppositely signed curves. (b) An “ECH handleslide.”In either case, the homology is unaffected.
From this, we obtain the desired corollary:
Corollary 1.9.
Let α and β be nondegenerate admissible orbit sets and J P J p Y, λ q begeneric. If deg p α, β q ą and I p α, β q “ then the mod 2 count Z M J p α, β q “ . If α and β are associated to λ ε as in Lemma 3.1 and A p α q , A p β q ă L p ε q then xB L p ε q α, β y “ . Because we are using Z -coefficients, we will not sort through the signs. J -holomorphic currentscontributes to B L then the currents must consist of trivial cylinders together with one cylin-der, which is the union of fibers over a Morse flow line in the base Σ g contributing to theMorse differential. We obtain the main result, Theorem 1.1 as the counts of such cylindersequal the counts of the Morse flow lines which are their images under p , cf. Proposition 4.7.We show in § H is perfect, that the lattercounts are all zero.We conclude with a brief discussion of Proposition 1.8, which is proven in §
6. In Propo-sition 1.8(a) the mod 2 counts of curves in M t i ´ ε p α, β q and M t i ` ε p α, β q are the same. Thedifferential can change at an ECH handleslide, at which a sequence of Fredholm and ECHindex 1 curves t C p t qu breaks into a holomorphic building in the sense of [BEHWZ] into com-ponents consisting of an ECH and Fredholm index 0 curve, an ECH and Fredholm index 1curve, and some “connectors,” which are Fredholm index 0 branched covers of a trivial cylin-der γ ˆ R . In §
5, we demonstrate that connectors cannot appear at the top most or bottommost level of the building via intersection theory arguments similar to [HN16, § § In §
7, as a step in the proof of Theorem 1.1, we prove the second, stronger conclusion ofTheorem 1.1 in terms of the total H p Y q classes Γ and the total multiplicity of representativesof the ECH homology classes, namely that as Z -graded Z -modules ECH ˚ p Y, ξ, Γ q – à d P Z ě Λ Γ `p´ e q d H ˚ p Σ g ; Z q . (1.6)Here we are abusing notation on the right hand side by considering Γ as the element of t , . . . , ´ e ´ u corresponding to its homology class (see Lemma 3.7 (i)). We illustrate the Z -grading (1.3) in an example. Example 1.10.
The following table organizes the first several generators in the case g “ , e “ ´ , Γ “ I “ ´ I “ ´ I “ I “ I “ I “ I “ H Λ e ´ h i e ` Λ e ´ e ´ h i e ´ e ` , h i h j h i e ` e ` Λ e ´ e ´ h i e ´ e ` , e ´ h i h j e ´ h i e ` , h i h j h k e ´ e ` , h i h j e ` Λ e ´ We use I to denote I p¨ , Hq , and we use h i to denote any hyperbolic generator (there are atmost four). Whenever any two or three h i appear together, they are all different.8otice that for ˚ ě
1, the
ECH ˚ groups are isomorphic to Z . This is a special case ofthe following more general result, which is a corollary of Theorem 1.1. It relies on the degree ´ U : ECH ˚ p Y, ξ, Γ q Ñ ECH ˚´ p Y, ξ, Γ q induced by a chain map which counts index 2 J -holomorphic curves passing through a basepoint. Corollary 1.11 (Stability of ECH) . For ˚ sufficiently large and g ą , the groups ECH ˚ p Y, ξ, Γ q are isomorphic to Z f p g q , where f p g q “ g ´ . (For the computation of all of the ECH groups for g “
0, see Example 1.2.)
Proof.
Firstly we show that for ˚ sufficiently large, all of the homology groups are isomorphic.Seiberg-Witten Floer cohomology, denoted z HM ‚ (see § Y with spin-c structures s , for which Taubes [T10I]-[T10IV] has shownthat z HM ´˚ p Y, s ξ, Γ q – ECH ˚ p Y, ξ, Γ q (the notation s ξ, Γ denotes a spin-c structure naturallyassociated to p ξ, Γ q as in [HT13]). It fits into a long exact sequence with two other homologytheories ¨ ¨ ¨ Ð ~ HM ‚ p Y, s q Ð z HM ‚ p Y, s q Ð HM ‚´ p Y, s q Ð ¨ ¨ ¨ where HM ‚ contains z HM ‚ as a subcomplex, and is constructed to satisfy the property U : HM ‚ p Y, s ξ, Γ q Ñ HM ‚` p Y, s ξ, Γ q is an isomorphismwhere U is a map on HM ‚ equalling the image of the U -map in ECH under the isomorphism z HM ´˚ p Y, s ξ, Γ q – ECH ˚ p Y, ξ, Γ q . Since ~ HM ‚ is zero for ‚ sufficiently small as in the proofof [KM, Cor. 35.1.4], the isomorphism from HM descends to an isomorphism U : z HM ´˚ p Y, s ξ, Γ q Ñ z HM ´˚` p Y, s ξ, Γ q thus to an isomorphism between ECH ˚ p Y, ξ, Γ q and ECH ˚´ p Y, ξ, Γ q for ˚ sufficiently large.We will compute the dimension of a group ECH i p Y, ξ, Γ q for a well-chosen value of i ,where if Γ ‰ i to be the index relative to e Γ ´ . We first make two observations, bothfollowing directly from the index formula (1.5):(i) Amongst all generators e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` with m ´ ` m ` ¨ ¨ ¨ ` m g ` m ` “ M , thehighest value of I p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ q is realized by e M ` , and the lowest by e M ´ . Wealso have I p e M ` , e M ´ q “ M. (ii) We have I p e Γ ´p d ` q e ´ , e Γ ´ de ` q “ χ p Σ g q . By (i) and (ii), we can take d large enough compared to g so that all generators e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` with I p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ de ´ q “ M have m ´ ` m ` ¨ ¨ ¨ ` m g ` m ` “ M “ Γ ´ ed. (1.7)9herefore, to complete the proof we will count all generators with I p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ de ´ q “ M which satisfy (1.7). Because e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` satisfies (1.7), we obtain I p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ de ´ q “ m ` ´ m ´ ` Γ ´ de. (1.8)By (1.7) and (1.8) it suffices to count all generators with m ´ “ m ` and 2 m ´ ` m `¨ ¨ ¨` m g “ M . We can choose d even larger if necessary so that d ą g .If Γ ´ de is even, then the set of such generators can be grouped into those with0 , , , . . . , k, . . . , g hyperbolic generators. There are ` g k ˘ generators with 2 k hyperbolicgenerators, therefore there are ˆ g ˙ ` ˆ g ˙ ` ˆ g ˙ ` ¨ ¨ ¨ ` ˆ g k ˙ ` ¨ ¨ ¨ ˆ g g ˙ “ g ´ (1.9)generators with m ´ “ m ` and 2 m ´ ` m ` ¨ ¨ ¨ ` m g “ M in total. Similarly, if Γ ´ de is odd,then the set of such generators can be grouped into those with 1 , , . . . , k ` , . . . , g ´ ˆ g ˙ ` ˆ g ˙ ` ¨ ¨ ¨ ` ˆ g k ` ˙ ` ¨ ¨ ¨ ˆ g g ´ ˙ “ g ´ (1.10)such generators in total. Both (1.9) and (1.10) are elementary equalities following from thesum of binomial coefficients n ÿ k “ ˆ nk ˙ x k “ p ` x q n (1.11)with n “ g . Setting x “ ` gk ˘ is 2 g , and adding (1.11) for x “ ´ x “ Conjecture 1.12.
For ˚ sufficiently large, the U map is an isomorphism on the chain levelof the ECH of prequantization bundles, for the chain complex of Proposition 1.5. When g “
0, Conjecture 1.12 can likely be proved using Example 1.2 and the methodsof [Hu14, § U -map at the chain level as it is crucial to the defini-tion of the “ECH spectrum.” This is a list where the k th term is the minimum action of ahomologically essential ECH cycle in ECH ˚ p Y, ξ, rHsq which is homologous to the preimageunder U k of the class of the empty orbit set. The ECH spectrum of a contact three-manifoldgoverns the embedding properties of its symplectic fillings, and we plan to use the U -mapof prequantization bundles and their generalization, Seifert fiber spaces, to understand theembedding properties of fillings of the simple singularities, among other examples. In par-ticular, we expect to directly recover that U is an isomorphism in sufficiently large degree,as is known to be true for Seiberg-Witten cohomology.We expect to be able to compute the U -map for general prequantization bundles followingthe computation in the g “ , e “ ´ Y “ S ) in [Hu14, § as a count of index two gradient flow lines for a Morse function on the base S , togetherwith a count of meromorphic multisections of the complex line bundle associated to the Hopffibration. As discussed in [Fa, § J -holomorphic curve in R ˆ Y which intersects each R ˆ t fiber u the same number of times can be interpreted asa meromorphic multisection of the line bundle associated to Y Ñ Σ g with its asymptoticends on Reeb orbits interpreted as either zeroes or poles. We plan to make this perspectiverigorous. Remark 1.13 ( U -map) . For prequantization bundles with negative Euler class e over aclosed surface of genus g , the U -map is given in terms of index two gradient flow linesand meromorphic multisections of the line bundle associated to Y , and for sufficiently largegrading induces an isomorphism on ECH.We will investigate an analogous result for Seifert fiber spaces. Here, the novel con-tribution will be to understand how to characterize the U -map in terms of the orbifoldGromov-Witten invariants of the base, and should follow ideas in [Ri14], which computesthe symplectic cohomology of negative line bundles via Gromov-Witten theory of the baseorbifold.In addition, we hope to extend our methods to compute the knot filtered ECH of Seifertfiber spaces considered as prequantization bundles over orbifolds in the sense of [Tho76].Knot filtered ECH was introduced in [Hu16] to study disk symplectomorphisms via ECH,and extended in [Wei] to study annulus symplectomorphisms via similar methods. Sinceperiodic surface symplectomorphisms can be realized as the return map of open book de-compositions on Seifert fibered contact manifolds [CH], our computations will allow us tostudy the dynamics of such symplectomorphisms, which include all symplectomorphisms ongenus zero surfaces. Along the way to knot filtered ECH, understanding the standard ECHof Seifert fiber spaces should allow us to recover the computation in [MOY97, Cor. 1.0.6] ofthe irreducible Seiberg-Witten Floer homology of Brieskorn manifolds. Acknowledgements.
We thank Michael Hutchings for numerous elucidating conversationsand comments on an earlier draft of this paper.
Let Y be a closed contact 3-dimensional manifold equipped with a nondegenerate contactform λ ; let ξ “ ker p λ q denote the associated contact structure. We say that a closed Reeborbit γ is elliptic if the eigenvalues of the linearized return map P γ are on the unit circle, positive hyperbolic if the eigenvalues of P γ are positive real numbers, and negative hyperbolic if the eigenvalues of P γ are negative real numbers.An orbit set is a finite set of pairs α “ tp α i , m i qu , where the α i are distinct embeddedReeb orbits and the m i are positive integers. We call m i the multiplicity of α i in α . Thehomology class of the orbit set α is defined by r α s “ ÿ i m i r α i s P H p Y q . α is admissible if m i “ α i is positive or negative hyperbolic.The ECH chain complex is generated by admissible orbit sets. The differential countsECH index one J -holomorphic currents in R ˆ Y . The definition of the ECH index dependson three components: the relative first Chern class c τ , which detects the contact topologyof the curves; the relative intersection pairing Q τ , which detects the algebraic topology ofthe curves; and the Conley-Zehnder terms, which detect the contact geometry of the Reeborbits. Further details are provided in the subsequent sections. Given a punctured compact Riemann surface p ˙Σ , j q , with a partition of its punctures S into a positive subset S ` and negative subset S ´ we consider asymptotically cylindrical J -holomorphic maps of the form C : p ˙Σ , j q Ñ p R ˆ Y, J q , dC ˝ j “ J ˝ dC, subject to the following asymptotic condition. If γ is a (possibly multiply covered Reeborbit), a positive end of C at γ is a puncture near which C is asymptotic to R ˆ γ as s Ñ 8 and a negative end of C at γ is a puncture near which C is asymptotic to R ˆ γ as s Ñ ´8 . This means there exist holomorphic cylindrical coordinates identifying a puncturedneighborhood of z with a respective positive half-cylinder Z ` “ r ,
8q ˆ S or negative half-cylinder Z ´ “ p´8 , s ˆ S and a trivial cylinder C γ z : R ˆ S Ñ R ˆ Y such that C p s, t q “ exp C γz p s,t q h z p s, t q for | s | sufficiently large , (2.1)where h z p s, t q is a vector field along C γ z satisfying | h z p s, ¨q| Ñ s Ñ ˘8 . Boththe norm and the exponential map are assumed to be defined with respect to a translation-invariant choice of Riemannian metric on R ˆ Y . To obtain a moduli space of J -holomorphiccurves from the above asymptotically cylindrical maps, we mod out by the usual equivalencerelation p ˙Σ , j, S , C q „ p ˙Σ , j , S , C q , which exists whenever there exists a biholomorphism φ : p ˙Σ , j q Ñ p ˙Σ , j q taking S to S with the ordering preserved, i.e. φ p S ` q “ S and φ p S ´ q “ S , such that C “ C ˝ φ. Definition 2.1.
An asymptotically cylindrical pseudoholomorphic curve C : p ˙Σ , j q Ñ p R ˆ Y, J q is said to be multiply covered whenever there exists a pseudoholomorphic curve C : p ˙Σ , j q Ñ p R ˆ Y, J q , and a holomorphic branched covering ϕ : p ˙Σ , j q Ñ p ˙Σ , j q with S “ ϕ p S ` q and S “ ϕ p S ´ q such that C “ C ˝ ϕ, deg p ϕ q ą , allowing for ϕ to not have any branch points. The multiplicity of C is given by deg p ϕ q .
12n asymptotically cylindrical pseudoholomorphic curve C is called simple whenever it isnot multiply covered. In [Ne15, § somewhere injective , meaning for some z P ˙Σ,which is not a puncture, dC p z q ‰ C ´ p C p z qq “ t z u . A point z P ˙Σ with this property is called an injective point of C .Let α “ tp α i , m i qu and β “ tp β j , n j qu be orbit sets in the same homology class ř i m i r α i s “ ř j n j r β j s “ Γ P H p Y q . We define a J -holomorphic current from α to β to be a finite set ofpairs C “ tp C k , d k qu , where the C k are distinct irreducible somewhere injective J -holomorphiccurves in R ˆ Y , the d k are positive integers, C is asymptotic to α as a current as the R -coordinate goes to `8 , and C is asymptotic to β as a current as the R -coordinate goes to ´8 . This last condition means that the positive ends of the C k are at covers of the Reeborbits α i , the sum over k of d k times the total covering multiplicity of all ends of C k at coversof α i is m i , and analogously for the negative ends.Let M J p α, β q denote the set of J -holomorphic currents from α to β . A holomorphiccurrent C is said to be somewhere injective if d k “ k . A somewhere injectivecurrent is said to be embedded whenever each C k is embedded and the C k are pairwisedisjoint.Let H p Y, α, β q denote the set of relative homology classes of 2-chains Z in Y such that B Z “ ÿ i m i α i ´ ÿ j n j β j , modulo boundaries of 3-chains. The set H p Y, α, β q is an affine space over H p Y q , and every J -holomorphic current C P M J p α, β q defines a class r C s P H p Y, α, β q . Let C be an asymptotically cylindrical curve with k positive ends at (possibly multiplycovered) Reeb orbits γ ` , ..., γ ` k and (cid:96) negative ends at γ ´ , ..., γ ´ (cid:96) . We denote the set ofequivalence classes by M J p γ ` , γ ´ q : “ M J p γ ` , ..., γ ` k ; γ ´ , ..., γ ´ (cid:96) q Note that R acts on M J p γ ` , γ ´ q by translation of the R factor in R ˆ Y .The Fredholm index of C is defined byind p C q “ ´ χ p C q ` c τ p C q ` k ÿ i “ CZ τ p γ ` i q ´ (cid:96) ÿ j “ CZ τ p γ ´ j q . (2.2)The Euler characteristic of the domain ˙Σ of C is denoted by χ p C q “ p ´ g p Σ q ´ k ´ (cid:96) q . The remaining terms are a bit more involved to define as they depend on the choice of atrivialization τ of ξ over the Reeb orbits γ ` i and γ ´ j , which is compatible with dλ , and aredefined in the following subsections. 13he significance of the Fredholm index is that the moduli space of somewhere injectivecurves M J p γ ` , γ ´ q is naturally a manifold near C of dimension ind p C q . In particular, wehave the following results. Theorem 2.2. [Wen-SFT, Theorem 8.1]
Fix a nondegenerate contact form λ on a closedmanifold Y . Let J p Y, λ q be the set of all λ -compatible almost complex structures on R ˆ Y .Then there exists a comeager subset J reg p Y, λ q Ă J p Y, λ q , such that for every J P J reg p Y, λ q ,every curve C P M J p γ ` , γ ´ q with a representative C : p ˙Σ , j q Ñ p R ˆ Y, J q that has aninjective point z P int p ˙Σ q satisfying π ξ ˝ dC p z q ‰ is Fredholm regular. The above result also holds for the set of somewhere injective curves in completed exactsymplectic cobordisms p W, J q ; see [Wen-SFT, Theorem 7.2]. Moreover, we have that Fred-holm regularity implies that a neighborhood of a curve admits the structure of a smoothorbifold. Theorem 2.3 (Theorem 0, [Wen10]) . Assume that C : p ˙Σ , j q Ñ p W, J q is a non-constantcurve in M J p γ ` , γ ´ q . If C is Fredholm regular, then a neighborhood of C in M J p γ ` , γ ´ q naturally admits the structure of a smooth orbifold of dimension ind p C q “ ´ χ p C q ` c τ p C q ` k ÿ i “ CZ τ p γ ` i q ´ (cid:96) ÿ j “ CZ τ p γ ´ j q , whose isotropy group at C is given by Aut : “ t ϕ P Aut p ˙Σ , j q | C “ C ˝ ϕ u . Moreover, there is a natural isomorphism T C M J p γ ` , γ ´ q “ ker D ¯ B J p j, C q{ aut p ˙Σ , j q . The trivialization τ determines a trivialization of ξ | C over the ends of C up to homotopy.We denote the set of homotopy classes of symplectic trivializations of the 2-plane bundle γ ˚ ξ over S by T p γ q . After fixing trivializations τ ` i P T p γ ` i q for each i and τ ´ j P T p γ ´ j q , wedenote this set of trivialization choices by τ P T p α, β q .We are now ready to define the relative first Chern number of the complex line bundle ξ | C with respect to the trivialization τ , which we denote by c τ p C q “ c p ξ | C , τ q . Let π Y : R ˆ Y Ñ Y denote projection onto Y . We define c p ξ | C , τ q to be the algebraic countof zeros of a generic section ψ of ξ | r π Y C s which on each end is nonvanishing and constantwith respect to the trivialization on the ends. In particular, given a class Z P H p Y, α, β q we represent Z by a smooth map f : S Ñ Y where S is a compact oriented surface withboundary. Choose a section ψ of f ˚ ξ over S such that ψ is transverse to the zero section14nd ψ is nonvanishing over each boundary component of S with winding number zero withrespect to the trivialization τ . We define c τ p Z q : “ ψ ´ p q , where ‘ α, β, τ, and Z . If Z P H p Y, α, β q is another relative homology class then c τ p Z q ´ c τ p Z q “ x c p ξ q , Z ´ Z y . (2.3)Our sign convention is that if τ , τ : γ ˚ ξ Ñ S ˆ R are two trivializations then τ ´ τ “ deg ` τ ˝ τ ´ : S Ñ Sp p , R q – S ˘ . (2.4)Thus, given another collection of trivialization choices τ “ ` t τ i u , t τ j u ˘ over the orbit setsand the convention (2.4), we have c τ p Z q ´ c τ p Z q “ ÿ i m i ´ τ ` i ´ τ ` i ¯ ´ ÿ j n j ´ τ ´ j ´ τ ´ j ¯ . (2.5) Next we define the
Conley-Zehnder index of an embedded nondegenerate Reeb orbit γ withrespect to the trivialization τ. Pick a parametrization γ : R { T Z Ñ Y. The choice of trivialization τ of ξ over γ is an isomorphism of symplectic vector bundles τ : γ ˚ ξ » ÝÑ p R { T Z q ˆ R . Let t ϕ t u t P R denote the one-parameter group of diffeomorphisms of Y given by the flow ofthe Reeb vector field R . With respect to τ , the linearized flow p dϕ t q t Pr ,T s induces an arc ofsymplectic matrices P : r , T s Ñ Sp p q defined by P t “ τ p t q ˝ dϕ t ˝ τ p q ´ . To each arc of symplectic matrices t P t u t Pr ,T s with P “ P T nondegenerate, there is anassociated Conley Zehnder index CZ pt P t u t Pr ,T s q P Z . We define the Conley-Zehnder index of γ with respect to τ by CZ τ p γ q “ CZ ` t P t u t Pr ,T s ˘ . lliptic case: In the elliptic case each trivialization is homotopic to one whose linearized flow t ϕ t u can be realized as a path of rotations. If we take τ to be one of these trivializationsso that each ϕ t is rotation by the angle 2 πϑ t P R then ϑ t is a continuous function of t P r , T s satisfying ϑ “ ϑ : “ ϑ T P R z Z . The number ϑ P R z Z is the rotationangle of γ with respect to the trivialization and CZ τ p γ k q “ (cid:98) kϑ (cid:99) ` . Hyperbolic case:
Let v P R be an eigenvector of φ T . Then for any trivialization used,the family of vectors t ϕ t p v qu t Pr ,T s , rotates through angle πr for some integer r . Theinteger r is dependent on the choice of trivialization Φ, but is always even in the positivehyperbolic case and odd in the negative hyperbolic case. We obtain CZ τ p γ k q “ kr. The Conely-Zehnder index depends only on the Reeb orbit γ and homotopy class of τ in the set of homotopy classes of symplectic trivializations of the 2-plane bundle γ ˚ ξ over S “ R { T Z . Given two trivializations τ and τ we have that CZ τ p γ k q ´ CZ τ p γ k q “ k p τ ´ τ q , (2.6)maintaining our sign convention (2.4).In our later expression of the ECH index, we will use the following shorthand for an orbitset α : CZ Iτ p α q “ ÿ i m i ÿ k “ CZ τ p α ki q . (2.7)Another set of trivialization choices τ for α yields CZ Iτ p α q ´ CZ Iτ p α q “ ÿ i p m i ` m i qp τ i ´ τ i q . (2.8)We will also abbreviate the following Conley-Zehnder contributions to the Fredholm indexand ECH index for a curve C P M p α, β ; J q by: CZ Iτ p C q “ CZ Iτ p α q ´ CZ Iτ p β q CZ indτ p C q “ k ÿ i “ CZ τ p α m i i q ´ (cid:96) ÿ j “ CZ τ p β n j j q . (2.9) The ECH index depends on the relative intersection pairing, which is related to the asymp-totic writhe and linking number. We review these notions now and summarize the relativeadjunction formula. 16 .3.1 Asymptotic writhe and linking number
Given a somewhere injective curve C P M J p γ ` , γ ´ q , we consider the slice u X pt s u ˆ Y q . If s "
0, then the slice u X pt s u ˆ Y q is an embedded curve which is the union, over i , of a braid ζ ` i around the Reeb orbit γ ` i with m i strands. The trivialization τ can be used to identifythe braid ζ ` i with a link in S ˆ D . We identify S ˆ D with an annulus cross an interval,projecting ζ ` i to the annulus, and require that the normal derivative along γ ` i agree withthe trivialization τ .We define the writhe of this link, which we denote by w τ p ζ ` i q P Z , by counting thecrossings of the projection to R ˆ t u with (nonstandard) signs. Namely, we use the signconvention in which counterclockwise rotations in the D direction as one travels counter-clockwise around S contribute positively.Analogously the slice u X pt s u ˆ Y q for s ! j of a a braid ζ ´ j aroundthe Reeb orbit β j with n j strands and we denote this braid’s writhe by w τ p ζ ´ i q P Z .The writhe depends only on the isotopy class of the braid and the homotopy class of thetrivialization τ . If ζ is an m -stranded braid around an embedded nondegenerate Reeb orbit γ and τ P T p γ q is another trivialization then w τ p ζ q ´ w τ p ζ q “ m p m ´ qp τ ´ τ q because shifting the trivialization by one adds a full clockwise twist to the braid.If ζ and ζ are two disjoint braids around an embedded Reeb orbit γ we can define their linking number (cid:96) τ p ζ , ζ q P Z to be the linking number of their oriented images in R . Thelatter is by definition one half of the signed count of crossings of the strand associated to ζ with the strand associated to ζ in the projection to R ˆ t u . If the braid ζ k has m k strandsthen a change in trivialization results in the following formula (cid:96) τ p ζ , ζ q ´ (cid:96) τ p ζ , ζ q “ m m p τ ´ τ q . The writhe of the union of two braids can be expressed in terms of the writhe of the individualcomponents and the linking number: w τ p ζ Y ζ q “ w τ p ζ q ` w τ p ζ q ` (cid:96) τ p ζ , ζ q . (2.10)If ζ is a braid around an embedded Reeb orbit γ which is disjoint from γ we define the winding number to be the linking number of ζ with γ : η τ p ζ q : “ (cid:96) τ p ζ, γ q P Z . We have the following bound on writhe [Hu14, § Proposition 2.4. If C P M p α, β ; J q is somewhere injective then, w τ p C q ď CZ Iτ p C q ´ CZ indτ p C q , where the Conley-Zehnder shorthand is given by (2.9) . .3.2 Admissible representatives In order to speak more “globally” of writhe and winding numbers associated to a curve, weneed the following notion of an admissible representative for a class Z P H p Y, α, β q , as in[Hu09, Def. 2.11]. Given Z P H p Y, α, β q we define an admissible representative of Z to bea smooth map f : S Ñ r´ , s ˆ Y , where S is an oriented compact surface such that1. The restriction of f to the boundary B S consists of positively oriented covers of t uˆ α i whose total multiplicity is m i and negatively oriented covers of t´ u ˆ β j whose totalmultiplicity is n j .2. The projection π Y : r´ , s ˆ Y Ñ Y yields r π p f p S qqs “ Z .3. The restriction of f to int p S q is an embedding and f is transverse to t´ , u ˆ Y .If S is an admissible representative for any Z P H p Y, α, β q then we say S is an admissiblesurface .The utility of the notion of an admissible representative S for Z can be seen in thefollowing. For ε ą S X pt ´ ε u ˆ Y q consists of braids ζ ` i with m i strands in disjoint tubular neighborhoods of the Reeb orbits α i , which are well defined upto isotopy. Similarly, S X pt´ ` ε u ˆ Y q consists of braids ζ ´ j with n j strands in disjointtubular neighborhoods of the Reeb orbits α i , which are well defined up to isotopy.Thus an admissible representative of Z P H p Y ; α, β q permits us to define the total writhe of a curve interpolating between the orbit sets α and β by w τ p S q “ ÿ i w τ ` i p ζ ` i q ´ ÿ j w τ ´ j p ζ ´ j q . Here ζ ` i are the braids with m i strands in a neighborhood of each of the α i obtained bytaking the intersection of S with t s u ˆ Y for s close to 1. Similarly, the ζ ´ j are the braidswith n j strands in a neighborhood of each of the β j obtained by taking the intersection of S with t s u ˆ Y for s close to ´
1. Bounds on the writhe in terms of the Conley-Zehnderindex are given in [HN16, § § § S is an admissible representative of Z P H p Y, α , β q such that the interior of S does not intersect the interior of S near the boundary, with braids ζ ` i and ζ ´ j we candefine the linking number of S and S to be (cid:96) τ p S, S q : “ ÿ i (cid:96) τ p ζ ` i , ζ ` i q ´ ÿ j (cid:96) τ p ζ ´ j , ζ ´ j q . Above the orbit sets α and α are both indexed by i , so sometimes m i or m i is 0, similarlyboth β and β are indexed by j and sometimes n j or n j is 0. The trivialization τ is atrivialization of ξ over all Reeb orbits in the sets α, α , β, and β .18 .3.3 The relative intersection pairing The relative intersection pairing can be defined using an admissible representative, which ismore general than the notion of a τ -representative [Hu02b, Def. 2.3], as the latter uses thetrivialization to control the behavior at the boundary. Consequently, we see an additionallinking number term appear in the expression of the relative intersection pairing when weuse an admissible representative. Let S and S be two surfaces which are admissible repre-sentatives of Z P H p Y, α, β q and Z P H p Y, α , β q whose interiors ˙ S and ˙ S are transverseand do not intersect near the boundary. We define the relative intersection pairing by thefollowing signed count Q τ p Z, Z q : “ ´ ˙ S X ˙ S ¯ ´ (cid:96) τ p S, S q . (2.11)Moreover, Q τ p Z, Z q is an integer which depends only on α, β, Z, Z and τ . If Z “ Z thenwe write Q τ p Z q : “ Q τ p Z, Z q .For another collection of trivialization choices τ , Q τ p Z, Z q ´ Q τ p Z, Z q “ ÿ i m i m i p τ ` i ´ τ ` i q ´ ÿ i n j n j p τ ´ i ´ τ ´ i q . We recall how [Hu09, § Y . An admissible representative S of Z P H p Y, α, β q is said to be nice whenever the projection of S to Y is an immersion and the projection of the interior˙ S to Y is an embedding which does not intersect the α i ’s or β j ’s. Lemma 3.9 from [Hu09]establishes that if none of the α i equal the β j then every class Z P H p Y, α, β q admits a niceadmissible representative.If S is a nice admissible representative of Z with associated braids ζ ` i and ζ ´ j then wecan define the winding number η τ p S q : “ ÿ i η τ ` i ` ζ ` i ˘ ´ ÿ j η τ ´ j ` ζ ´ j ˘ . Lemma 2.5 (Lemma 3.9 [Hu09]) . Suppose that S is a nice admissible representative of Z .Then Q τ p Z q “ ´ w τ p S q ´ η τ p S q . In this section we review the relative adjunction formulas of interest, which are used later in §
5. This is taken from [Hu02b, §
3] and is stated for pseudoholomorphic curves interpolatingbetween orbit sets α and β in symplectizations. As explained in [Hu09, § Lemma 2.6.
Let u P M J p α, β q be somewhere injective, S be a representative of Z P H p Y, α, β q , and τ P T p α, β q . Let N S be the normal bundle to S . (i) If u is further assumed to be embedded everywhere then c τ p Z q “ χ p S q ` c τ p N S q . (2.12)19ii) For general embedded representatives S , e.g. ones not necessarily coming from pseuo-holomorphic curves, (2.12) holds mod 2 and c τ p N S q “ w τ p S q ` Q τ p Z, Z q . (2.13)(iii) If u is embedded except at possibly finitely many singularities then c τ p Z q “ χ p u q ` Q τ p Z q ` w τ p u q ´ δ p u q , (2.14) where δ p u q is a sum of positive integer contributions from each singularity. Remark 2.7. If u is a closed pseudoholomorphic curve, then there is no writhe term ortrivialization choice, and (2.14) reduces to the usual adjunction formula x c p T W q , r u sy “ χ p u q ` r u s ¨ r u s ´ δ p u q . We are now ready to give the definition of the ECH index.
Definition 2.8 (ECH index) . Let α “ tp α i , m i qu and β “ tp β j , n j qu be Reeb orbit sets inthe same homology class, ř i m i r α i s “ ř j n j r β j s “ Γ P H p Y q . Given Z P H p Y, α, β q . Wedefine the ECH index to be I p α, β, Z q “ c τ p Z q ` Q τ p Z q ` CZ Iτ p α q ´ CZ Iτ p β q . where CZ I is the shorthand defined in (2.7). When α and β are clear from context, we usethe notation I p Z q .The Chern class term is linear in the multiplicities of the orbit sets and the relative in-tersection term is quadratic. The “total Conley-Zehnder” index term CZ Iτ typically behavesin a complicated way with respect to the multiplicities. We also have the following generalproperties of the ECH index, as proven in [Hu02b, § Theorem 2.9.
The ECH index has the following basic properties: (i) (Well Defined)
The ECH index I p Z q is independent of the choice of trivialization. (ii) (Index Ambiguity Formula) If Z P H p Y, α, β q is another relative homology class, then I p Z q ´ I p Z q “ x Z ´ Z , c p ξ q ` p Γ qy . (iii) (Additivity) If δ is another orbit set in the homology class Γ , and if W P H p Y, β, δ q ,then Z ` W P H p Y, α, δ q is defined and I p Z ` W q “ I p Z q ` I p W q . If α and β are generators of the ECH chain complex, then p´ q I p Z q “ ε p α q ε p β q , where ε p α q denotes ´ to the number of positive hyperbolic orbits in α . To learn more about the wonders of the ECH index see [Hu09, § Remark 2.10.
We will also use the notation I p α, β, C q and I p C q for C “ tp C k , d k qu P M J p α, β q to denote I p α, β, r π Y C sq , where r¨s denotes equivalence in H p Y, α, β q , as well as c τ p C q and Q τ p C q , following [Hu09, Notation 4.7]. In addition, we will occasionally use thenotation Q τ p S q for S an admissible surface in r´ , s ˆ Y to denote Q τ pr π Y S sq (recall thatwe are using the notation π Y to denote both the projections from R ˆ Y and from r´ , s ˆ Y to Y ).The ECH index inequality (cf. § trivial cylinder is a cylinder R ˆ γ Ă R ˆ Y where γ isan embedded Reeb orbit. Proposition 2.11 (Prop. 3.7 [Hu14]) . Suppose J is generic. Let α and β be orbit setsand let C P M J p α, β q be any J -holomorphic current in R ˆ Y , not necessarily somewhereinjective. Then: (i) We have I p C q ě with equality if and only if C is a union of trivial cylinders withmultiplicities. (ii) If I p C q “ then C “ C \ C , where I p C q “ , and C is embedded and has ind p C q “ I p C q “ . (iii) If I p C q “ , and if α and β are generators of the chain complex ECC ˚ p Y, λ, Γ , J q , then C “ C \ C , where I p C q “ , and C is embedded and has ind p C q “ I p C q “ . The ECH partition conditions are a topological type of data associated to the pseudoholo-morphic curves (and currents) which can be obtained indirectly from certain ECH indexrelations. In particular, the covering multiplicities of the Reeb orbits at the ends of thenontrivial components of the pseudoholomorphic curves (and currents) are uniquely deter-mined by the trivial cylinder component information. The genus can be determined by thecurrent’s relative homology class.
Definition 2.12. [Hu14, § γ be an embedded Reeb orbit and m a positive integer. Wedefine two partitions of m , the positive partition P ` γ p m q and the negative partition P ´ γ p m q as follows. • If γ is positive hyperbolic, then P ` γ p m q : “ P ´ γ p m q : “ p , ..., q . Previously the papers [Hu02b, Hu09] used the terminology incoming and outgoing partitions. If γ is negative hyperbolic, then P ` γ p m q : “ P ´ γ p m q : “ " p , ..., q m even, p , ..., , q m odd. • If γ is elliptic then the partitions are defined in terms of the quantity ϑ P R z Z forwhich CZ τ p γ k q “ (cid:98) kϑ (cid:99) `
1. We write P ˘ γ p m q : “ P ˘ ϑ p m q , with the right hand side defined as follows.Let Λ ` ϑ p m q denote the highest concave polygonal path in the plane that starts at p , q ,ends at p m, (cid:98) mϑ (cid:99) q , stays below the line y “ ϑx , and has corners at lattice points. Thenthe integers P ` ϑ p m q are the horizontal displacements of the segments of the path Λ ` ϑ p m q between the lattice points.Likewise, let Λ ´ ϑ p m q denote the lowest convex polygonal path in the plane that startsat p , q , ends at p m, (cid:100) mϑ (cid:101) q , stays above the line y “ ϑx , and has corners at latticepoints. Then the integers P ´ ϑ p m q are the horizontal displacements of the segments ofthe path Λ ´ ϑ p m q between the lattice points.Both P ˘ ϑ p m q depend only on the class of ϑ in R z Z . Moreover, P ` ϑ p m q “ P ´´ ϑ p m q . Example 2.13.
If the rotation angle for elliptic orbit γ satisfies ϑ P p , { m q then P ` ϑ p m q “ p , ..., q P ´ ϑ p m q “ p m q . The partitions are quite complex for other ϑ values, see [Hu14, Fig. 1].We end this section with the ECH index inequality [Hu09, Theorem 4.15] in symplecticcobordisms. As before we take α “ tp α i , m i qu and β “ tp β j , n j qu to be Reeb orbit sets in thesame homology class. Let C P M J p α, β q . For each i let a ` i denote the number of positiveends of C at α i and let t q ` i,k u a ` i k “ denote their multiplicities. Thus ř a ` i k “ q ` i,k “ m i . Likewise,for each j let b ´ i denote the number of negative ends of C at β j and let t q ´ j,k u b ´ j k “ denote theirmultiplicities; we have ř b ´ j k “ q ´ j,k “ n j . Theorem 2.14 (ECH index inequality) . Suppose C P M J p α, β q is somewhere injective.Then ind p C q ď I p C q ´ δ p C q . Equality holds only if t q ` i,k u “ P ` α i p m i q for each i and t q ´ j,k u “ P ´ β j p n j q for each j . If α and β are orbit sets and k is an integer, define M Jk p α, β q “ t C P M J p α, β q | I p C q “ k u . α is a generator of the ECH chain complex we define the differential B on ECC ˚ p Y, λ, Γ , J q by B α “ ÿ β p M J p α, β q{ R q β. The sum is over chain complex generators β , and ‘ R acts on M J p α, β q by translation of the R -coordinate on R ˆ Y . By Proposition 2.11 thequotient M J p α, β q{ R is a discrete set. By the arguments in [Hu14, § M J p α, β q{ R is finite. Finally, because the differential decreases the symplectic actionand since any nondegenerate contact form has only finitely many Reeb orbits with boundedsymplectic action, we have that the ECH differential is well-defined.Proving that B “ p ECC ˚ p Y, λ, Γ , J q , Bq is independent of J andof λ up to c p ξ q requires Seiberg-Witten theory. In light of this invariance we denote thishomology by ECH ˚ p Y, ξ, Γ q , and call it the embedded contact homology , or ECH, of p Y, ξ, Γ q .Since the ECH index depends on a choice of relative second homology class Z , for general p Y, ξ, Γ q we can only expect a relative Z d grading, where d is the divisibility of the class c p ξ q ` PD p Γ q in H p Y ; Z q mod torsion. This is because of the index ambiguity propertyof the ECH index, Theorem 2.9 (ii), whereby we set I p α, β q : “ r I p α, β, Z qs P Z d . In the setting of prequantization bundles, we will have d “
0, because c p ξ q ` PD p Γ q istorsion, as we now explain. If g “
0, then H p Y q – H p Y q is torsion, as shown in § g ą
0, then we will demonstrate in Lemma 3.11 that the divisibility of c p ξ q is zero. In § § Z grading.We often refine our relative Z grading to an absolute Z grading by setting a chosengenerator to have grading zero. In particular, if Γ “ rHs , we will choose H to have gradingzero. There is a filtration on ECH which enables us to compute it via successive approximations,as explained in § § symplectic action or length of an orbit set α “ tp α i , m i qu is A p α q : “ ÿ i m i ż α i λ. If J is λ -compatible and there is a J -holomorphic current from α to β , then A p α q ě A p β q by Stokes’ theorem, since dλ is an area form on such J -holomorphic curves. Since B counts J -holomorphic currents, it decreases symplectic action, i.e., xB α, β y ‰ ñ A p α q ě A p β q . Let
ECC L ˚ p Y, λ, Γ; J q denote the subgroup of ECC ˚ p Y, λ, Γ; J q generated by orbit sets ofsymplectic action less than L . Because B decreases action, it is a subcomplex. It is shown in23HT13, Theorem 1.3] that the homology of ECC ˚ p Y, λ, Γ; J q is independent of J , thereforewe denote its homology by ECH L ˚ p Y, λ, Γ q , which we call filtered ECH .Given L ă L , there is a homomorphism ι L,L : ECH L ˚ p Y, λ, Γ q Ñ ECH L ˚ p Y, λ, Γ q , induced by the inclusion ECC L ˚ p Y, λ, Γ; J q (cid:44) Ñ ECC L ˚ p Y, λ, Γ; J q and independent of J , asshown in [HT13, Theorem 1.3]. The ι L,L fit together into a direct system pt ECC L ˚ p Y, λ, Γ qu L P R , ι L,L q .Because taking direct limits commutes with taking homology, we have ECH ˚ p Y, λ, Γ q “ H ˚ ´ lim L Ñ8 ECC L ˚ p Y, λ, Γ; J q ¯ “ lim L Ñ8 ECH L ˚ p Y, λ, Γ q . The inclusion maps are compatible with certain cobordism maps as follows. An exactsymplectic cobordism from p Y ` , λ ` q to p Y ´ , λ ´ q is a pair p X, λ q where X is a four-manifoldwith B X “ Y ` ´ Y ´ and λ a one-form on X with dλ symplectic and λ | Y ˘ “ λ ˘ .Define ECH ˚ p Y, λ q : “ à Γ P H p Y q ECH ˚ p Y, λ, Γ q , (2.15)and define ECH L ˚ p Y, λ q similarly.Exact symplectic cobordisms induce maps on filtered ECH. The properties of these cobor-dism maps will allow us to compute the ECH of prequantization bundles via a nondegenerateperturbation of the contact form, as discussed in § §
7. For now we state the resultsfrom [HT13] on these cobordism maps which we will need in order to understand § §
6; amore in-depth discussion and detour through Seiberg-Witten theory is postponed to §
7. Wedo not need the following two results in their full strength, and so we paraphrase below. Inparticular, we do not need the full notion of “composition” of exact symplectic cobordisms,so we cite the composition property in the following theorem only in the case which we willneed. The fact that if ε ă ε then pr ε , ε s ˆ Y, p ` s p ˚ H q λ q (where s is the coordinate on r ε , ε s ) is an exact symplectic cobordism from p Y, λ ε q to p Y, λ ε q is addressed in the proof ofProposition 3.2 in § Theorem 2.15 ([HT13, Theorem 1.9, Remark 1.10]) . Let λ ˘ be contact forms on closed, ori-ented, connected three-manifolds Y ˘ , with p X, λ q an exact symplectic cobordism from p Y ` , λ ` q to p Y ´ , λ ´ q . There are maps of ungraded Z -modules: Φ L p X, λ q : ECH L ˚ p Y ` , λ ` q Ñ ECH L ˚ p Y ´ , λ ´ q , for each real number L such that:(Inclusion) If L ă L then the following diagram commutes: ECH L p Y, λ ` , Γ q Φ L p X,λ q (cid:47) (cid:47) ι L,L (cid:15) (cid:15) ECH L p Y, λ ´ , Γ q ι L,L (cid:15) (cid:15) ECH L p Y, λ ` , Γ q Φ L p X,λ q (cid:47) (cid:47) ECH L p Y, λ ´ , Γ q (2.16)24 Composition) Given ε ă ε ă ε , Φ L pr ε , ε s ˆ Y, p ` s p ˚ H qq “ Φ L pr ε , ε s ˆ Y, p ` s p ˚ H qq ˝ Φ L pr ε , ε s ˆ Y, p ` s p ˚ H qq . Furthermore, the maps Φ L p X, λ q respect the decomposition (2.15) in the following sense:the image of ECH ˚ p Y ` , λ ` , Γ ` q has a nonzero component in ECH ˚ p Y ´ , λ ´ , Γ ´ q only if Γ ˘ P H p Y ˘ q map to the same class in H p X q . In order to understand the impact of these cobordism maps on the chain level in certainwell-behaved cobordisms, we will also use the simplification expressed in Lemma 2.16 of twotechnical lemmas and a definition from [HT13].
Lemma 2.16 ([HT13, Lemma 3.4 (d), Lemma 5.6, and Definition 5.9]) . Given a real number L , let λ t and J t be smooth 1-parameter families of contact forms on Y and λ t -compatiblealmost complex structures such that • The contact forms λ t are of the form f t λ , where f : r , s ˆ Y Ñ R ą satisfies B f B t ă everywhere. • All Reeb orbits of each λ t of length less than L are nondegenerate, and there are noorbit sets of λ t of action exactly L (This condition is referred to in [HT13] as λ t being“ L -nondegenerate.”) • Near each Reeb orbit of length less than L the pair p λ t , J t q satisfies the conditions of[T10I, (4-1)]. (This condition is referred to in [HT13] as p λ t , J t q being “ L -flat.”) • For orbit sets of action less than L , the ECH differential B is well-defined on admissibleorbit sets of action less than L and satisfies B “ . (This is a condition on thegenericity of J t described in [HT09], and referred to in [HT13] as J t being “ ECH L -generic.”)Then pr´ , s ˆ Y, λ ´ t q is an exact symplectic cobordism from p Y, λ q to p Y, λ q , and for all Γ P H p Y q , the cobordism map Φ L pr´ , s ˆ Y, λ ´ t q agrees with the map ECC L ˚ p Y, λ , Γ; J q Ñ ECC L ˚ p Y, λ , Γ; J q , determined by the canonical bijection on generators. In this section we compute the ECH index for certain orbit sets in perturbations of pre-quantization bundles. The canonical contact form associated to a prequantization bundleis degenerate, so it is not possible to compute its ECH directly. Instead we introduce theperturbation λ ε “ p ` ε p ˚ H q λ, (3.1)where H is a perfect Morse function on the base Σ g . As explained in § Lemma 3.1.
Fix a Morse function H such that | H | C ă . i) For each L ą , there exists ε p L q ą such that for all ε ă ε p L q , all Reeb orbits with A p γ q ă L are nondegenerate and project to critical points of H , where A is computedusing λ ε .(ii) The action of a Reeb orbit γ kp of R ε over a critical point p of H is proportional to thelength of the fiber, namely A p γ kp q “ ż γ kp λ ε “ kπ p ` ε p ˚ H q . By Lemma 3.1 (i), it is possible to choose ε p L q based on L so that the embedded orbitscontributing to the generators of ECC L ˚ p Y, λ ε p L q , Γ; J q consist only of fibers above criticalpoints of H . In the proof of Lemma 3.1 in § ε p L q „ L . Therefore, by Lemma3.1 (ii), we can choose L large enough so that the generators of ECC L ˚ p Y, λ ε p L q q , Γ; J q includeany given orbit set whose embedded Reeb orbits are fibers above critical points.To capture all these filtered complexes, we prove the following result in § Proposition 3.2.
With
Y, λ , and ε p L q as discussed above, for any Γ , there is a directsystem consisting of all the ECH L ˚ p Y, λ ε p L q , Γ q . The direct limit lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q is the homology of the chain complex generated by orbit sets tp α i , m i qu where the α i are fibersabove critical points of H and ř i m i r α i s “ Γ . Proposition 3.2 will allow us to prove Theorem 1.1 in §
7, where we will relate the directlimit to the ECH of the original prequantization bundle. In light of Proposition 3.2, in § § Lemma 3.3.
The fibers above the index zero and two critical points of H are embeddedelliptic orbits, while the fibers above the index one critical points of H are embedded positivehyperbolic orbits. Remark 3.4 (Notation) . For technical reasons, we need to assume H is perfect. We denotethe corresponding embedded Reeb orbits by e ´ , e ` , and h i , respectively, and throughout therest of the paper consider generators of the form e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` where m ˘ , m i P Z ě tp e ´ , m ´ q , p h , m q , . . . , p h g , m g q , p e ` , m ` qu When specifying a particular orbit set with multiplicative notation, we will follow the con-vention that m ˘ , m i ą
0, omitting the term γ m if m “
0. When using multiplicative notationto denote an unspecified or general orbit set, however, we will allow m ˘ , m i “
0, and it willcorrespond to the orbit set in the usual notation with the pair p e ˘ , m ˘ q or p h i , m i q removed.Given orbit sets α “ e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` and β “ e n ´ ´ h n ¨ ¨ ¨ h n g g e n ` ` , let d “ M ´ N ´ e , where M : “ m ´ ` m ` ¨ ¨ ¨ ` m g ` m ` and similarly N : “ n ´ ` n ` ¨ ¨ ¨ ` n g ` n ` . (Notethat d corresponds to the degree of any curves counted in xB α, β y , as proved in § roposition 3.5. Let p Y, λ q be a prequantization bundle over a surface Σ g with Euler class e P Z ă . The ECH index in ECH L ˚ p Y, λ ε p L q , Γ q satisfies the following formula for any Γ , L ,and orbit sets α and β . I p α, β q “ χ p Σ g q d ´ d e ` dN ` m ` ´ m ´ ´ n ` ` n ´ . (3.2)In particular, note that the ECH index I p α, β q depends only on the generators α and β and not on a relative homology class Z P H p Y, α, β q .We will obtain Proposition 3.5 in § § § § Let p Y, λ q be the prequantization bundle over p Σ g , ω q with Euler class e “ ´ π r ω s ă ξ “ ker p λ q . Recall that the Reeb orbits of λ consist of the S fibers of p : Y Ñ Σ g , all of which have action 2 π . We take: λ ε “ p ` ε p ˚ H q λ. (3.3)A standard computation, cf. [Ne20, Prop. 4.10], yields the following. Lemma 3.6.
The Reeb vector field of λ ε is given by R ε “ R ` ε p ˚ H ` ε r X H p ` ε p ˚ H q , (3.4) where X H is the Hamiltonian vector field on Σ and r X H is its horizontal lift. We now prove Lemma 3.1.
Proof of Lemma 3.1.
To prove (i), note that the horizontal lift r X H is determined by dh p q q r X H p q q “ X εH p h p q qq and λ p r X H q “ . Thus those orbits which do not project to p P Crit p H q must project to X H . We have ε p ` ε q ă ε p ` ε p ˚ H q ă ε p ´ ε q A Taylor series expansion shows that the k -periodic orbits of X H give rise to orbits of ε r X H p ` ε p ˚ H q which are Cε -periodic for some C . We note that C and k must be bounded awayfrom 0 since X H is time autonomous. Nondegeneracy of Reeb orbits γ such that A p γ q ă L follows from the proof of Theorem 13 in Appendix A of [ABW10].We obtain (ii) because the period of an orbit of R ε over a critical point p of H must be1 ` εH p p q times the period of the orbit of R over p by (3.4). We use the convention ω p X H , ¨q “ dH. .2 Homology of prequantization bundles In this subsection we review preliminaries on the homology of prequantization bundles. Let Y be a prequantization bundle over a two-dimensional surface Σ g of negative Euler class e .The second page of its Leray-Serre spectral sequence has terms E p,q “ H p p Σ g ; t H q p Y x quq “ H p p Σ g ; Z q for q “ ,
1. Since B : E p,q Ñ E p ´ ,q ` , the only differential on the second page whichneither starts nor ends at a trivial group is from E , “ H p Σ g ; Z q to E , “ H p Σ g ; Z q ; thisdifferential sends the element of E , corresponding to a closed 2-cell in Σ to the obstructionto finding a section over Σ g , and so the image of B in E , is e Z . The other groups areunchanged.Since all higher differentials will either start or end at a trivial group, we obtain H ˚ p Y ; Z q “ $’’’&’’’% Z ˚ “ Z g ˚ “ Z g ‘ Z ´ e ˚ “ Z ˚ “ Lemma 3.7.
Let Y be a prequantization bundle over a two-dimensional surface Σ g of neg-ative Euler class e . (i) Let f p denote the fiber over p P Σ g . Its k -fold cover represents the class k mod p´ e q inthe Z ´ e summand of H p Y q . (ii) Each H p Y q class is represented by the union of fibers over a representative of an H p Σ g q class.Proof. Because Σ g is a CW complex, the Leray-Serre spectral sequence can be constructedusing the filtration on C ˚ p Y q where F p p C ˚ p Y qq is the subcomplex consisting of singular chainssupported in the preimage under p of the p -skeleton of Σ g .To show (i), we first show that E , is generated by the fiber. Because E , is generatedby the fiber and E , is generated by 2-chains of Y over the 1-skeleton of Σ g , the image of E , under B is zero. Therefore E , is also generated by the fiber.Secondly, we show that E , is generated by a section of Y over Σ g ´ t pt u . It follows fromthe definitions that E , is generated by a section of Y over Σ g ´ t pt u , and every such sectionis in the kernel of B because its boundary is a 1-chain in π ´ pt pt uq . Therefore E , “ E , .The differential B takes the generator of E , to f ep , so (i) is proved.To show (ii), note that E , consists of 2-chains over the 1-skeleton of Σ g whose boundariesdo not wrap around fibers and which are not the boundary of a 3-chain in the preimage under p of the 1-skeleton of Σ g . Therefore elements of E , can be represented by preimages under p of representatives of H p Σ g q classes.In an abuse of notation, we will often refer to elements of the subgroup t u ˆ Z ´ e of H p Y q simply as elements of Z ´ e . 28 .3 Trivialization and Conley Zehnder index We will use the constant trivialization as considered in [ ? , § q P p ´ p p q , a fixed trivialization of T p Σ g allows us to trivialize ξ q because ξ q – T p Σ. This trivialization is invariant under the linearized Reeb flow and canbe thought of as a constant trivialization over the orbit γ p because the linearized Reeb flow,with respect to this trivialization, is the identity map.Using this constant trivialization, we have the following result regarding the Robbin-Salamon index, see [vKnotes, Lem. 3.3], [Ne20, Lem. 4.8]. Lemma 3.8.
Let p Y, λ q p Ñ p Σ g , ω q be a prequantization bundle of negative Euler class e .Then for the constant trivialization τ along the circle fiber γ “ π ´ p p q , we obtain RS τ p γ q “ and RS τ p γ k q “ , where RS denotes the Robbin-Salamon index of the k -fold iterate of thefiber. We also have the following formula for the Conley-Zehnder indices of iterates of orbitswhich project to critical points p of H . We denote the k -fold iterate of an orbit whichprojects to p P Crit p H q by γ kp . Lemma 3.9.
Fix L ą and H a Morse-Smale function on Σ which is C close to 1.Then there exists ε ą such that all periodic orbits γ of R ε with action A p γ q ă L arenondegenerate and project to critical points of H . The Conley-Zehnder index such a Reeborbit over p P Crit p H q is given by CZ τ p γ kp q “ RS τ p γ k q ´ ` index p H, “ index p H ´ . Detailed definitions and computations of the Conley-Zehnder and Robbin-Salamon indexas well as the proofs of the preceding standard computations can be found in [Ne20, § We explain why generators of the form e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` are all we need to consider until §
7. Our focus through § L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ; J q . In Theorem 7.1 we will show thatlim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q “ ECH ˚ p Y, ξ, Γ q We will then prove the main theorem in § L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q ,as we understand it via Proposition 3.2, to the Morse homology of the base, which will re-quire the analysis of § §
6. This section will be devoted to understanding the ECH index ofthe generators which Proposition 3.2 tells us are relevant, i.e., those whose embedded orbitsare fibers above critical points of H . 29 emark 3.10. We require that H be perfect, so that H has exactly as many critical pointsof index i as its i th Betti number. Let e ` denote the orbit whose image is the fiber above theindex two critical point of H . Similarly, let e ´ denote the orbit above the index zero criticalpoint and let h , . . . , h g denote the orbits above the index one critical points.The notation is derived from the fact that the orbits e ˘ are elliptic, with slightly posi-tive/negative rotation numbers, respectively, in the constant trivialization discussed in § h i are positive hyperbolic. This is the content of Lemma 3.3. Heuristically, it is truebecause the linearized return map of an orbit projecting to a critical point p of H approxi-mately agrees with a lift of the linearized flow of εX H on T p Σ g . However, we prove Lemma3.3 using the Conley Zehnder index instead.We next prove Proposition 3.2. Proof of Proposition 3.2.
Given ε ą ε there is an exact symplectic cobordism p X ε,ε , λ ε,ε q : “pr ε , ε s ˆ Y, p ` s p ˚ H q λ q from p Y, λ ε q to p Y, λ ε q . (It is symplectic because dλ ε,ε is a positivemultiple of ds ^ λ ^ dλ .)Thus we have cobordism maps Φ L p X ε,ε , λ ε,ε q as in Theorem 2.15, inclusion maps ι L,L as in [HT13, Thm. 1.3], and a commutative diagram ECH L ˚ p Y, λ ε , Γ q Φ L p X ε,ε ,λ ε,ε q (cid:47) (cid:47) ι L,L (cid:15) (cid:15) ECH L ˚ p Y, λ ε , Γ q ι L,L (cid:15) (cid:15) ECH L ˚ p Y, λ ε , Γ q Φ L p X ε,ε ,λ ε,ε q (cid:47) (cid:47) ECH L ˚ p Y, λ ε , Γ q (3.6)by adapting (2.16) from the Inclusion property of Theorem 2.15. (Because X is a productof Y with an interval, the cobordism maps respect the splitting.)Because if L ă L then ε p L q ą ε p L q , from either path on the commutative diagram (3.6)we get a well-defined map ECH L ˚ p Y, λ ε p L q , Γ q Ñ ECH L ˚ p Y, λ ε p L q , Γ q . (3.7)For the ECH L ˚ p Y, λ ε p L q , Γ q to form a direct system, it remains to show that the maps (3.7)compose. In the following denote by Φ L p ε, ε q the cobordism map Φ L p X ε,ε , λ ε,ε q . It is enoughto show that for L ą L ą L and ε ă ε ă ε , the composition ECH L ˚ p Y, λ ε , Γ q ι L,L Ñ ECH L p Y, λ ε , Γ q Φ L p ε ,ε q˝ Φ L p ε,ε q ÝÑ ECH L p Y, λ ε , Γ q ι L ,L Ñ ECH L p Y, λ ε , Γ q equals either Φ L p ε, ε q˝ ι L,L or ι L,L ˝ Φ L p ε, ε q . This follows from the four-fold commutativediagram consisting of the versions of (3.6) for p L, ε q to p L , ε q , p L, ε q to p L , ε q , p L , ε q to p L , ε q , and p L , ε q to p L , ε q in concert. In this four-fold commutative diagram, the pathacross the top and down the right side equals Φ L p ε, ε q ˝ ι L,L , by the Composition propertyof Theorem 2.15, and similarly the path down the left side and across the bottom equals ι L,L ˝ Φ L p ε, ε q .It remains to show that the direct limit is the homology of the chain complex generatedby orbit sets whose embedded orbits are fibers above the critical points of H and whosemultiplicities can be any element of Z ą . 30he embedded orbits contributing to the generators of any ECC L ˚ p Y, λ ε p L q , Γ; J q mustbe orbits over critical points of H by Lemma 3.1 (i). And by Lemma 3.1 (ii), for any pair p γ, m γ q where γ is an orbit above a critical point of H and m γ P Z ą , there is some (possiblyvery large) L for which m γ A p γ q ă L when A is computed using λ ε p L q .To complete the proof we need to know that the maps (3.7) are induced by the obviousinclusion of chain complexes ECC L ˚ p Y, λ ε p L q , Γ; J q Ñ ECC L ˚ p Y, λ ε p L q , Γ; J q . Because ι L,L is induced by inclusion, it suffices to show that the map Φ L p X, λ ε,ε q is alsoinduced by inclusion. (There is no need to check the cobordism map Φ L p X, λ ε,ε q becausethe diagram commutes.)That Φ L p X, λ ε,ε q is induced by inclusion follows if there is a smooth 1-parameter family λ t where λ t “ λ ε ´p ε ´ ε q t which, for λ t -compatible almost complex structures J t , the pairs p λ t , J t q satisfy the hypotheses of Lemma 2.16 for L . The first and second bullet points followimmediately from the construction. The fourth bullet point generically holds. The thirdbullet point can then be accomplished by a deformation as in [T10I, Prop. B.1] (see also[HT13, Lem. 3.6]).We prove Lemma 3.3, which classifies the orbits we consider. Proof of Lemma 3.3.
By Lemma 3.9, we have CZ τ p γ kp q “ index p H ´ , (3.8)because in our case n “ RS τ p γ k q “ λ ε is a small perturbation of λ , all linearized return maps of Reeb orbits mustbe close to the identity. Therefore there can be no negative hyperbolic orbits.Positive hyperbolic orbits have even Conley Zehnder indices, so the e ˘ , with ConleyZehnder indices ˘ h i , which all have Conley Zehnderindex zero by (3.8), must be positive hyperbolic. In this subsection we prove Proposition 3.5. We note that the structure of the proof follows[Fa, § Y be a prequantization bundle over a surface Σ g of negative Euler class e , andlet L P R be large. Let Γ be a torsion element in the t u ˆ Z ´ e subgroup of H p Y q . In thissubsection we prove Proposition 3.5. We first introduce some notation which we will usethroughout the rest of this section in our computation of the ECH index.Given generators α “ e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` and β “ e n ´ ´ h n ¨ ¨ ¨ h n g g e n ` ` of ECC L ˚ p Y, λ ε p L q , Γ q ,let M : “ m ´ ` m ` ¨ ¨ ¨ ` m g ` m ` and N : “ n ´ ` n ` ¨ ¨ ¨ ` n g ` n ` . Because r α s “ r β s “ Γ, there is some d P Z so that M “ N ` p´ e q d. (3.9)Throughout the proof of the index formula, which occupies the rest of this section, we willassume d ě
0; the d ď emma 3.11. Given α and β as above and Z P H p Y, α, β q , the ECH index I p α, β, Z q doesnot depend on Z .Proof. Let A P H p Y q and Z P H p Y, α, β q . From the index ambiguity formula, Theorem2.9 (ii), we have I p α, β, Z ` A q ´ I p α, β, Z q “ x c p ξ q ` P D p Γ q , A y “ c p ξ qp A q ` ¨ A. Assume g ą
0. Recall from Lemma 3.7 (ii) that H p Y q “ Z g , and if a , b , . . . , a g , b g generate H p Σ q , then the unions of fibers over representatives of a and b will generate H p Y q .The class Γ can be represented by a single fiber, so a representative of Γ can be isotoped notto intersect a representative of A . Thus Γ ¨ A “
0. Moreover, we have c p ξ qp A q “
0, via c p ξ qp A q “ c p T Σ g qp p ˚ p A qq “ , because p ˚ will send a representative of A to a representative of a 1-cycle in Σ.If g “
0, then we have H p Y q “
0, and because H p Y, α, β q is affine over H p Y q , there isno possibility for index ambiguity.Therefore I p α, β, Z q is independent of Z , and will from now on be denoted I p α, β q .Similarly, we will use c τ p α, β q and Q τ p α, β q to denote c τ p Z q and Q τ p Z q .We will now compute the relative first Chern class and relative self intersection termsin the ECH index. Lemmas 3.9 and 3.3 allow us to compute the Conley Zehnder indexterm in the final proof of Proposition 3.5. Throughout the computation we use the constanttrivialization τ from § Y Let α and β be homologous orbit sets, thus satisfying (3.9). Before computing c τ and Q τ ,we will define surfaces in Y representing r α s “ r β s to be used in both the computation of therelative first Chern class in § § α “ tp α k , m k qu and β “ tp β l , n l qu ; in particular,the m k and n l are not necessarily equal to the multiplicities of hyperbolic orbits.Let p p α q and p p β q denote the sets of points t p p α k qu and t p p β l qu , respectively, where thepoint p p α k q appears with multiplicity m k and p p β l q appears with multiplicity n l . Chooseany subset of p p α q of total multiplicity N and denote it p p α q β ; such a subset exists becausewe are assuming d ě p p α k q in p p α q β does not haveto equal m k , though it is at most m k . Denote the set of points in Σ g underlying p p α q β by t p p α q β u .First we explain how to obtain a surface in Y connecting a set of orbits from α of totalmultiplicity N with β . Choose a graph G N embedded in Σ with vertices t p p α q β u Y t p p β qu ,where the degree of each vertex equals its multiplicity as part of p p α q β or p p β q . Furthermore,we require that the edges of G N partition t p p α q β u Y t p p β qu into p p α q β and p p β q in the sensethat each edge of G N can be labeled with a pair in p p α q β ˆ p p β q where the edge of G N connects the underlying pair in Σ g , and all points in p p α q β Y p p β q are connected in thisway. Finally, we require that the edges of G N intersect only transversely, including at theirendpoints (meaning that if x is an endpoint with degree at least two, the one-sided limits of32he tangent vectors to those edges form a basis for T x Σ g ). In particular, if x P p p α q β X p p β q then the graph can include transversely intersecting loops from x to x . Let ˇ S N denote theunion of the fibers above G N .Now we explain how to obtain a surface in Y with boundary homologous to the remaining p´ e q d orbits in α (counted with multiplicity). Denote p p α q ´ p p α q β by p p α q α . Divide thepoints in p p α q α into d subsets, each of total multiplicity ´ e . Denote each subset by p p α q kα ,for k “ , . . . , d , and denote the underlying set of points by t p p α q kα u . Let mult k p x q denote themultiplicity of x as an element of p p α q kα . For each k choose a section ˇ S k of Y over Σ ´ t p p α q kα u whose boundary forms a mult k p x q -fold cover of the fiber over x , for each x P t p p α q kα u .For z R t p p α q kα u , denote the point of ˇ S k above z by ˇ S k p z q . We compute the relative first Chern class c τ p α, β q of orbit sets α and β . Lemma 3.12.
Given orbit sets α and β satisfying (3.9), their relative first Chern class c τ p α, β q satisfies the following formula: c τ p α, β q “ χ p Σ q d. (3.10) Proof.
We will use the surfaces ˇ S N and ˇ S k of § c τ that if S and S are two admissible surfaces, then c τ p S Y S q “ c τ p S q Y c τ p S q . Therefore, we have c τ p α, β q “ c τ p ˇ S N Y ˇ S ¨ ¨ ¨ Y ˇ S d q “ c τ p ˇ S N q ` d ÿ k “ c τ p ˇ S k q (3.11)Since ξ “ p ˚ T Σ, the first Chern class of ξ is p ˚ c p T Σ q . Since p p ˇ S N q “ G N representszero in H p Σ; Z q , c τ p ˇ S N q “ . (3.12)Since r p p ˇ S k qs “ r Σ s in H p Σ; Z q , c τ p ˇ S k q “ χ p Σ q . (3.13)Combining equations (3.11), (3.12), and (3.13) yields the desired result. We compute the relative self intersection number Q τ p α, β q of orbit sets α and β . Lemma 3.13.
Given orbit sets α and β satisfying (3.9), their relative self-intersection num-ber Q τ p α, β q satisfies the following formula: Q τ p α, β q “ ´ ed ` dN. (3.14)33 roof. We will first lift the surfaces ˇ S N and ˇ S k in Y from § r´ , s ˆ Y to use in our computation. To lift ˇ S N to an admissible surface S N Ă r´ , s ˆ Y ,parameterize the edges of G N by r´ , s from p p β q to p p α q β so that they do not intersect asparameterized curves. Denote these parameterizations by g i . The non-intersecting require-ment means that if g , g parameterize two edges of G N which intersect in Σ, then we have g p t q “ g p t q only if t ‰ t . Let S N be the surface N ď i “ p t, p ´ p g i p t qqq . To construct an admissible surface with boundary on the remaining p´ e q d componentsof α , we will define a family of lifts for each ˇ S k to an admissible surface S k Ă r´ , s ˆ Y .The lifts are isotopic, so the relative self-intersection number will not depend on our choicewithin the family. We will need this flexibility in order to guarantee transverse intersections.Choose a disc neighborhood D x for each x P t p p α q α u which do not pairwise intersect,and parameterize each D x with radial function 0 ď r x ď
2. For any choice of functions (cid:15), l : t , . . . , d u Ñ r , q with 0 ď (cid:15) p k q ă l p k q ă δ : t , . . . , d u Ñ R ą , let f k : Σ g Ñ R be a smooth function for which f k p z q “ $’&’% δ p k q r x ď r x ď (cid:15) p k q δ p k q l p k q l p k q ď r x ď l p k q outside Ť x P p p α q kα D x and f k ą (cid:15) p k q δ p k q ď r x ă l p k q For each k “ , . . . , d , we define S k Ă r´ , s ˆ Y to be the surface S k : “ p ´ f k p z q , ˇ S k p z qq . Heuristically, S k lifts to r´ , s near 1 by the negative of each radial direction r x times δ p k q ,until the r´ , s coordinate reaches (cid:15) p k q . After some smooth interpolation depending withinthe D x discs, the rest of S k simply equals t ´ l p k qu ˆ ´ ˇ S k ´ Ť x P p p α q kα D x ¯ .Expanding [Hu09, (3.11)], we have Q τ p S N Y S Y ¨ ¨ ¨ Y S d q “ Q τ p S N q ` d ÿ k “ Q τ p S N , S k q ` Q τ p S Y ¨ ¨ ¨ Y S d q . (3.15)We will compute each term separately.First, we have Q τ p S N q “ , (3.16)because the graph G N has self-intersection zero as a parameterized graph. That is, anyintersections between the edges of G N , including self-intersections, can occur away from p p α q β and p p β q , and the parameterizations can be adjusted so as to avoid intersection in r´ , s ˆ Y . In particular, the self-linking of the braids S N X t ´ (cid:15) u ˆ Y is zero because G N can be isotoped so that its edges do not intersect near p p α q β and p p β q , even as non-parameterized curves. Alternately, one could show that S N is a “ τ -representative” of r p Y p S N qs , following [Hu02b], an alternateconstruction of Q τ for which there is no need to consider boundary self-linking. Q τ p S N , S k q . We can choose the parameterizations g p t q of the edgesof G N so that when t “ ´ l p k q , the point g p t q is outside all disks D x , the derivative g p ´ l p k qq ‰
0, and if g p t q has an end at x , that when t ě ´ (cid:15) p x q δ p x q , the parameterization g p t q ” x .The points p ´ l p k q , ˇ S k p z qq P p ´ l p k q , p ´ p g p ´ l p k qqqq will then be the only points ofintersection between ˇ S k and the edges of G N . Each contributes to the count of intersectionswith sign ` tB s , R, B , B u for R s ˆ Y , where tB , B u is an oriented basis for Σ and B equals the tangent vector to the edge in question (i.e., B “ g p ´ l p k qq ), an oriented basis for T S N ‘ T S k at their point of intersection is tp , , , q , p , , , q , p , , , q , p , , , qu . Therefore Q τ p S N , S k q “ N. (3.17)Finally we consider the self-intersection of the union of S k s. We have Q τ p S Y ¨ ¨ ¨ Y S d q “ d ÿ k “ Q τ p S k q ` ÿ k ‰ k Q τ p S k , S k q We will show that Q τ p S k , S k q does not depend on the k i (even if k “ k ). Therefore,because d ` ˆ d ˙ “ d ` d !2 p d ´ q ! “ d ` d p d ´ q “ d , we will get Q τ p S Y ¨ ¨ ¨ Y S d q “ d Q τ p S q . To compute Q τ p S k , S k q for any k , k , let δ i , (cid:15) i , l i denote δ p k i q , etc. Choose δ ą δ and (cid:15) δ ą l .Because (cid:15) δ ą l , all intersections between S k and S k must occur at points whose pro-jection to Σ lies within the disk neighborhoods of p p α q k α .Assume x P t p p α q k α u but x R t p p α q k α u . In the local product coordinates B s , R, B r , B θ determined by the section ˇ S k , the intersection S k X pt ´ l u ˆ Y q consists of a p , mult k p x qq torus knot about the fiber over x in the s “ ´ l level of r´ , s with r x “ l δ , and S k consists of the zero section of p , so is parameterized by p ´ l , , r, θ q . In particular, by T , mult k p x q we are referring to B θ as the meridional coordinate and R as the longitudinalcoordinate.Since in oriented bases for T S k and T S k near any intersection in the s “ ´ l sliceonly the first basis vector for T S k will have any B s component, and it will be positive, theintersection number in r´ , s ˆ Y will agree with the intersection number of the projectionsto Y in Y . These projections will consist of the T , mult k p x q torus knot in the r “ l δ torusand the disk obtained by projecting off the Reeb direction.Similarly, only the first basis vector for T S k will have any B r component, and it willbe positive. Therefore the intersection number of the projections to Y will agree with the35ntersection number in the r x “ l δ torus of the T , mult k p x q torus knot and the meridian,parameterized by θ . Their intersection number can easily be computed via a matrix: (cid:12)(cid:12)(cid:12)(cid:12) mult k p x q
01 1 (cid:12)(cid:12)(cid:12)(cid:12) “ mult k p x q . (3.18)Now assume x P t p p α q k α u X t p p α q k α u . Let 0 ă (cid:15) ă (cid:15) . Using local coordinates s , theReeb direction coming from p ´ p x q , and polar coordinates r x , θ on the base, the intersections S k i X pt ´ (cid:15) u ˆ Y q consist of T , mult ki p x q torus knots in the tori r x “ (cid:15)δ i , respectively. Because δ ą δ , the T , mult k p x q torus knot lies on a torus nested “inside” the torus of the T , mult k p x q torus knot, where “inside” refers to the component of Y ´ T containing p ´ p x q . From knotdiagrams of the image in R under the diffeomorphism of § (cid:96) τ p π Y p S k X pt ´ (cid:15) u ˆ Y qq , π Y p S k X pt ´ (cid:15) u ˆ Y qqq “ mult k p x q . (3.19)Equations (3.18) and (3.19) show that each x P t p p α q k α u contributes to Q τ p S k , S k q according to its multiplicity. Since there are no other intersections or boundary components,we obtain Q τ p S k , S k q “ ´ e. (3.20)Combining equations (3.15), (3.16), (3.17), and (3.20) yields the desired result. Proof of Proposition 3.5.
Combining Lemmas 3.3, 3.9 tells us that CZ Iτ p α q ´ CZ Iτ p β q “ m ` ´ m ´ ´ p n ` ´ n ´ q . Adding (3.10) and (3.14) proves the result.Checking that our formula satisfies the additivity property of Theorem 2.9 (iii) is straight-forward. Checking that our formula satisfies the index parity property of Theorem 2.9 (iv)requires relating the sums m ` ¨ ¨ ¨ ` m g and n ` ¨ ¨ ¨ ` n g to m ˘ and n ˘ via the formula M “ N ` p´ e q d defining d . J In this section we work towards proving that B L only counts cylinders which are the unionof fibers over Morse flow lines in Σ. One can count cylinders with a generic fiberwise S -invariant almost complex structure J : “ p ˚ j Σ g , the S -invariant lift of j Σ g , but unfortunatelywe cannot use J for higher genus curves because it cannot be independently perturbed atthe intersection points of π Y C with a given S -orbit by an S -invariant perturbation; see § J P J p Y, λ q for moduli spaces of nonzerogenus curves, but we cannot assume that this J is S -invariant.To resolve this issue, we employ Farris’ strategy [Fa, §
6] of using a family of S -invariantdomain dependent almost complex structures, J , for higher genus curves, which was modeled36n Cieliebak and Mohnke’s approach for genus zero pseudoholomorphic curves in [CM07].To an ( S -invariant) domain dependent almost complex structure J P J S ˙Σ and a map C : p ˙Σ , j q Ñ p R ˆ Y, J q where g p ˙Σ q ą
0, we associate the p , q -form B j, J : “ p dC ` J p z, C q ˝ dC ˝ j q , which at the point z P ˙Σ is given by B j, J C p z q : “ p dC p z q ` J p z, C p z qq ˝ dC p z q ˝ j p z qq . We say that C is J -holomorphic whenever B j, J C “
0. There are two new phenomena to beaccounted for in the case of higher genus J -holomorphic curves. The first is that higher genusRiemann surfaces have have finite nontrivial symmetry groups, so the moduli space M g,n is an orbifold, and therefore the moduli spaces of J -holomorphic curves are also orbifolds.The second is that, even when using domain dependent almost complex structures, a nodalcurve with a constant component of positive genus cannot be perturbed away to achievetransversality. However in dimension 4, we will show how Farris’s index considerationsobstruct the latter configurations from arising.Our scheme for obtaining regularity will be that if z, z P ˙Σ map under C to the same S orbit in Y , then we will perturb J independently at z and z while preserving J ’s S -invariance. We will exploit this construction to prove the existence of regular S -invariantdomain dependent almost complex structures in § S -invariant domain dependent almost complex structure J , the modulispaces of curves of nonzero genus with ECH index 1 are empty. In § S -invariant) J and a generic λ -compatible almost complexstructure J , permitting us to conclude that the only contributions to an appropriately filteredECH differential are from cylinders which project to Morse flow lines. We first review the notion of a degree of a completed projection for a pseudoholomorphiccurve in the symplectization of a prequantization bundle.
Definition 4.1 (Degree of a completed projection) . If we compose the J -holomorphic curve C : ˙Σ Ñ R ˆ Y, with the projection p : Y Ñ Σ g , then we obtain a map p ˝ π Y C : ˙Σ Ñ Σ g , which has a well-defined non-negative degree because p ˝ π Y C extends to a map of closedsurfaces. We define the degree of C , denoted deg p C q , to be the degree of this map.37 emark 4.2. One should not confuse degree with multiplicity . Recall that in Definition2.1 that the multiplicity of a pseudoholomorphic curve C is given by degree of the holomor-phic branched covering map between the domain of C and the domain of the underlyingsomewhere injective curve. The multiplicity of a somewhere injective curve is always 1.We can relate the degree of the completed projection map p ˝ π Y C to the number ofpositive and negative ends via the dλ ε -energy and Stokes’ Theorem as follows. First we notethe following. Remark 4.3.
The action of a Reeb orbit γ kp of R ε over a critical point p of H is proportionalto the length of the fiber, namely A p γ kp q “ ż γ kp λ ε “ kπ p ` ε p ˚ H q , because p ˚ H is constant on critical points p of H . Proposition 4.4.
For all λ ε , we have the following relation between the degree deg p C q of acurve C P M J p α, β q and the total multiplicity of the Reeb orbits at the positive and negativeends: M ´ N “ | e | deg p C q Proof.
Note that equality of the total homology classes of α and β forces M ´ N “ | e | Denote by H ˘ the values of H at p p e ˘ q , respectively, and denote by H i the value of H at p p h i q .Recall that the dλ ε -energy (equivalently, contact area) A p C q of a J -holomorphic curve C is given by A p C q : “ ż ˙Σ p π Y C q ˚ p dλ ε q . Stokes’ Theorem yields A p C q “ ż ˙Σ p π Y C q ˚ p dλ ε q“ ż Bp π Y C ˚ r ˙Σ sq λ ε “ π p M ´ N ` ε pp m ´ ´ n ´ q H ´ ` p m ´ n q H ` ¨ ¨ ¨ ` p m g ´ n g q H g ` p m ` ´ n ` q H ` qq On the other hand, we have dλ ε “ ε p ˚ dH ^ λ ` p ` ε p ˚ H q dλ where p ˚ ω “ dλ and ω r Σ g s “ π | e | . Therefore A p C q “ ε ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q ` ż ˙Σ p ` εH ˝ p ˝ π Y C qp π Y C q ˚ p p ˚ ω q“ ε ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q ` ω rp p ˝ π Y C q ˚ ˙Σ s ˜ ` ε ż Σ g H ¸ “ ε ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q ` π | e | deg p C q because ż Σ g H “ ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q “ π pp m ´ ´ n ´ q H ´ `p m ´ n q H `¨ ¨ ¨`p m g ´ n g q H g `p m ` ´ n ` q H ` q (4.1)from which we obtain the conclusion by setting the two values computed for A p C q equal toone another.(4.1) also follows from Stokes’ theorem. Again because ş Σ g H “
0, we have ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q “ ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q ` ω rp p ˝ π Y C q ˚ ˙Σ s ż Σ g H “ ż ˙Σ p π Y C q ˚ p p ˚ dH ^ λ q ` ż p π Y C q ˚ r ˙Σ s p ˚ p Hω q“ ż ˙Σ p π Y C q ˚ p d p H ˝ p q ^ λ ` p H ˝ p q dλ q“ ż ˙Σ p π Y C q ˚ d pp H ˝ p q λ q“ ż Bp π Y C q ˚ r ˙Σ sq p H ˝ p q λ which equals the right hand side of (4.1).As a consequence of Proposition 4.4, the degree of any two curves in M J p α, β q must beequal, and therefore we make the following definition. Definition 4.5.
The degree of a pair of ECH generators p α, β q , denoted deg p α, β q , isdeg p α, β q : “ M ´ N | e | A curve C which contributes nontrivially to the ECH differential is degree zero if andonly if it is a cylinder, as we now explain. Lemma 4.6.
Let C P M J p α, β q be a J -holomorphic curve with domain p ˙Σ , j q and with I p C q “ . Then deg p C q “ if and only if ˙Σ the union of cylinders.Proof. If ˙Σ is a cylinder and deg p C q ą
0, then the completion of p ˝ π Y C has S as itsdomain. The Riemann-Hurwitz formula (5.3) tells us2 “ deg p C q χ p Σ g q ´ ÿ p P ˙Σ p e p p q ´ q (4.2)Since we are assuming g ě
1, the right hand side is at most zero, therefore (4.2) is acontradiction. Therefore, if ˙Σ is a cylinder, then C has degree zero.For the opposite implication, assume deg p C q “
0. Because I p C q “
1, the curve C isembedded and has ind p C q “
1, by Proposition 2.11. By the ECH index inequality, Theorem39.5, the positive and negative partitions of the ends of C must equal the positive and negativepartitions defined in Definition 2.12.By Example 2.13 and the analogous result for ϑ slightly smaller than zero, i.e. P ` ϑ p m q “ p m q P ´ ϑ p m q “ p , . . . , q and the fact that in admissible orbit sets hyperbolic orbits have multiplicity at most one, wefind that ˙Σ has exactly 1 ` m `¨ ¨ ¨` m g ` m ` positive ends and exactly n ´ ` n `¨ ¨ ¨` n g ` CZ indτ p C q “ p m ` ´ q ´ p ´ n ´ q Therefore,1 “ ind p C q“ ´ ` g p ˙Σ q ` M ´ m ´ ` ` N ´ n ` ` ` ` m ` ´ ´ ` n ´ by (3.10) “ ´ ` g p ˙Σ q ` M ´ m ´ ´ n ` ` ` m ` ` n ´ because M “ N (4.3)Note that I p C q “ M ´ N | e | “ deg p C q “
0, along with the index formula (3.2), imply1 “ I p C q “ m ` ´ m ´ ´ n ` ` n ´ (4.4)Combining (4.3) and (4.4) gives0 “ ´ ` g p ˙Σ q ` M ô “ g p ˙Σ q ` M Since M ą
0, we must have M “ N “ g p ˙Σ q “
0, therefore ˙Σ must consist of aunion of cylinders.In particular we note that fiberwise S -invariant J -holomorphic cylinders have degree 0.We have the following correspondence between J -holomorphic cylinders asymptotic to Reeborbits which project to critical points of H and downward gradient flow lines of H , whichwill be key later in relating the filtered ECH differential to the Morse differential on the base. Proposition 4.7.
For suitable orientation choices, if p and q are critical points of H , thenthere is an orientation-preserving bijection between the moduli space of J -holomorphic cylin-ders M J p γ kp , γ kq q and the moduli space M Morse p p, q q of downward gradient flow lines of H from p to q , modulo reparametrization. Furthermore, each of the holomorphic cylinders is a k -foldcover which is cut out transversely. Complete details on this correspondence are found in [Ne20, § § § § emark 4.8. If a J -holomorphic curve C : ˙Σ Ñ R ˆ Y has degree d “ , S -invariant J exists. If d ě π Y C has intersection number d with a given S -orbit, and hence has at least d intersections, which are counted with multiplicity, since C could be a nontrivial branchedcovering of its image. The complex structure J cannot be independently perturbed at these d ě S -invariant perturbation. By Lemma 4.6, a curve C which contributesnontrivially to the ECH differential is degree zero if and only if it the union of cylinders. Let J p Y, λ q denote the set of λ -compatible almost complex structures and let J S p Y, λ q Ă J p Y, λ q denote the subset of S -invariant λ -compatible almost complex structures. Since J P J S p Y, λ q is always obtained from p ˚ j Σ g , there is a correspondence between the S -invariant complex structures on ξ and the complex structures on T Σ g . Fix a “generic” J P J S p Y, λ q , let t N γ u be a disjoint union of tubular neighborhoods associated to the setof Reeb orbits t γ u , and set N “ \ γ N γ . We define J N : “ t J P J p Y, λ q | J q “ p J q q , for q P N u , (4.5)to be the subset of λ -compatible almost complex structures which agree with J on N ,and let J S N Ă J N consist of the S invariant elements of J N . The elements of J N are incorrespondence with complex structures on T Σ g which agree with a fixed J “ p ˚ j on N .We first note that we have regularity for ECH index 1 curves for generic λ -compatible J satisfying the constraint J | N “ J . Proposition 4.9.
Let α and β be nondegenerate admissible orbit sets in the same homologyclass with I p α, β q “ . If J P J N is generic then M J p α, β q is cut out transversely.Proof. Regularity follows from the subclaim in the proof of [Hu02b, Lemma 9.12]. Wehave that in the absence of trivial cylinders, there is a nonempty set U Ă C away froma neighborhood of of the periodic orbits with action ď A p α q , such that for each x P U , π ´ Y p π Y p x qq “ t x u , C is nonsingular, and that a certain projection of the derivative of B j,J p C q with respect to J is surjective on U . The proof actually shows that U is an opendense subset of C , and because the intersection with C ´ p R ˆ p Y z N qq contains a nonemptyopen set, the result holds.When ˙Σ ‰ R ˆ S , it is not possible to achieve regularity via a generic choice of S -invariant J . Instead, we will use an S -invariant domain dependent family J of λ -compatiblealmost complex structures. To define such a J , we consider a certain class of functions onthe domain ˙Σ that are independent of reparameterization, meaning that these functions areto be defined on isomorphism classes of punctured Riemann surfaces, e.g. elements of M g,n ,where we view the punctures S of ˙Σ as the n marked points and g “ g p ˙Σ q . M g,n To ensure that we have well-defined, nontrivial functions on ˙Σ (from which we will constructdomain dependent almost complex structures), we need ˙Σ to be stable , meaning that χ p ˙Σ q ă
41, e.g. 2 g ` n ě
3, where n is the number of punctures of ˙Σ. If ˙Σ is stable then M g,n is anorbifold with dim p M g,n q “ g ´ g ` n, and the associated automorphism groupAut p ˙Σ , j q : “ t ϕ P Diff ` p Σ , S q | ϕ ˚ j “ j u is finite for any j P J p Σ q . Here J p Σ q is the set of smooth complex structures on Σ that inducethe given orientation and Diff ` p Σ , S q is the group of orientation preserving diffeomorphismson Σ that fix the set of punctures S . Our class of domain dependent almost complexstructures on ˙Σ must respect the orbifold structure of M g,n , meaning they must be invariantwith respect to the finite symmetry groups of the orbifold points of M g,n . While the derivativeof such an invariant function (giving rise to J ) will have nontrivial kernel at an orbifold pointof M g,n , the set of orbifold points has positive complex codimension, meaning that there issufficient flexibility in the normal direction. Remark 4.10.
Following [Wen10, § “ R ˆ S and˙Σ “ C . We previously showed that we can use a domain independent S -invariant J tocount pseudoholomorphic cylinders, so it remains to consider when ˙Σ “ C . Since anypseudoholomorphic map C : C Ñ R ˆ Y must be asymptotic to a Reeb orbit γ , we canconsider its projected completion to the base p Σ g , ω q of the prequantization bundle C : S Ñ Σ g , which is null homotopic for g ą
0. Since C is null homotopic when g ą
0, if we consider asufficiently small perturbation, C must be close to constant, which means that C is “concen-trated” near its limiting Reeb orbit, and thus cannot bound C , because otherwise we wouldobtain a contradiction to the fact that far away fibers of p : p Y, λ q Ñ p Σ g , ω q are linked. Forthis reason, in § g is zero or positive. When g “
0, weshow that the differential vanishes for grading reasons, which permits us to use a generic λ -compatible J without appealing to the results of § J and take limits of sequences of J -holomorphiccurves in the sense of [BEHWZ], we must actually work with the Deligne-Mumford com-pactification M g,n , a compact and metrizable topological space containing M g,n as an opensubset, which consists of connected stable nodal Riemann surfaces with n marked points,(presupposing that 2 g ` n ě Definition 4.11.
Recall that an element of M g,n is stable whenever 2 g ` n ě
3. A stablenodal Riemann surface is an element p ˙Σ , j q P M g,n , which is itself a disjoint union of elements p ˙Σ i , j i q P M g i ,n i ` m i , where ˙Σ i is a stable curve whose n i ` m i marked points consist of a subset n i of the marked points of ˙Σ (hence ř n i “ n ), with the induced ordering, and m i nodes.Every node z P ˙Σ i is paired with another node z P ˙Σ i , with the stipulation that i ‰ i for atleast one of the nodes of each C i . We thus obtain a connected singular surface by gluing z to z for every pair t z, z u of nodes. 42ny sequence of curves t ˙Σ p k qu P M g,n Ă M g,n has a subsequence whose limit is a nodalcurve ˙Σ P M g,n . Furthermore, if z i p k q P ˙Σ p k q is a marked point, passing to a subsequencemeans z i p k q converges to some marked point z P ˙Σ, hence z P ˙Σ i , where ˙Σ i P M g i ,n i ` m i forsome ˙Σ i Ă ˙Σ. We recall the following result regarding the topology of the nodal limit, notingthat further details can be found in [Wen-SFT, § Lemma 4.12 (Lem. 6.1.1 [Fa]) . If ˙Σ P M g,n then for each component ˙Σ i P M g i ,n i ` m i of ˙Σ we have that g i ď g and if g i “ g then n i ` m i ă n .Proof. The nodal limit ˙Σ is obtained topologically from any smooth sequence t ˙Σ p k qu viathe following types of degenerations. The first degeneration is that (cid:96) marked points in somecomponent ˙Σ j can collide and form a bubble attached to ˙Σ j . The genus of ˙Σ j does notchange, but it loses (cid:96) marked points and gains a node where the bubble arises. Thus, thetotal number of marked and nodal points on ˙Σ j decreases by (cid:96) ´
1. The bubble itself isa genus 0 component with (cid:96) marked points and one node. If the original smooth curve˙Σ p k q had genus 0, then, then it must have had more than (cid:96) marked points. Thus each newcomponent resulting from iterative bubbling has either genus 0 or g .The second kind of degeneration comes from letting the complex structure on the curvedegenerate. Topologically, this results in a simple closed curve on some component, i.e. thevanishing cycle, being crushed to a point. If the vanishing cycle is a non-separating curve, itreduces the genus of a component by 1 without creating new components. If the vanishingcycle is a separating curve, it breaks a component into two pieces, whose genera sum to thegenus of the original component. The case where one component has genus 0 and the otherhas full genus is topologically identical to bubbling, cf. [MS J -hol].Lemma 4.12 induces an ordering on pairs p g, n q wherein p g , n q ă p g, n q whenever g ă g or g “ g and n ă n . The boundary B M g,n is a stratified space, and each stratum containinga nodal curve ˙Σ is the product over the M g ,n for each component ˙Σ i of ˙Σ with ˙Σ i P M g ,n .Moreover, we have that if M g ,n is a factor in a stratum of M g,n , then p g , n q ă p g, n q andwe can distinguish one of the components of a given ˙Σ P M g,n whenever it contains the n th marked point, which we will always denote as z . This prescribes an inductive means ofcoherently defining functions on all strata of M g,n simultaneously, though before we get intothis we need to briefly review [CM07, § In the following discussion, we consider the Deligne-Mumford space M g,n ` for 2 g ` n ě n ` z , ..., z n . We will see momentarily that the point z plays a specialrole, as it serves as the variable for holomorphic maps and is key in the proof of Theorem 4.18.First consider the case when g “
0. We call a decomposition I “ p I , ..., I (cid:96) q of t , ..., n u stable whenever I “ t u and | I | : “ (cid:96) ` ě
3. We will always order the I j such that theintegers i j : “ min t i | i P I j u satisfy 0 “ i ă i ă ... ă i (cid:96) . Denote by M I Ă M ,n ` the union over stable trees that give rise to the stable decomposition I . The M I are submanifolds of M ,n ` with M ,n ` “ Y I M I M J is a union of certain strata M I with | I | ď | J | . The above ordering ofthe I j determines a projection p I : M I Ñ M | I | , sending a stable curve to the special points on the component S α . Definition 4.13. [CM07, Definition 1.3] Let Z be a Banach space and n ě
3. We call acontinuous map J ,n ` : M g,n ` Ñ Z coherent if it satisfies the following two conditions:(a) J ,n ` ” M I with | I | “ I with | I | ě
4, there exists a smooth map J I : M , | I | Ñ Z such that J | M I “ J I ˝ p I : M I Ñ Z. More generally, let Z ˚ Ă Z be an open neighborhood of 0 and let I be a collection ofstable decompositions. Then we call a continuous map J : Y I P I M I Ñ Z ˚ coherent if itsatisfies (a) and (b) and in addition:(c) The image of J is contractible in Z ˚ .The space of coherent maps from M ,n ` to Z is equipped with the C -topology on M ,n ` and the C -topology on each M I via the projection p I .For g ą
0, we deploy Lemma 4.12, obtaining an inductive means of coherently definingfunctions on all strata of M g,n simultaneously. In particular, assume for all p g , n q ă p g, n q we have defined continuous maps J g ,n : M g ,n Ñ Z Each element of B M g,n is a nodal curve ˙Σ with n marked points . If p n lies on ˙Σ P M g ,n ,then by Lemma 4.12, p g , n q ă p g, n q , and by hypothesis there is a function J g ,n defined on M g ,n . We can define J g,n p ˙Σ q : “ J g ,n p ˙Σ q . The collection t J g ,n u p g ,n qăp g,n q thus determines t J g ,n u M g,n and we can extend J g,n to theinterior M g,n . We can continue this procedure, defining J g,N on M g,N for all N ą n , then f g ` , n for all n , and so on. This inductive procedure provides the definition for our contin-uous maps with g ą coherent .Before we can use this class of to define domain dependent almost complex structures, weneed to review a few details regarding Banach manifolds of (parametrized) almost complexstructures. For a symplectic vector space p V, ω q , denote by J p V, ω q the space of ω -tamed almost complexstructures. The space J p V, ω q is a manifold with tangent space T J J p V, ω q : “ t Y P End p V q | J Y J “ Y u . V is equipped with a Euclidean metric g then trace p Y t Y q defines a Riemannian metricon J p V, ω q . The exponential map of this metric defines embeddings from the open ball ofradius ρ p g, J q ą g and J :exp J : T J J p V, ω q Ą B p , ρ p g, J qq (cid:44) Ñ J p V, ω q . We review the construction of the
Floer C ε -space [Fl88b], which circumvents the issue thatnaturally arising spaces of smooth functions are not Banach spaces; see also [Wen-SFT,Appendix B]. For a vector bundle E Ñ X over a closed manifold X , we denote the space ofFloer’s C ε -sections in E by C ε p X, E q : “ s P Ω p X, E q ˇˇˇˇ ÿ i “ ε i || s || C i ă 8 + . Here ε “ p ε i q i P N is a fixed sequence of positive numbers and || ¨ || C i is the C i -norm withrespect to some connection on E . By [Fl88b, Lemma 5.1], if the ε i converge sufficientlyfast to zero, then C ε p X, E q is a Banach space consisting of smooth sections and containingsections with support in arbitrarily small sets in X .Next let p X, ω q be the symplectization of a closed contact manifold Y (or exact symplecticcobordism). Fix a λ -compatible almost complex structure J on p X, ω q , e.g. a smooth sectionin the bundle J p T X, ω q Ñ X with fibers J p T x X, ω x q . Let g be the canonical Riemannianmetric on X defined via ω and J . Let T J J p T X, ω q Ñ X be the vector bundle with fibers T J p x q J p T x X, ω x q and set T J J ε : “ C ε p X, T J J p T X, ω qq J ε : “ J ε p X, ω q : “ exp J p B q , where B : “ t Y P T J J ε | Y p x q P B p , ρ p g p x q , J p x qqqu . Thus J ε is the space of λ -compatiblealmost complex structures of p X, ω q that are of class C ε over J via exp J . We can regard J ε as a Banach manifold with a single chart exp J .Next we consider spaces of parametrized complex structures. A complex structure on p X, ω q parametrized by a manifold P is a smooth section in the pullback bundle J p T X, ω q Ñ P ˆ X . Fix J as above and let T J J P p T X, ω q Ñ P ˆ X be the vector bundle with fibers T J p p,q q J p T x X, ω x q and set T J J εP : “ C ε p P ˆ X, T J J P p T X, ω qq J εP : “ J εP p X, ω q : “ exp J p B P q , where B P : “ t Y P T J J εP | Y p p, x q P B p , ρ p g p x q , J p x qqqu . We may think of J P J εP as amap P Ñ J ε . For an open subset U Ă P , we denote by T J J εU Ă T J J εP the subspace of thosesection having compact support in U .We will be interested in the spaces of domain dependent almost complex structures J ε ˙Σ : “ J ε M g,n and J ˙Σ : “ J M g,n parametrized by the Deligne-Mumford space M g,n .45 efinition 4.14. A domain dependent almost complex structure J is a coherent collectionof C (cid:96) , (cid:96) ą J “ t J g,n : M g,n Ñ J N u , (recall J N was defined in (4.5)) that additionally satisfy the following condition. If given asequence t ˙Σ p k qu P M g,n converging to ˙Σ P B M g,n and ˙Σ n P M g ,n is the component of ˙Σ containing the n th marked point, thenlim k Ñ8 J g,n p ˙Σ p k qq “ J g,n p ˙Σ q “ J g,n p ˙Σ n q . We denote the set of all such C (cid:96) domain dependent almost complex structures by J (cid:96) ˙Σ . Wecall a domain dependent almost complex structure J generic if for every p g, n q , the extensionof J g,n from the boundary, where the values are determined by J g ,n , to the interior of M g,n is a generic C (cid:96) map.The extension to nodal maps follows from [CM07, § J ε ˙Σ is a Banach manifold. When it is understood that we should be using the Floer C ε -space we will drop ε from the notation, and use J ˙Σ . Remark 4.15.
Since the target of our collection of functions on ˙Σ is J N , we have that if J P J ˙Σ and if C : p ˙Σ , j q Ñ p R ˆ Y, J q is a p j, J q -holomorphic curve, then C | C ´ p R ˆ N q is p j, J q -holomorphic. Since J is domainindependent, the subset C p ˙Σ q X R ˆ N Ă C p ˙Σ q satisfies intersection positivity, which will beimportant in the proof of Proposition 5.12.Before we can conclude that this algorithm for constructing domain dependent almostcomplex structures is well-defined, it remains to discuss two technicalities, the first beingthat M g,n is an orbifold, while the other concerns the special role of the “last” marked point z . We elucidate these points in the following remarks. Remark 4.16 (Orbifold structure of M g,n ) . A neighborhood of a point in an k -dimensionalorbifold is modeled on the quotient of R k by the linear action of some finite group G , anda C (cid:96) function on an orbifold in a neighborhood modeled on R k { G is a C (cid:96) function on R k which is invariant under the group action G . For g ą
1, the locus of points on M g,n withoutautomorphisms (e.g. the action of G on R k is nontrivial) has real codimension at least two.Thus, a generic curve of genus g ą g “
0, everystable curve has a trivial automorphism group. For g “ n “
1, dim R M g, “ M g, have additionalautomorphisms, hence functions on M g, have no constraints at generic points and respectthe additional symmetries at the isolated points admitting the extra automorphisms.The derivative of a G -invariant function always has nontrivial kernel, but on any tangentspace T z M g,n ` , there is a subspace of real dimension at least two on which the derivativehas no constraints. Hence there exists a map from a neighborhood of any point in M g,n ` toa neighborhood of any point x in a manifold X with dim R X ě , sending a two dimensionalsubspace of the unconstrained subspace to any two dimensional subspace of T x X . If we fix j and the marked points z , ..., z n on M g,n ` , we may view this as a map T z ˙Σ Ñ T x X .46 emark 4.17 (The role of the special marked point z ) . Recall that there is a forgetful map π : M g,n ` Ñ M g,n which forgets the special marked point z and collapses any resulting unstable components,which are necessarily of genus zero. The fiber over ˙Σ P M g,n is itself isomorphic to ˙Σ. To seewhy this holds over the marked and nodal points, note the following. The fiber above the k th marked point z k is a single nodal curve which has a genus zero component containingthe marked points z k and z as well as a node, which is glued to z k P ˙Σ. This componentcollapses when the marked point z is removed. The fiber above a node resulting from gluing z P ˙Σ to z P ˙Σ is a single curve which has a genus zero component containing two nodesand the marked point z , attached by the first node to ˙Σ at z and to ˙Σ at z by the secondnode. This genus zero component similarly collapses when the marked point is removed.Thus, a point of M g,n ` is equivalent to a pair p ˙Σ , z q , where ˙Σ P M g,n and z P ˙Σ.If ˙Σ P M g,n , we can delete the first n marked points to obtain an n -times puncturedRiemann surface with one marked point z . Fix the n marked points corresponding to the n punctures and let J : “ t J g ,n u be a domain dependent almost complex structure. Byrestricting J g,n ` to ˙Σ – π ´ p ˙Σ q Ă M g,n ` , we obtain a family of almost complex structureson ξ parametrized by ˙Σ, which we denote by J ˙Σ . Rather than writing J ˙Σ , we will work underthe assumption that in the Cauchy-Riemann equation below, the domain of J is restrictedto π ´ p ˙Σ q , where p ˙Σ , j q P M g,n . Returning to the perspective of ˙Σ as a n -times puncturedRiemann surface, a map C : p ˙Σ , j q Ñ p R ˆ Y, J q can be associated with the p , q -form B j, J : “ p dC ` J p z, C q ˝ dC ˝ j q , which at the point z P int p ˙Σ q is given by B j, J C p z q : “ p dC p z q ` J p z, C p z qq ˝ dC p z q ˝ j p z qq . We say that C is J -holomorphic whenever B j, J C “ S -invariant domain dependent J In this section, we prove that a generic S -invariant domain dependent almost complexstructure J is regular. We note that the weaker statement that a generic J P J ˙Σ is regularfollows similarly. Theorem 4.18 (Thm. 6.2.1 [Fa]) . Let α and β be nondegenerate orbit sets with deg p α, β q ą . If J P J S ˙Σ is generic and ˙Σ does not include C or a union of cylinders, then anynonconstant holomorphic curve C P M J p α, β q is regular, meaning that the linearization D B J p C q is surjective and a neighborhood of C P M J p α, β q naturally admits the structure ofa smoooth orbifold of dimension given by the Fredholm index ind p C q , whose isotropy groupat C is Aut p C q : “ t ϕ P Aut p ˙Σ , j q | C “ C ˝ ϕ u and there is a natural isomorphism T C M J p α, β q “ ker D B J p j, C q{ aut p ˙Σ , j q . emark 4.19. At an orbifold point C P M J p α, β q , C has a nontrivial automorphism groupwith respect to which C is invariant, so C factors through the branched covering ˙Σ Ñ ˙ΣAut p ˙Σ ,j q .Additional multiple covers may arise which do not come from automorphisms of the domain,but the use of domain-dependent almost complex structures permits us to perturb away themultiple covers of the latter type by choosing different perturbations at the different pointsin C ´ p C p q qq . However, multiple covers coming from automorphisms of the domain remainbecause the functions from which we defined domain dependent almost complex structuresare invariant with respect to the orbifold symmetry groups of M g,n . Since the subset oforbifold points of M g,n has real codimension at least 2 in M g,n , hence we can conclude thatthe subset of J -holomorphic curves in the moduli space whose domains are orbifolds also hasreal codimension at least 2. Thus a generic J -holomorphic curve is not an orbifold point inits moduli space, and a generic path of J -holomorphic curves avoids the locus of orbifoldpoints.Before giving the proof, we provide the corollary, which demonstrates that positive degreecurves do not contribute to ECH index 1 moduli spaces.
Corollary 4.20 (Cor. 6.2.3 [Fa]) . Let α and β be nondegenerate admissible orbit sets and J P J S ˙Σ be generic. If deg p α, β q ą and I p α, β q “ then M J p α, β q “ H .Proof. Given a generic J P J S ˙Σ consider a J -holomorphic curve C : ˙Σ Ñ R ˆ Y withdeg p α, β q ą
0. Take J P J ˙Σ to be generic and sufficiently close to J then ind p C J q “ ind p C J q . Moroever, if we take J P J reg p Y, λ q to be sufficiently close to J then ind p C J q “ ind p C J q . By the definition of degree, the domain cannot be a union of cylinders, so S actslocally freely on M J p α, β q . Regularity of J guarantees that Fredholm index 1 curves do notexist unless they are fixed by the S -action. Since R acts freely on M J p α, β q and, becausethese actions commute, we have that dim M J p α, β q ě M J p α, β q ‰ H . SinceTheorem 4.18 guarantees that M J p α, β q is cut out transversely and has dimension equalto the Fredholm index, we have that dim M J p α, β q ě M J p α, β q ‰ H . We nowobtain a contradiction because if I p α, β q “ p C J q ď
1, by the ECH index inequality,Theorem 2.14: ind p C J q “ ind p C J q ď I p C J q “ I p α, β q “ § § kp ą
2, let B k,p,δ : “ W k,p,δ p ˙Σ , R ˆ Y ; α, β q Ă C p ˙Σ , R ˆ Y q be the usual smooth, separable, and metrizable Banach manifold of exponentially weightedSobolev spaces of maps which are asymptotically cylindrical curves to the orbit sets α and β at the ends. The tangent space to B k,p,δ at C P B k,p,δ can be written as T C B k,p,δ “ W k,p,δ p C ˚ p R ˆ Y qq ‘ V S , where V S Ă Γ p C ˚ p R ˆ Y qq is a non-canonical choice of a 2 | S | -dimensional vector space ofsmooth sections asymptotic at the punctures to constant linear combinations of the vectorfields spanning the canonical trivialization of the first factor in T p R ˆ Y q “ (cid:15) ‘ ξ . The space V S appears due to the fact that two distinct elements of B k,p,δ are generally asymptotic48o collections of trivial cylinders that differ from each other by | S | pairs of constant shifts p a, b q P R ˆ S .Fix J P J ε ˙Σ . The nonlinear Cauchy-Riemann operator is then defined as a smooth section B j, J : B k,p,δ Ñ E k ´ ,p,δ ; C ÞÑ T C ` J ˝ T C ˝ j of a Banach space bundle E k ´ ,p,δ Ñ B k,p,δ with fibers E k ´ ,p,δC “ W k ´ ,p,δ p Hom C p T ˙Σ , C ˚ p R ˆ Y qqq . The zero set of B j, J is the set of all maps C P B k,p,δ that are J -holomorphic.More generally, the universal Cauchy-Riemann operator is the section B : B k,p,δ ˆ J ε ˙Σ Ñ E k ´ ,p,δ ; p C, J q ÞÑ B j, J C of a Banach space bundle E k ´ ,p,δ Ñ B k,p,δ ˆ J ε ˙Σ with fibers E k ´ ,p,δC “ W k ´ ,p,δ p Hom C p T ˙Σ , C ˚ p R ˆ Y qqq . The zero section gives rise to the universal moduli space : M p J ε ˙Σ q : “ tp C, J q | J P J ε ˙Σ , B j, J C “ u . Arguments similar to [Wen-SFT, Lemma 7.15] demonstrate that the universal moduli space M p J ε ˙Σ q is a smooth separable Banach manifold, and the projection M p J ε ˙Σ q Ñ J ε ˙Σ ; p C, J q ÞÑ J is smooth.For any C P B ´ j, J p q , the linearization D B j, J : T J J ε ˙Σ ˆ T C B k,p,δ Ñ E k ´ ,p,δC defines a bounded linear operator D : W k,p,δ p C ˚ T p R ˆ Y qq ‘ T J J ε ˙Σ ‘ V S Ñ W k ´ ,p,δ p Hom C p T ˙Σ , C ˚ p R ˆ Y qqq Since V S is finite dimensional, D will be Fredholm if and only if its restriction to the firsttwo factors is Fredholm; denote this restriction by D : “ D C ` D J : W k,p,δ p C ˚ T p R ˆ Y qq ‘ T J J ε ˙Σ Ñ W k ´ ,p,δ p Hom C p T ˙Σ , C ˚ p R ˆ Y qqq We will view the n punctures of the domain ˙Σ of C as fixed, with j varying on int p ˙Σ q ,so that the tangent space to M g,n ` at a point p Σ , j, z , z , ..., z n q is T j J p ˙Σ q ‘ T z ˙Σ. If V “ p a, A q P T J p J S ,ε ˙Σ q , where A : T j p ˙Σ q Ñ T J J S ,ε ˙Σ and a P End j p T ˙Σ q , then D J p V q “ A ˝ du ˝ j ˙Σ ` J ˝ du ˝ a. roof of Theorem 4.18. We begin by recalling a few observations in the proof of [Fa, Theorem6.2.1]. Since deg p α, β q ą
0, the domain ˙Σ cannot solely consist of a union of cylinders. TheECH index is additive and positive for pseudoholomorphic curves which are not themselvescylinders, so there is a unique noncylindrical component ˙Σ of ˙Σ. Trivial cylinders arealways cut out transversely [Wen-SFT, Proposition 8.2], as are somewhere injective cylinders[Wen-SFT, § “ ˙Σ and ˙Σ is stable. Since C is not a trivial cylinder, C ´ p R ˆ p Y z N qq contains a nonempty open set of ˙Σ.Next we show that C cannot be a nodal curve with a constant component of positivegenus, which crucially relies on dim p R ˆ Y q “
4, noting this is in part why [CM07] restrictsto genus 0 curves. Suppose to the contrary that C is the union of a nodal curve C with aconstant component of positive genus C and that ind p C q “
1, thenind p C q ` ind p C q “ . Since C | ˙Σ is constant, the restricted pullback C ˚ p T p R ˆ Y q| ˙Σ is trivial, thus c p C ˚ p T p R ˆ Y qq “ c p C ˚ p T p R ˆ Y qq| ˙Σ ` c p C ˚ p T p R ˆ Y qq| ˙Σ “ c p C ˚ p T p R ˆ Y qq| ˙Σ . Let c | ˙Σ “ c p C ˚ p T p R ˆ Y qq| ˙Σ . Because C maps ˙Σ to a constant, all the punctures must lie on ˙Σ . Thus the Fredholmindex contribution of the Conley-Zehnder indices of the orbits asymptotic to the ends of ˙Σand ˙Σ must agree. Denote this contribution by CZ indτ p C q . By hypothesis, we have that g p ˙Σ q ą g p ˙Σ q ă g p ˙Σ q , and χ p ˙Σ q ą χ p ˙Σ q . Thus1 “ ind p C q “ ´ χ p C q ` c | ˙Σ ` CZ indτ p C q , ind p C q “ ´ χ p C q ` c | ˙Σ ` CZ indτ p C q , hence ind p C q ă ind p C q “
1. Therefore ind p C q “ ´ ind p C q ą
0. Since we assumed that J is a generic S -invariant domain dependent almost complex structure, we have that all ofits restrictions to B M g,n are generic, which determine the almost complex structure on ˙Σ and ˙Σ . However, for generic almost complex structures, positive index curves of positivegenus do not exist. Thus C is not constant on a component of positive genus. Since constantcomponents of genus 0 can be eliminated by reparametrization, we can assume without lossof generality that C is not constant on any component of ˙Σ, hence the zeros of dC areisolated. Note that the above argument also holds if ind p C q ą
1. The remainder of theargument is similar to that of [CM07, Lemmas 4.1, 5.4, 5.6], [MS J -hol, Proposition 3.4.2],[Wen-SFT, § C : ˙Σ Ñ R ˆ Y be a J -holomorphic map. The set of regular points z of ˙Σ such that π Y C p z q is a regular value of π Y ˝ C form an open dense subset of ˙Σ. If we intersect theset of regular points with the set of points z P ˙Σ where im p dC z q “ ξ π Y C p z q , it remains openand dense because the projection π Y is already open and dense by the nonintegrability of ξ . Denote the further intersection of these sets with C ´ p R ˆ p Y z N qq by U . Note that U contains a nonempty open set. 50fter fixing p C, J q P M p J S ˙Σ q , we want to show that the linearization D “ D C ` D J issurjective. Since D C is Fredholm, D has closed range and hence surjectivity is equivalent tothe triviality of the annihilator of Im p D q . We prove the result for k “
1, noting that k ą J -hol, Theorem C.2.3]. When k “
1, we have that thedual space of any space of sections of class L p,q can be identified with sections of class L q, ´ δ for p ` q “ d p e r λ q on R ˆ Y wecan use it to define a nondegenerate L -pairing x¨ , ¨y “ L p,δ ˆ L q, ´ δ . Moreover ` L p,δ ˘ ˚ – L q,δ , so we can consider the formal adjoint D ˚ C of D C . Let η P coker p D q ,then the splitting and dualization yield that orthogonality to Im p D q amounts to the equations x D C p ζ q , η y “ x D J p V q , η y “ ζ P T C B k,p,δ and V P T J p J S ˙Σ q . By the first equation, η P ker p D ˚ C q . By ellipticregularity, η is smooth. The second equation implies that if η vanishes on an open set, then η ” J -hol, Lemma 3.4.7].The remainder of the argument is similar to the original proof by Farris. Assume that η z ‰ z P U Ă ˙Σ. This implies that η z P Hom J p z,C p z qq p T z ˙Σ , T C p z q T p R ˆ Y qq and dC z ˝ j z P Hom J p z,C p z qq p T z ˙Σ , T C p z q T p R ˆ Y qq are injective maps. Thus given any 0 ‰ v P T z ˙Σ we have that η z p v q ‰ , dC z ˝ j z p v q ‰ . Next we find some A z P End J p z,C p z qq p T p R ˆ Y q , dλ C p z q q such that A z p dC z ˝ j z p v qq “ η z . On the set U , we have that ξ C p z q and im p dC z q are distinct complex subspaces which span T C p z q p R ˆ Y q . Hence the codomain of D , admits the following splitting:Hom J p z,C p z qq p T z ˙Σ , T C p z q T p R ˆ Y qq “ Hom J p z,C p z qq p T z ˙Σ , ξ C p z q q ‘ End J p z,C p z qq p T z ˙Σ q . We split η z accordingly: η z “ η ˙Σ ` η ξ , where η ξ “ η ξ C p z q ,η ˙Σ “ η T z ˙Σ . Since p J ˝ dC q z is injective, for any given ν z P T z ˙Σ, we can choose a z P End j p z q p T z ˙Σ q so that p J ˝ du ˝ a q z p ν z q “ η ˙Σ . We are now dropping the ε from the notation, as it is understood we should be working with the Floer C ε -space. ξ -component. Since End J p z,C p z qq p ξ C p z q q “ T J p z,C p z qq J S ˙Σ is complex onedimensional, and for any given v z , w z P ξ z , there is an element B z P End J p z,C p z qq p ξ C p z q q sending v z to w z . Hence we take B z : T z ˙Σ Ñ T J p z,C p z qq J S ˙Σ sending p du ˝ j q z p v z q to η ξ . Thus A z : “ p a z , B z q : p ν z , v z q ÞÑ p η ˙Σ , η ξ q , as desired.We now need to suitably extend A z to an A P T J p z,C p z qq J S ˙Σ . When J g,n is restricted to˙Σ P M g,n ` , V P T J p J S ˙Σ q depends on the special marked point z P ˙Σ and p p π Y C p z qq P Σ g .In order to extend A to all of T J J S ˙Σ , we must let it vary with the complex structure j on ˙Σ.The domain of V p p, q q is M g,n ` ˆ Σ g .Define a smooth cutoff function κ : Σ g Ñ R which is nonnegative, takes the value oneat p p π Y C p z qq , and the value zero outside some open neighborhood of p p π Y C p z qq whichdoes not contain any critical points of the perfect Morse function used to define λ ε . Let ν : M g,n ` Ñ R be a smooth nonnegative function which is one at p ˙Σ , j, z , z , ..., z n q andzero outside an appropriately small open neighborhood of this point. The neighborhoodsof the z i should not intersect each other, and the neighborhood U of the special markedpoint z should not contain any preimages of p p π Y C p z qq besides z itself. Note that thesepreimages are finite in number, as otherwise they would accumulate, forcing C to be globallyconstant.Choose an arbitrary smooth extension A of A z , shrinking neighborhoods as necessaryto ensure that x A p j, z , z , ..., z n , q q ˝ dC z ˝ j z , η z y ą q P supp p κ q and p j, z , ..., z n q P supp p ν q . We define A p j, z , ..., z n , q q : “ κ p q q ν p j, z , ..., z n q A p j, z , ..., z n , q q . Since x D J p A q z , η z y ą
0, obtain a contradiction to the assumption that η P coker p D q . Thus η ” D is surjective as claimed. It follows from surjectivity of D B J p C q that M J p α, β q naturally admits the structure of a smooth orbifold of dimension given by the Fredholmindex by way of a virtual repeat of [Wen10, Theorem 0]. In this section we carry out some index calculations which allow us to classify connectors C ,which are defined to be branched and unbranched covers of a union of trivial cylinders \ i γ i ˆ R . We will also use intersection theory to show, under sufficient genericity assumptions, thatcertain sequences of holomorphic curves cannot converge to a building which has a connectorat the top most or bottom most level. Subsequently in §
6, we use these classification results toinvoke the obstruction bundle gluing theorems [HT07, HT09] and prove that the appearanceof ECH handleslides does not impact the homology.52 .1 Buildings and connectors
As in § C is a J -holomorphic curve with positive ends at Reeb orbits α , . . . , α k andnegative ends at Reeb orbits β , . . . , β l , then the Fredholm index of C is given by the formulaind p C q “ ´ χ p C q ` c τ p C q ` k ÿ i “ CZ τ p α i q ´ l ÿ j “ CZ τ p β j q . (5.1)Here χ p C q denotes the Euler characteristic of the domain of C , so if C is irreducible of genus g then χ p C q “ ´ g ´ k ´ l. (5.2)Next we recall the definition of a pseudoholomorphic building from [BEHWZ]; see also[Wen-SFT, § Definition 5.1.
For our purposes, a holomorphic building is an m -tuple p u , . . . , u m q , forsome positive integer m , of (possibly disconnected) J -holomorphic curves u i in R ˆ Y , called levels . Although our notation does not indicate this, the building also includes, for each i P t , . . . , m ´ u , a bijection between the negative ends of u i and the positive ends of u i ` ,such that paired ends are at the same Reeb orbit . If m ą i , at least one component of u i is not a trivial cylinder . A positive end of the building p u , . . . , u m q is a positive end of u , and a negative end of p u , . . . , u m q is a negative end of u m . The genus of the building p u , . . . , u m q is the genus of the Riemann surface obtained bygluing together negative ends of the domain of u i and positive ends of the domain of u i ` bythe given bijections (when this glued Riemann surface is connected).We define the Fredholm index of a holomorphic building byind p u , . . . , u m q “ m ÿ i “ ind p u i q . We recall the Riemann-Hurwitz theorem, in part to fix notation.
Theorem 5.2 (Hartshorne, Corollary IV.2.4) . Let ϕ : r ˙Σ Ñ ˙Σ be a compact k -fold cover ofthe punctured Riemann surface ˙Σ . Then χ p r ˙Σ q “ kχ p ˙Σ q ´ ÿ p P r ˙Σ p e p p q ´ q , (5.3) where e p p q ´ is the ramification index of ϕ at p . At unbranched points p we have e p p q ´ “
0, thus for any q P ˙Σ, ÿ p P ϕ ´ p q q e p p q “ k. One might also want a holomorphic building to include appropriate gluing data when Reeb orbits aremultiply covered, but we will not need this. A trivial cylinder is a J -holomorphic cylinder R ˆ γ in R ˆ Y where γ is a Reeb orbit, which is notrequired to be embedded. In this section we investigate the relation between low ECH and Fredholm index connectors C and the configurations of Reeb orbits at the ends of the components of each C . Recallthat a connector C is a branched cover of a union of trivial cylinders, and all or some of thecomponents may be unbranched.First, we recall that in a symplectization of a contact 3-manifold, all covers of trivialcylinders have non-negative Fredholm index. Lemma 5.3 (Lem. 1.7 [HT07]) . Let C P M J p α, β q be a branched or unbranched cover of atrivial cylinder R ˆ γ , where γ is an embedded Reeb orbit. Then ind p C q ě , with equalityonly if (a) Each component of the domain ˙Σ of C has genus 0. (b) If γ is hyperbolic, then the covering C : ˙Σ Ñ R ˆ γ has no branch points. The remainder of this section concerns the proof of the following result.
Lemma 5.4 (Lem. 7.2.1 [Fa]) . Let C : ˙Σ Ñ R ˆ Y be a connector, where C “ Ť i C i andeach C i is connected. The ECH index I p C q “ and the genus of each component C i iszero.Further assuming that the Fredholm index ind p C q P t , u , then: (i) If ind p C q “ then ind p C i q “ for all i , and either a. C i is an unbranched cover of a trivial cylinder. b. C i is branched, covers R ˆ e ` , and has a single positive end. c. C i branched, covers R ˆ e ´ , and has a single negative end. (ii) If ind p C q “ then C “ C Y Ť i C i where ind p C q “ , ind p C i q “ , and C is abranched cover of R ˆ h j for some j P t , . . . , g u with either one positive end andtwo negative ends, or two positive ends and one negative end. Each of the C i is anunbranched cover of a trivial cylinder.Proof. Let m ˘ , m j denote the multiplicities of the ends of C at the orbits e ˘ , h j : the mul-tiplicities at the positive and negative ends will be the same because C covers a union oftrivial cylinders. In particular, the difference between the total multiplicities at the positiveand negative ends of C will be zero. Therefore, from the index formula (3.2), we have I p C q “ χ p Σ q ¨ ´ e ` ¨ ¨ ˜ m ` ` ÿ j m j ` m ´ ¸ ` m ` ´ m ´ ´ m ` ` m ´ “ c τ p C q “
0. These formulas also hold for each component C i of C .54et p ˘ p C i q denote the number of positive and negative ends of C i , respectively, andlet g p C i q denote the genus of C i . Recall that the Euler characteristic of a surface with p punctures is 2 ´ g ´ p . Case (i)
If ind p C q “
0, then ind p C i q “ i by Lemma 5.3. Case (i.a)
Assume u p C i q is a branched cover of R ˆ h j . Because c τ p C i q “ “ ind p C i q “ ´ χ p C i q The Euler characteristic of a cylinder is 0, therefore the Riemann-Hurwitz Theorem (5.3)gives us 0 “ χ p C i q “ ´ ÿ p P ˙Σ p e p p q ´ q Because e p p q ě p , each term e p p q ´ ě
0, so we must have e p p q “ p .Therefore, C i is unbranched. Moreover,0 “ g p C i q ´ ` p ` p C i q ` p ´ p C i q ô “ g p C i q ` p ` p C i q ` p ´ p C i q . Because C i is a cover of a cylinder, p ˘ p C i q ě
1. Therefore g p C i q “
0, both p ˘ p C i q “
1, and C i unbranched cover of a cylinder. Case (i.b)
Because the Conley-Zehnder index of a cover of e ` is always 1, we have0 “ ind p C i q“ g p C i q ´ ` p ` p C i q ` p ´ p C i q ` p ` p C i q ´ p ´ p C i q , (5.4)hence 1 “ g p C i q ` p ` p C i q . Therefore, because p ` p C i q ě
1, we have g p C i q “ p ` p C i q “ Case (i.c)
By the same argument as for 1.(b), using the fact that the Conley-Zehnderindex of a cover of e ´ is always ´
1, we get g p C i q “ p ´ p C i q “ Case (ii)
If ind p C q “ p C i q ď i . Because ind p C i q ě i by Lemma5.3, there must be one component C with ind p C q “ C i have ind p C i q “ C were a branched cover of R ˆ e ` , then setting the analogue of the right hand sideof (5.4) equal to ind p C q would imply that1 “ p g p C q ´ ` p ` p C qq a contradiction. Similarly C being a branched cover of R ˆ e ´ would lead to a contradiction.Therefore C must be a branched cover of R ˆ h j . In this case, because hyperbolic orbitshave Conley-Zehnder index zero, we have1 “ g p C q ´ ` p ` p C q ` p ´ p C q ô “ g p C q ` p ` p C q ` p ´ p C q Because p ˘ p C q ě
1, this implies 1 ě g p C q , requiring g p C q “
0. Therefore either p p ` p C q , p ´ p C qq “ p , q or p p ` p C q , p ´ p C qq “ p , q .55 .3 Classification of connectors arising in buildings In this section we use intersection theory and higher asymptotics of holomorphic curvesto rule out connectors from appearing at the top-most and bottom-most level of a buildingarising as a limit of a (sub)sequence of holomorphic curves defined in terms of a one parameterfamily of domain dependent almost complex structures, cf. Proposition 5.12. This resultwill be key in §
6. We begin by recalling some needed results about the asymptotics ofholomorphic curves from [HN16, § γ be an embedded Reeb orbit, and let N be a tubular neighborhood of γ . We canidentify N with a disk bundle in the normal bundle to γ , and also with ξ | γ . Let ζ be abraid in N , i.e. a link in N such that that the tubular neighborhood projection restricts to asubmersion ζ Ñ γ . Given a trivialization τ of ξ | γ , one can then define the writhe w τ p ζ q P Z .To define this one uses the trivialization τ to identify N with S ˆ D , then projects ζ to anannulus and counts crossings of the projection with (nonstandard) signs; see § § § C be a J -holomorphic curve in R ˆ Y . Suppose that C has a positive endat γ d which is not part of a multiply covered component. Results of Siefring [Si08, Cor.2.5 and 2.6] show that if s is sufficiently large, then the intersection of this end of C with t s u ˆ N Ă t s u ˆ Y is a braid ζ , whose isotopy class is independent of s . We will need boundson the writhe w τ p ζ q , which are provided by the following lemma. Lemma 5.5 (Lemma 3.2 [HN16]) . Let γ be an embedded Reeb orbit, let C be a J -holomorphiccurve in R ˆ Y with a positive end at γ d which is not part of a trivial cylinder or a multiplycovered component, and let ζ denote the intersection of this end with t s u ˆ Y . If s " , thenthe following hold: (a) ζ is the graph in N of a nonvanishing section of ξ | γ d . Thus, using the trivialization τ towrite this section as a map γ d Ñ C zt u , it has a well-defined winding number around , which we denote by wind τ p ζ q . (b) wind τ p ζ q ď (cid:4) CZ τ p γ d q{ (cid:5) . (c) If J is generic, CZ τ p γ d q is odd, and ind p u q ď , then equality holds in (b). (d) w τ p ζ q ď p d ´ q wind τ p ζ q . Symmetrically to Lemma 5.5, we also have the following:
Lemma 5.6.
Let γ be an embedded Reeb orbit, let C be a J -holomorphic curve in R ˆ Y witha negative end at γ d which is not part of a trivial cylinder or multiply covered component,and let ζ denote the intersection of this end with t s u ˆ Y . If s ! , then the following hold: (a) ζ is the graph of a nonvanishing section of ξ | γ d , and thus has a well-defined windingnumber wind τ p ζ q . (b) wind τ p ζ q ě (cid:6) CZ τ p γ d q{ (cid:7) . (c) If J is generic, CZ τ p γ d q is odd, and ind p u q ď , then equality holds in (b). w τ p ζ q ě p d ´ q wind τ p ζ q . Remark 5.7.
Lemma 5.5(b),(d) imply that w τ p ζ q ď p d ´ q (cid:4) CZ τ p γ d q{ (cid:5) . In fact one can improve this to w τ p ζ q ď p d ´ q (cid:4) CZ τ p γ d q{ (cid:5) ´ gcd ` d, (cid:4) CZ τ p γ d q{ (cid:5) ˘ ` , (5.5)see [Si11]. Recent work of Cristofaro-Gardiner - Hutchings - Zhang obtains equality in (5.5)in the following situation. Lemma 5.8 ([CGHZ, Cor. 5.3]) . Let γ be an embedded Reeb orbit, let C be a J -holomorphiccurve in R ˆ Y with only one positive end at γ d , and let ζ denote the intersection of thisend with t s u ˆ Y for s " . Suppose CZ τ p γ d q is odd, the index of u is at most 2, and J isgeneric. Then ζ is isotopic to the braid given by a regular end and w τ p ζ q “ p d ´ q (cid:4) CZ p γ d q{ (cid:5) ´ gcd ` d, (cid:4) CZ p γ d q{ (cid:5) ˘ ` . The definition of a regular end is lengthy, see [CGHZ, Def. 1.3]. It ensures that thetopology of the braid near an embedded Reeb orbit is completely determined by the totalmultiplicity of the orbit and the corresponding partition numbers. However, [CGHZ, Thm.1.4], guarantees that for generic J , every generic curve has regular positive and negativeends. Symmetrically to Lemma 5.8 we have the following result for a negative end. Lemma 5.9.
Let γ be an embedded Reeb orbit, let C be a J -holomorphic curve in R ˆ Y with only one negative end at γ d , and let ζ denote the intersection of this end with t s u ˆ Y for s ! . Suppose CZ τ p γ d q is odd, the index of u is at most 2, and J is generic. Then ζ isisotopic to the braid given by a regular end and w τ p ζ q “ p d ´ q (cid:6) CZ p γ d q{ (cid:7) ` gcd ` d, (cid:6) CZ p γ d q{ (cid:7) ˘ ´ . The proof of the main classification result, Proposition 5.12, requires the following directcomputation of asymptotic writhes and linking numbers, which uses the preceding lemmas.
Lemma 5.10.
Let J be generic. Let ζ i , ζ j be connected braids about an embedded Reeb orbit γ with multiplicities d i , d j . If both ζ i , ζ j arise from either the positive or the negative ends ofa curve which covers γ , then (i) Assuming γ “ e ` : a. There is only one positive end ζ ` , and w τ p ζ ` q “ ´ d ` . b. If the ζ i , ζ j are negative ends, then w τ p ζ i q “ d i ´ , w τ p ζ j q “ d j ´ , and (cid:96) τ p ζ i , ζ j q “ min p d i , d j q . (ii) Assuming γ “ e ´ : If the ζ i , ζ j are positive ends, then w τ p ζ i q “ ´ d i , w τ p ζ j q “ ´ d j , and (cid:96) τ p ζ i , ζ j q “ ´ min p d i , d j q . b. There is only one negative end ζ ´ , and w τ p ζ ´ q “ d ´ ´ .Proof. We proceed casewise.
Case (i.a)
By Lemma 5.4 (i.b), the end ζ ` is the only positive end. Therefore Lemma5.8 applies, giving us w τ p ζ ` q “ p d ` ´ q (cid:4) CZ p γ d ` q{ (cid:5) ´ gcd ` d ` , (cid:4) CZ p γ d ` q{ (cid:5) ˘ ` “ p d ` ´ q (cid:22) (cid:23) ´ gcd ˆ d ` , (cid:22) (cid:23) ˙ ` “ ´ gcd p d ` , q ` “ ´ d ` . Case (i.b)
Firstly, we immediately have wind τ p ζ i q “ τ p ζ i q “ (cid:6) CZ τ p ζ d i i q{ (cid:7) “ (cid:24) (cid:25) “ p d i , wind τ p ζ i qq “ gcd p d i , q “
1, which is a sub-case in the proof of [Hu02b,Lemma 6.7]. There the equality w τ p ζ i q “ p d i ´ q wind τ p ζ i q is proven by showing that the ζ i are isotopic to p d i , q torus braids when gcd p d i , wind τ p ζ i qq “
1. Therefore w τ p ζ i q “ d i ´ λ i denote the smallest eigenvalue of L d i in the expansionof ζ i . The proof of [Hu02b, Lemma 6.9] is proceeds by considering three cases: withoutloss of generality, when λ i ă λ j , when λ i “ λ j and the coefficients of the correspondingeigenfunctions are different, and when λ i “ λ j and the coefficients of the correspondingeigenfunctions are the same. We are guaranteed by [HT09, Proposition 3.9] that we are ineither of the first two cases, while the proof of [Hu02b, Lemma 6.9] gives the equality (cid:96) τ p ζ i , ζ j q “ min t d i , d j u in both of those cases, which is stronger than its general result. Case (ii.a)
We immediately have wind τ p ζ i q “ ´ τ p ζ i q “ (cid:4) CZ τ p ζ d i i q{ (cid:5) “ (cid:22) ´ (cid:23) “ ´ . The proof that w τ p ζ i q “ p d i ´ q wind τ p ζ i q and hence that w τ p ζ i q “ ´ d i is a virtualrepeat of the proof for negative ends from [Hu02b, Lemma 6.7] as in Case (i.b).For the claim on linking, we can repeat the proof in Case (i.b). Note that [Hu02b, Lemma6.9] only applies to negative ends, but the proof will work using the asymptotic expansion of58 positive end from [HWZ96], written in our notation as [Hu14, Lemma 5.2]. If λ i ă λ j , or λ i “ λ j with corresponding eigenfunctions having different multiplicities in the ζ i , we knowthat the braid ζ j must be nested inside ζ i , therefore (cid:96) τ p ζ i , ζ j q “ wind τ p ζ i q d j “ ´ d j . We have ´ d j “ ´ min t d i , d j u because, by pulling back both ζ i to covers of γ d i d j , we multiply their winding numbers by d j and d i , respectively, and can apply the analytic perturbation theory of [HWZ95, § d j wind τ p ζ i q ě d wind τ p ζ j q ô d j ď d i . Case (ii.b)
By Lemma 5.4(i.c), the end ζ ´ is the only negative end. Therefore Lemma5.9 applies, giving us w τ p ζ ´ q “ p d ´ ´ q (cid:6) CZ p γ d ´ q{ (cid:7) ` gcd ` d , (cid:6) CZ p γ d ´ q{ (cid:7) ˘ ´ “ p d ´ ´ q (cid:24) ´ (cid:25) ` gcd ˆ d ´ , (cid:24) ´ (cid:25) ˙ ´ “ ` gcd p d ´ , q ´ “ d ´ ´ . Finally, we need the following inequality from intersection theory of holomorphic curves,cf. [HN16, § γ be an embedded Reeb orbit with tubular neighborhood N , and let τ be atrivialization of ξ | γ . Lemma 5.11.
Let C be a J -holomorphic curve in r s ´ , s ` s ˆ N with no multiply coveredcomponents and with boundary ζ ` ´ ζ ´ where ζ ˘ is a braid in t s ˘ u ˆ N . Then χ p C q ` w τ p ζ ` q ´ w τ p ζ ´ q “ δ p C q ě , where χ p C q denotes the Euler characteristic of the domain of C and δ p C q is a count of thesingularities of C in Y with positive integer weights. With these preliminaries in place, we are now ready to prove the key classification resultwhich excludes connectors from appearing in the top most or bottom most level of a buildingarising as a limit in the sense of [BEHWZ].
Proposition 5.12.
Let t J t u t Pr , s be a generic family of domain dependent almost complexstructures and α and β be admissible orbit sets with I p α, β q “ . Let C p t q P M J t p α, β q be asequence of Fredholm index 1 curves, which, as t Ñ , converges in the sense of [BEHWZ] toa building with n levels given by C i P M J p γ i ´ , γ i q , i “ , ..., n , where γ “ α and γ n “ β .Then neither the top most level C nor the bottom most level C n are connectors. roof. We assume that the proposition is false and set up some notation. Suppose to geta contradiction that there exists a sequence of t J t u -holomorphic curves t C p t qu P M J t p α, β q which converges in the sense of [BEHWZ] to a n -level building which has either C or C n as aconnector. Recall that C i is an equivalence class of holomorphic curves in R ˆ Y , where twoholomorphic curves are equivalent iff they differ by R -translation in R ˆ Y . In the following,we will choose a representative of this equivalence class and still denote it by C i . If necessary,translate the holomorphic curve C upward and C n downward so that Lemmas 5.5-5.9 apply,cf. [HN16, § C P M J p α, α q . Consider an embedded Reeb orbit γ appearing inthe orbit set α . Let N γ be a tubular neighborhood of the Reeb orbit γ . For some sufficientlylarge s " t close to 1, the intersection C p t q X pr s ,
8q ˆ N γ q can be identifiedwith the union of components of C that cover R ˆ γ . Denote both by C . Note that as asubset of C p t q , C is not a trivial cylinder, but rather an embedding in the complement of afinite number of singular points.While intersection positivity is not true in general for domain dependent almost complexstructures, by Remark 4.15 if C : p ˙Σ , j q Ñ p R ˆ Y, J q is a p j, J q -holomorphic curve, then C | C ´ p R ˆ N q is p j, J q -holomorphic. Since J is domainindependent, the subset C p ˙Σ q X R ˆ N Ă C p ˙Σ q satisfies intersection positivity. Thus wewill be in a situation to apply relative adjunction as in Lemma 5.11 because intersectionpositivity holds. Moreover, the count of singularities of C , satisfies δ p C q ě C is embedded.We will show that if C arises from a nontrivial connector appearing at the top most level,then relative adjunction as in Lemma 5.11 will imply that δ p C q ă
0, a contradiction. Notethat a connector cannot be trivial in the sense that it exclusively consists of unbranchedcomponents, e.g. trivial cylinders, as explained in [Wen-SFT, Remark 9.26].There are three cases to consider, corresponding to connectors containing componentssatisfying the conclusions of Lemma 5.4, (i.b), (i.c), and (ii).
Case (i.b)
Assume C is a component of a connector covering R ˆ e ` . Then by Lemma5.4 we have g p C q “ C has a single positive end. Let d ` denotethe covering multiplicity of this end, and let d i denote the covering multiplicity of the i th negative end of C . Because C covers a trivial cylinder, d ` “ ř p ´ p C q i “ d i . For t sufficientlyclose to 1, there is a representative C with the following properties.1. C ´ pr , Y q is an annulus with one puncture, which is mapped by C to r , N e ` C ´ pp´8 , s ˆ Y q consists of as many half cylinders C i as there are p ´ p C q negativeends of C .3. C p C i q is contained in p´8 , s ˆ N e ` and C p C i q X pt u ˆ N q is a braid ζ i which projectsto e ` with degree d i and has distance at most εp ´ p C q` from e ` .Also let ζ ` denote the braid corresponding to the positive end of C at e d ` ` . It follows that60he union Ť i ζ i is a braid. We obtain a contradiction:2 δ p C q “ ´ p ` p C q ´ p ´ p C q ` w τ p ζ ` q ´ w τ ˜ď i ζ i ¸ “ ´ p ´ p C q ` p ´ d ` q ´ ˜ p ´ p C q ÿ i “ p d i ´ q ` ÿ i ‰ j min p d i , d j q ¸ by Lemma 5.10(i) and (2.10) “ ´ d ` ´ ÿ i ‰ j min p d i , d j qď ´ ř i ‰ j min p d i , d j q accounts for the factor of two in (2.10).) In theinequality we have used the fact that ř i ‰ j min p d i , d j q ě d i ě
1. There can never be just one negative end lest C betopologically a cylinder and therefore unbranched, by the Riemann-Hurwitz Theorem. Case (i.c)
Assume C is a component of a connector covering R ˆ e ´ . Then by Lemma5.4 we have g p C q “ C has a single negative end. Let d ´ denotethe covering multiplicity of this end, and let d i denote the covering multiplicity of the i th positive end of C . For t sufficiently close to 1, there is a representative C with the followingproperties.1. C ´ pp´8 , s ˆ Y q is an annulus with one puncture, which is mapped by C to p8 , s ˆ N e ´ C ´ pr ,
8q ˆ Y q consists of as many half cylinders C i as there are p ` p C q positive endsof C .3. C p C i q is contained in r ,
8q ˆ N e ´ and C p C i q X pt u ˆ N q is a braid ζ i which projectsto e ´ with degree d i and has distance at most εp ` p C q` from e ´ .Also let ζ ´ denote the braid corresponding to the negative end of C at e d ´ ´ . It follows thatthe union Ť i ζ i is a braid. We obtain a contradiction:2 δ p C q “ ´ p ` p C q ´ p ´ p C q ` w τ ˜ď i ζ i ¸ ´ w τ p ζ ´ q“ ´ p ` p C q ` ˜ p ` p C q ÿ i “ p ´ d i q ´ ÿ i ‰ j min p d i , d j q ¸ ´ p d ´ ´ q by Lemma 5.10(ii) and (2.10) “ ´ d ´ ´ ÿ i ‰ j min p d i , d j qď ´ p ` p C q ě
2, hence ř i ‰ j min p d i , d j q ě Case (ii)
If a branched component of the connector at the top (respectively, the bottom)covers R ˆ h , where h is hyperbolic, then by Lemma 5.4(ii), its ends must be asymptotic to h . Therefore α (respectively, β ) must include the pair p h, m q with m ě
2, which contradictsthe fact that α (respectively, β ) is an ECH chain complex generator.61 From domain dependent J to domain independent J In Corollary 4.20, we saw that for a generic S -invariant domain dependent almost complexstructure J P J S ˙Σ that ECH index one moduli spaces of nonzero genus curves are empty.However ECH is defined using a domain independent generic λ -compatible J , so we mustprove the analogous result when J is a generic λ -compatible almost complex structure. Inorder to do so, we consider a generic one parameter family t J t u t Pr , s of domain dependentalmost complex structures interpolating between a generic J : “ J P J S ˙Σ and a domainindependent generic λ -compatible J : “ J P J reg p Y, λ q and show that the computation ofECH is not affected. Our main result is the following.
Proposition 6.1.
Let α and β be admissible orbit sets with I p α, β q “ and deg p α, β q ą . For generic paths t J t u t Pr , s connecting J : “ J P J S ˙Σ and J : “ J P J reg p Y, λ q , the modulispace M t : “ M J t p α, β q is cut out transversely save for a discrete number of times t , ..., t (cid:96) Pp , q . At each such t i , the ECH differential can change either by: (a) The creation or destruction of a pair of oppositely signed curves. (b) An ECH handleslide.However, in either case, the homology is unaffected.
For each J t we consider the moduli space M t p α, β q of J t -holomorphic curves where α and β are admissible orbit sets satisfying I p α, β q “ p α, β q ą
0. That deg p α, β q ą J t -holmorphic cylinders, for which the domain dependent almostcomplex structures cannot be used, cf. Lemma 4.6. We have that M t p α, β q is cut outtransversely save a discrete number of times t i P r , s and at such a nonregular J t i , thedifferential can be impacted by either the creation or destruction of a pair of oppositelysigned curves or by an “ECH handleslide.” In the former case, the signed and mod 2 counts ofcurves in M t i ´ ε and M t i ` ε are the same. The differential can change at an ECH handleslide,at which a sequence of Fredholm and ECH index 1 curves t C p t qu breaks into a holomorphicbuilding in the sense of [BEHWZ] into components consisting of an ECH and Fredholm index0 curve, an ECH and Fredholm index 1 curve, and some “connectors,” which are Fredholmindex 0 branched covers of a trivial cylinder R ˆ γ . In § § § Because we are using Z -coefficients, we will not sort through the signs. orollary 6.2. Let α and β be nondegenerate admissible orbit sets and J P J p Y, λ q begeneric. If deg p α, β q ą and I p α, β q “ then the mod 2 count Z M J p α, β q “ . If α and β are associated to λ ε as in Lemma 3.1 and A p α q , A p β q ă L p ε q then xB L p ε q α, β y “ . In § § t i , a sequence of J t -holomorphic curves t C k | ind p C k q “ u breaks intoan ECH handleslide building p C ` , C , C ´ q wherein(i) The top most curve C ` has either index 1 or index 0(ii) Connectors C with ind p C q “ C ´ has ind p C ´ q “ ´ ind p C ` q .Moreover, the index 0 curve occurring at either the top most or bottom most level cannotcontain any connectors. Definition 6.3.
We define an
ECH handleslide to be the index 0 curve which is not aconnector in an
ECH handleslide building p C ` , C , C ´ q , in analogy with Morse theory.As observed in [Fa, § t i . If weassume that the ECH handleslide is C ´ , then as explained in Remark 6.13 we obtain: M t i ` (cid:15) p α, β q “ M t i ´ (cid:15) p α, β q ` G p C ` , C ´ q ¨ M t i p α, γ ` q , (6.1)where γ is another (admissible) orbit set such that I p α, γ ` q “ C ` P M t i p α, γ ` q . Notethat the connector C P M t i p γ ` , γ ´ q and the ECH handleslide curve C ´ P M t i p γ ´ , β q . Asexplained in § G p C ` , C ´ q P Z , based on the negative asymptotic ends of C ` , the positive asymptotic ends of C ´ , andthe partitions associated to the ends of the connectors C . For each embedded Reeb orbit γ , the total covering multiplicity of Reeb orbits covering γ in the list γ ` is the same asthe total for γ ´ . (In contrast, for the usual form of Floer theory gluing, one would assumethat γ ` “ γ ´ .) In § G p C ` , C ´ q as we obtain M t i p α, γ ` q “ γ ` such that I p α, γ ` q “ An ECH handleslide building is a building arising as a limit of I p C q “
1, ind p C q “ R ˆ Y as the complex structure varies through domain-dependent almost complex structures.The terminology arises from the fact that such a building might include levels with I p C q “ emma 6.4 (Configuration of an ECH handleslide) . Fix admissible orbit sets α and β with deg p α, β q ą and I p α, β q “ . Let t J t u t Pr , s be a one generic parameter family of almostcomplex structures. Consider the corresponding moduli spaces M t : “ M J t p α, β q ; label thetimes at which M t is not cut out transversely by t , ..., t (cid:96) P p , q . Let C p t q P M t p α, β q with t Ñ t i . Then after passing to a subsequence, t C p t qu converges in the sense of [BEHWZ]either to a curve in M t i p α, β q or to an ECH handleslide building with (i)
An index 1 curve at the top most level C ` (or at bottom most level C ´ ); (ii) Connectors C with ind p C q “ ; (iii) An index 0 curve, the ECH handleslide, at the bottom most level C ´ (or at the top mostlevel C ` ).Proof. By the compactness theorem in [BEHWZ], any sequence of ECH and Fredholm index1 curves in M J t p α, β q has a subsequence which converges to some broken curve as t Ñ t i .Moreover, the indices of the levels of the broken curve sum to 1. By Proposition 5.12we cannot have connectors appear at the top most or bottom most level. Moreover bycompactness and the conservation of Fredholm index, a Fredholm index one connector cannotappear as a middle level in a handleslide building. If the sequence is close to breaking, cf.Definition 6.8, then by Lemma 5.3 and the definition of G δ , one of the following two scenariosoccurs:(i) The top most level of the broken curve contains the index 1 component C ` and somelower level contains the index 0 ECH handleslide C ´ .(ii) The bottom most level of the broken curve contains the index 1 component, C ´ , andthe top most level contains the index 0 ECH handleslide, C ` .Moreover, all other components of all levels are index zero branched covers of R -invariantcylinders, e.g. connectors. By analogy with condition (d) in the definition of a gluing pair,Definition 6.7, any covers of R -invariant cylinders in the top and bottom levels of the brokencurve must be unbranched.Finally, we review the possible bifurcations that appear in a generic 1-parameter family t J t u t Pr , s : Proposition 6.5.
Fix a nondegenerate contact form λ . Then for a 1-parameter family t J u t Pr , s of λ -compatible domain dependent almost complex structures with fixed endpointswe may arrange that the only possible bifurcations are: (a) A cancellation of two oppositely signed holomorphic curves. (b)
An ECH handleslide.
In the case of Morse theory, the corresponding transversality statement is [Hu02a, Lemma2.11(b)]. In the context of Seiberg-Witten Floer homology, Taubes completes this bifurcationanalysis at the end of [T02]. Note that cancellation of two oppositely signed curves does notchange the differential. The presence of an ECH handleslide does, change the differential,but in § .3 Recap of obstruction bundle gluing In this section we collect the results from [HT07] that will be used in the proof of Propo-sition 6.1. We state everything in the context considered in proving B “
0, and explain ina subsequent remark the difference and continued applicability in the setting under consid-eration. In connection with the index calculations for branched covered cylinders over anelliptic embedded Reeb orbit, cf. Lemma 5.3, we can define a partial order on the associatedset of partitions, which will be used in the construction of a gluing pair.
Definition 6.6 (Partial order ě ϑ ) . Let γ be a nondegenerate elliptic embedded Reeb orbitwith a fixed irrational rotation number ϑ , cf. § α “ p γ a , ..., γ a k q and β “ ` γ b , ..., γ b (cid:96) ˘ , consider C P M J p α, β q a branched cover of R ˆ γ . We say p a , ..., a k q ě ϑ p b , ..., b (cid:96) q whenever there exists an index zero branched cover of R ˆ γ P M J ` p γ a , ..., γ a k q , ` γ b , ..., γ b (cid:96) ˘˘ . Following [HT07, § G p C ` , C ´ q , and state the main obstruction bundle gluing theorem. Definition 6.7. A gluing pair is a pair of immersed J -holomorphic curves C ` p α, γ ` q and C ´ p γ ´ , β q such that:(a) ind p C ` q “ ind p C ´ q “ C ` and C ´ are not multiply covered, except that they may contain unbranched coversof R -invariant cylinders.(c) For each embedded Reeb orbit γ , the total covering multiplicity of Reeb orbits covering γ in the list γ ` is the same as the total for γ ´ . (In contrast, for the usual form of Floertheory gluing, one would assume that γ ` “ γ ´ .)(d) If γ is an elliptic embedded Reeb orbit with rotation angle ϑ , let m , ..., m k denote thecovering multiplicities of the R -invariant cylinders over γ in C ` and let n , ..., n j denotethe corresponding multiplicities in C ´ . Then under the partial order ě ϑ in Definition6.6, the partition p m , ..., m k q is minimal, and the partition p n , ..., n j q is maximal.Let p C ` , C ´ q be a gluing pair. The main gluing result of [HT07, HT09] computes aninteger G p C ` , C ´ q which, roughly speaking is a signed count of ends of the index twopart of the moduli space M J p α, β q{ R that break into C ` and C ´ along with some indexzero connectors (branched covers of R -invariant cylinders between them). When C ˘ containcovers of R -invariant cylinders, there are some subtleties which require the use of condition(d) above in showing that G p C ` , C ´ q is well-defined.Before giving the definition of the count G p C ` , C ´ q , we first define a set G δ p C ` , C ´ q of index two curves in M J p α, β q which, are close to breaking in the above manner. For thefollowing definition, choose an arbitrary product metric on R ˆ Y . Definition 6.8.
For δ ą
0, define C δ p C ` , C ´ q to be the set of immersed (except possibly atfinitely many singular points) surfaces in R ˆ Y that can be decomposed as C ´ Y C Y C ` such that the following hold: 65 There is a real number R ´ , and a section ψ ´ of the normal bundle to C ´ with | ψ ´ | ă δ ,such that C ´ is the set s ÞÑ s ` R ´ translate of the s ď δ portion of the image of ψ ´ under the exponential map. • Similarily, there is a real number R ` , and a section ψ ` of the normal bundle to C ` with | ψ ` | ă δ , such that C ` is the set s ÞÑ s ` R ` translate of the s ě ´ δ portion ofthe image of ψ ` under the exponential map. • R ` ´ R ´ ą δ . • C is contained in the union of the radius δ tubular neighborhoods of the cylinders R ˆ γ , where γ ranges over the embedded Reeb orbits covered by orbits in γ ˘ • B C “ B C ´ \ B C ` , where the positive boundary circles of C ´ agree with the negativeboundary circles of C , and the positive boundary circles of C agree with the negativeboundary circles of C ` .Let G δ p C ` , C ´ q denote the set of index two curves in M J p α, β q X C δ p C ` , C ´ q .To see that this definition works as expected, we have the following lemma. We include theproof, as it will better elucidate why we can invoke the obstruction bundle gluing formalismin the setting under consideration. Lemma 6.9.
Given a gluing pair p C ` , C ´ q , there exists δ ą with the following property.Let δ P p , δ q and let t C p k qu k “ , ,... be a sequence in G δ p C ` , C ´ q{ R . Then there is a subse-quence which converges in the sense of [BEHWZ] either to a curve in M J p α, β q{ R or to abroken curve in which the top level is C ` , the bottom level is C ´ , and all intermediate levelsare unions of index zero branched covers of R -invariant cylinders.Proof. By the compactness theorem in [BEHWZ], any sequence of index 2 curves in M J p α, β q{ R has a subsequence which converges to some broken curve. Moreover, the indices of the levelsof the broken curve sum to 2. If the sequence is in G δ p C ` , C ´ q{ R with δ ą G δ , one of the following two scenarios occurs:(i) One level of the broken curve contains the index 1 component of C ` , and some lowerlevel contains the index 1 component of C ´ .(ii) Some level contains two index 1 components or one index 2 component.Moreover, all other components of all levels are index zero branched covers of R -invariantcylinders. By condition (d) in the definition of a gluing pair, any covers of R -invariantcylinders in the top and bottom levels of the broken curve must be unbranched. It followsthat in Case (i), the top level is C ` and the bottom level is C ´ , while in Case (ii), there areno other levels. Definition 6.10.
Fix coherent orientations and generic λ -compatible J and let p C ` , C ´ q bea gluing pair. If δ P p , δ q , then by Lemma 6.9, one can choose an open set U Ă M J p α, β q{ R such that: • G δ p C ` , C ´ q{ R Ă U Ă G δ p C ` , C ´ q{ R for some δ P p , δ q .66 The closure U has finitely many boundary points.Define G p C ` , C ´ q P Z to be minus the signed count of boundary points of U . By Lemma6.9, this does not depend on the choice of δ or U .Note that by Lemma 5.3, if G p C ` , C ´ q ‰ γ , themultiplicities of the negative ends of C ` at covers of γ agree, up to reordering, with themultiplicities of the positive ends of C ´ at covers of γ . When this is the case, assume thatthe orderings of the negative ends of C ` and of the positive ends of C ´ are such that for eachpositive hyperbolic orbit γ , the aforementioned multiplicities appear in the same order for C ` and for C ´ . With this ordering convention, the statement of the main gluing theorem isas follows: Theorem 6.11. [HT07, Theorem 1.13]
Fix coherent orientations. If J is generic and if p C ` , C ´ q is a gluing pair then G p C ` , C ´ q “ (cid:15) p C ` q (cid:15) p C ´ q ź γ c γ p C ` , C ´ q . (6.2) Here the product is over embedded Reeb orbits γ such that C ` has a negative end at a coverof γ . The integer c γ p C ` , C ´ q depends only on γ and on the multiplicities of the R -invariantand non- R -invariant negative ends of C ` and positive ends of C ´ at covers of γ . We omit the discussion of the explicit computation of the gluing coefficient c γ p C ` , C ´ q ,as we will multiply this by zero; the budding obstruction bundle enthusiast can find furtherdetails in [HT07, § Remark 6.12.
We have that c γ p C ` , C ´ q “ § § § Remark 6.13.
Roughly speaking and along the lines of [Fl88a], [Hu02a, § (cid:15) small,the results of § B ¨˝ ď t Pr t i ´ (cid:15),t i ` (cid:15) s M t p α, β q ˛‚ “ M t i ` (cid:15) p α, β q ´ M t i ` (cid:15) p α, β q ¯ M t i p α, γ ` q ğ C P M ti p γ ` ,γ ´ q M t i p γ ´ , β q . However, technically speaking we cannot compute the boundary of the compactified modulispace on the left hand side. Instead, we must truncate this moduli space in order to invokethe obstruction bundle gluing theorem and obtain (6.1) similarly to the proof that the ECHdifferential squares to zero, cf. [HT07, Theorem 7.20]. In particular, if p C ` , C ´ q is a gluingpair arising from an ECH handleslide in which C ` P M t i p α, γ ` q and C ´ P M t i p γ ´ , β q , let V p C ` , C ´ q Ă M t p α, β q for t P r t i ´ (cid:15), t i ` (cid:15) s be an open set like the open set U in Definition67.10, but where the curves do not have any asymptotic markings or orderings of the ends.We truncate the interior of the cobordism by removing curves which are close to breaking, M : “ ď t Pr t i ´ (cid:15),t i ` (cid:15) s M t p α, β q z ğ p C ´ ,C ` q V p C ´ , C ` q By the analogue of [HT07, Lem. 7.23], namely Lemma 6.4, we have that M is compact.Because the handleslide is isolated, the signed count of truncated boundary points is0 “ B M “ ´ B V p C ` , C ´ q . The count G p C ` , C ´ q “ ´ B U distinguishes curves in B U that have different asymptoticmarkings and orderings of the ends, but represent the same element of B V p C ` , C ´ q , resultingin (6.1). Similarly to [Fa, § Proof of Proposition 6.1.
Number the ECH handleslides t , ..., t k . We omit the cancellationbifurcations as they do not change the curve counts, and note that one should occur beforethe first ECH handleslide, as otherwise the moduli space under consideration are empty byCorollary 4.20. Without loss of generality we will always assume that C ` is the index 1curve and C ´ is the index 0 ECH handleslide curve in an ECH handleslide building.By Remark 6.13 we have that at each handleslide t i , M t i ` (cid:15) p α, β q “ M t i ´ (cid:15) p α, β q ` G p C ` , C ´ q ¨ M t i p α, γ ` q , (6.3)where γ ` is another (admissible) orbit set such that I p α, γ ` q “ C ` P M t i p α, γ ` q .Note that the connector C P M t i p γ ` , γ ´ q and the ECH handleslide curve C ´ P M t i p γ ´ , β q .Since J : “ J is a generic S invariant domain dependent almost complex structure, byCorollary 4.20 M t ´ ε p α, β q “ H . If we can show for all possible γ that M t p α, γ q “ M t ` ε p α, β q “ M t ´ ε p α, β q ` “ . An inductive argument on k P , ..., (cid:96) , where each t k realizes an ECH handleslide, would thencomplete the proof.It remains to show (6.4); this will be done by a reductive degree argument. The degreeas defined in § p C p t qq “ deg p C ` q ` deg p C ´ q . C ´ has degree 0, then it is a union of branchedcovers of cylinders, at least one of which is not a trivial cylinder, as otherwise C ´ is aconnector. But a nontrivial cylinder, and hence the union of cylinders including it, haspositive Fredholm index. Thus the ECH handleslide curve C ´ must have positive degree(and positive genus) by Lemma 4.6. Hence, if at a handleslide, t C p t qu P M t p α, β q convergesto an ECH handleslide building p C ` , C , C ´ q thendeg p C ` q ă deg p C p t qq (6.5)Going back to the task at hand, consider M t p α, γ ` q , a smooth one dimensional modulispace of Fredholm and ECH index 1 curves. We wish to show that M t p α, γ ` q “ . This will be accomplished by inductive iteration:1. Consider another generic one parameter family t J t u t Pr ,t s of domain dependent almostcomplex structures from J to J t . Note that before the first handleslide, call it t , inthis new deformation M t ´ ε p α, γ ` q “ H .
2. Use (6.3) to relate the curve counts occurring immediately prior to and following thefirst appearance of an ECH handleslide at t in this new deformation t J t u t Pr ,t s : M t ` (cid:15) p α, γ ` q “ M t ´ (cid:15) p α, γ ` q ` G p C , C q ¨ M t p α, γ q“ G p C , C q ¨ M t p α, γ q .
3. Observe the degree reduction (6.5) for the resulting ECH index 1 component C P M t p α, γ q in the handleslide, namely,deg p C q ă deg p C ` q because the index 0 ECH handleslide curve must always have positive degree.We repeat this process until either the resulting index 1 curve C p k q` P M t p k q p α, γ p k q` q arisingfrom (6.3) associated to ! J p k q t ) for t P ” , t p k ´ q ı at the “next” handleslide at time t p k q canno longer degenerate via handleslides or is a degree 1 curve. (Note that we can also stopat degree 2, since transversality for degree 1 curves can be achieved by S -invariant domaindependent almost complex structures). This permits us to conclude that M t p α, γ ` q “ . for all admissible γ ` with I p α, γ ` q “ Computation of
E C H
In this section we prove Theorem 1.1.Before invoking the results in § § L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q to Λ ˚ H ˚ p Σ g , Z q , we mustfirst relate lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q to the ECH of the original prequantization bundle. In § Theorem 7.1.
With
Y, λ, ε p L q as in Lemma 3.1, for any Γ P H p Y ; Z q , lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q “ ECH ˚ p Y, ξ, Γ q . In § Z -graded isomor-phism of Z -modules à Γ P H p Y q lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q – Λ ˚ H ˚ p Σ g ; Z q . (7.1)The idea of the proof is as follows. As a consequence of Lemma 3.7 (i), the groups ECC L ˚ p Y, λ ε p L q , Γ; J q are all zero unless Γ is in the Z ´ e summand of H p Y q . We abuse notation by using Γ toalso indicate the corresponding element of Z ´ e . By Proposition 3.2 we can restrict attentionto orbits above critical points of the Morse function H , and by the analysis of § §
6, theECH differential “agrees” with the Morse differential in the sense that if a J -holomorphiccurve count contributing to the ECH differential were nonzero, then it must equal a countof gradient flow lines defining the Morse differential on Σ g . However, because H is perfect,all such counts of gradient flow lines are zero. By analyzing the implications of index parityand the fact that hyperbolic orbits can appear with multiplicity at most one, we prove thatthe chain complexes have zero differential and satisfylim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q – à d P Z ě Λ Γ `p´ e q d H ˚ p Σ g ; Z q (7.2)as Z -graded Z -modules, which implies (7.1).We conclude this introductory section by proving the third and fourth conclusions ofTheorem 1.1, assuming the first and second. Theorem 1.1, Z grading. In § p sends the orbits. Note that the images of generators with deg p α, e Γ ´ q “ d lie inΛ Γ `p´ e q d H ˚ p Σ g ; Z q . The Z -valued grading of the image of e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` under thismap is | e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` | ‚ “ m ` ¨ ¨ ¨ ` m g ` m ` “ M ` m ` ´ m ´ “ Γ ´ ed ` m ` ´ m ´ , where d “ deg p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ q “ M ´ Γ ´ e . On the other hand, I p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ q “ ´ ed ` p χ p Σ g q ` q d ` m ` ´ m ´ ` Γ “ ´ ed ` p χ p Σ g q ` q d ` | e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` | ‚ ` ed. .1 Direct limits and Seiberg-Witten Floer cohomology Because there are no Morse-Bott methods for ECH, we must compute
ECH ˚ p Y, ξ, Γ q byrelating it to lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q . Our discussion so far allows us to understand thelatter. In this section we prove Theorem 7.1, obtaininglim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q “ ECH ˚ p Y, ξ, Γ q . It is not possible to prove Theorem 7.1 solely via ECH. This is because we cannot relatethe filtered homologies computed for ε ą ε “
0, since thecontact form is degenerate and the filtered homology is not defined for any L . However, ECHis isomorphic to Seiberg-Witten Floer cohomology (including at the level of filtrations), andin the latter case the degeneracy of λ is not an issue. The key result is the last equation in[HT13, § § § § § § § [HT13, § § For a definition of Seiberg-Witten Floer cohomology in our setting, we refer the reader to[HT13, § § Y with Riemannian metrics g .Recall that a spin-c structure on Y is a rank two Hermitian vector bundle S on Y togetherwith a Clifford multiplication cl :
T Y Ñ End p S q . We denote a spin-c structure by s “ p S , cl q .Sections of S are called spinors . A spin-c connection on a spin-c structure s is a connection A S on S which is compatible with the Clifford multiplication, meaning that if v is a vectorfield on Y and ψ is a spinor, then ∇ A S p cl p v q ψ q “ cl p ∇ v q ψ ` cl p v q ∇ A S ψ, where ∇ v denotes the covariant derivative of v with respect to the Levi-Civita connection of g . Such a spin-c connection is equivalent to a Hermitian connection A on the determinantline bundle det p S q . The Dirac operator D A of A S is the composition C p Y ; S q ∇ AS Ñ C p Y ; T ˚ Y b S q cl Ñ C p Y ; S q , (7.3)where cl is the composition T ˚ Y g – T Y cl Ñ End p S q .Let η be an exact 2-form on Y . Let A be a connection on det p S q and Ψ be a spinor.Define a bundle map τ : S Ñ iT ˚ Y by τ p Ψ qp v q “ g p cl p v q Ψ , Ψ q . The Seiberg-Witten equations for p A , Ψ q with perturbation η are D A Ψ “ ˚ F A “ τ p Ψ q ` i ˚ η. (7.4)71eiberg-Witten Floer cohomology is generated by certain solutions p A , Ψ q to the (7.4). The gauge group G : “ C p Y ; S q acts on the set of all pairs p A , Ψ q by u ¨ p A ; Ψ q : “ p A ´ u ´ du, u Ψ q , and if p A , Ψ q is a solution to the Seiberg-Witten equations, so is u ¨ p A , Ψ q . Solutions are (gauge) equivalent if they are equivalent under the action of G . If η is generic then modulogauge equivalence there are only finitely many solutions with Ψ ı
0, and each is cut outtransversely. Such solutions are called irreducible (those with Ψ ” reducible ). Weassume η is sufficiently generic in this way.Denote by y CM ˚ irr the free Z -module generated by the irreducible solutions to the Seiberg-Witten equations, modulo gauge equivalence.We next describe the part of the Seiberg-Witten differential which maps y CM ˚ irr to itself.Let p A ˘ , Ψ ˘ q be solutions to the Seiberg-Witten equations. An instanton from p A ´ , Ψ ´ q to p A ` , Ψ ` q is a smooth one-parameter family p A p s q , Ψ p s qq parameterized by s P R , for which BB s Ψ p s q “ ´ D A p s q Ψ p s qBB s A p s q “ ´ ˚ F A p s q ` τ p Ψ p s qq ` i ˚ η (7.5)lim s Ñ˘8 p A p s q , Ψ p s qq “ p A ˘ , Ψ ˘ q . The gauge group and R both act on the space of instantons.If p A ˘ , Ψ ˘ q are irreducible, then the differential coefficient xBp A ` , Ψ ` q , p A ´ , Ψ ´ qy is acount of instantons from p A ´ , Ψ ´ q to p A ` , Ψ ` q , modulo the actions of G and R , living ina moduli space of local expected dimension one. This local expected dimension defines arelative Z { d p c p s qq grading on the chain complex, and the differential increases this gradingby one. More generally, so long as there is no moduli space of instantons to p A ` , Ψ ` q froma reducible solution to the Seiberg-Witten equations of local expected dimension one, then Bp A ` , Ψ ` q P y CM ˚ irr . For a discussion of further abstract perturbations necessary to obtainthe transversality required to fully define the differential, see [HT13, § Z by all solutions to theSeiberg-Witten equations, modulo gauge equivalence, whose differential extends the differ-ential from y CM ˚ irr to itself discussed above. We will not need to discuss this extension furtherhere, because the key to the proof of Theorem 7.1 is a filtered version of Seiberg-Witten Floercohomology whose generators are all irreducible, introduced in § z HM ˚ p Y, s ; g, η q . Because this homology is independent of the choices of p g, η q , we denotethe canonical isomorphism class of all such homologies by z HM ˚ p Y, s q .By s ξ, Γ we denote the spin-c structure s ξ ` P D p Γ q on Y , where s ξ is the spin-c structuredetermined by ξ as in [HT13, Example 2.1]. Taubes [T10I]-[T10IV] has shown ECH ˚ p Y, λ, Γ; J q – z HM ´˚ p Y, s ξ, Γ q ; (7.6)here we use the notation for ECH emphasizing the roles of λ and J , although inherent inthe result is the fact that both sides do not depend on λ or J but only on p Y, ξ, Γ q .72 .1.2 The contact form perturbation of the Seiberg-Witten equations If Y has a contact form λ , let J be an almost complex structure on ξ which extends to a λ -compatible almost complex structure on R ˆ Y . From λ and J we obtain a metric g forwhich g p R, R q “ g p R, ξ q “
0. In particular, on ξ , we have g p v, w q “ dλ p v, J w q . Anyspin-c structure s “ p S , cl q can be canoncially decomposed into eigenbundles of cl p λ q , i.e. S “ E ‘ ξE , where E is the i eigenbundle and concatenation denotes the tensor productof line bundles. In this decomposition, a connection A on det p S q “ ξE can be written A “ A ξ ` A for some connection A on E . Similarly to (7.3) we can define the Diracoperator D A of A .Let r ą µ be an exact 2-form satisfying the genericity conditions described in[HT13, § D A with D A and setting η “ ´ rdλ ` µ, ψ “ ? r Ψin (7.4) gives us the perturbed Seiberg-Witten equations for the pair p A, ψ q : D A Ψ “ ˚ F A “ r p τ p Ψ q ´ iλ q ´ ˚ F A ξ ` i ˚ µ. (7.7)Taking into account abstract perturbations as in [HT13, § y CM ˚ p Y, s ; λ, J, r q is generated by the solutions to (7.7), modulo gauge equivalence, and its homology is denoted z HM ˚ p Y, s ; λ, J, r q . As in the original chain complex, if p A ˘ , ψ ˘ q are irreducible solutions,then xBp A ` , ψ ` q , p A ´ , ψ ´ qy is a count (modulo the actions of G and R ) of solutions to theperturbed instanton equations BB s ψ p s q “ ´ D A p s q ψ p s qBB s A p s q “ ´ ˚ F A p s q ` r p τ p ψ p s qq ´ iλ q ´ ˚ F A ξ ` i ˚ µ (7.8)lim s Ñ˘8 p A p s q , ψ p s qq “ p A ˘ , ψ ˘ q , which live in a moduli space of local expected dimension one. Again we denote by y CM ˚ irr thecomponent of y CM ˚ p Y, s ; λ, J, r q generated by irreducible solutions to (7.7). Although thenotation y CM ˚ irr denotes two subcomplexes, it will be clear from context which it denotes. z HM ˚ Analogous to the action of Reeb orbit sets, the energy of a solution p A, ψ q to the perturbedSeiberg-Witten equations (7.7) is defined as E p A q : “ i ż Y λ ^ F A . In analogy to
ECH L ˚ , for L ą
0, define y CM ˚ L to be the submodule of y CM ˚ irr generated bythe irreducible solutions to (7.7) with E p A q ă πL .73ote that the energy of a reducible solution p A, q to (7.7) is a linear increasing functionin r , so if r is sufficiently large then the condition that elements of y CM ˚ L be elements of y CM ˚ irr is redundant: if E p A q ă πL then if r is large enough, the pair p A, q cannot be asolution to (7.7).We quote a lemma necessary for defining the homology of the submodule y CM ˚ L : Lemma 7.2 ([HT13, Lem. 2.3]) . Fix
Y, λ, J as above and L P R . Suppose that λ has noorbit set of action exactly L . Fix r sufficiently large, and a 2-form µ so that all irreduciblesolutions to (7.7) are cut out transversely. Then for every s and for every sufficiently smallgeneric abstract perturbation, y CM ˚ L p Y, s ; λ, J, r q is a subcomplex of y CM ˚ p Y, s ; λ, J, r q . When the hypotheses of Lemma 7.2 apply, we denote the homology of y CM ˚ L p Y, s ; λ, J, r q by z HM ˚ L p Y, λ, s q . In particular, if r is sufficiently large then this homology is independentof µ and r , and it is also independent of J , as shown in [HT13, Cor. 3.5]. We will use thenotation z HM ˚ L p Y ; λ, J, r q when we wish to emphasize the roles of J and r .Filtered Seiberg-Witten Floer cohomology is isomorphic to ECH: Lemma 7.3 ([HT13, Lem. 3.7]) . Suppose that λ is L -nondegenerate and J is ECH L -generic(see Lemma 2.16). Then for all Γ P H p Y q , there is a canonical isomorphism of relativelygraded Z -modules Ψ L : ECH L ˚ p Y, λ, Γ; J q – Ñ z HM ´˚ L p Y, λ, s ξ, Γ q . (7.9)Analogous to the cobordism maps on ECH L ˚ , there are cobordism maps on z HM ˚ L . Thefollowing is a modified version of [HT13, Cor. 5.3 (a)] which keeps track of the spin-cstructures in our setting. Note that therefore our notation for the cobordism maps on z HM ˚ L differs slightly from that of [HT13]. Lemma 7.4.
Let p X, λ q be an exact symplectic cobordism from p Y ` , λ ` q to p Y ´ , λ ´ q where λ ˘ is L -nondegenerate. Let s be a spin-c structure on X and let s ˘ denote its restrictionsto Y ˘ , respectively. Let J ˘ be λ ˘ -compatible almost complex structures. Suppose r is suffi-ciently large. Fix 2-forms µ ˘ and small abstract perturbations sufficient to define the chaincomplexes y CM ˚ p Y ˘ , s ˘ ; λ ˘ , J ˘ , r q . Then there is a well-defined map z HM ˚ L p X, λ, s q : z HM ˚ L p Y ` , s ` ; λ ` , J ` , r q Ñ z HM ˚ L p Y ´ , s ´ ; λ ´ , J ´ , r q , (7.10) depending only on X, s , λ, L, r, J ˘ , µ ˘ , and the perturbations, such that if L ă L and if λ ˘ are also L -nondegenerate, then the diagram z HM ˚ L p Y ` , s ` ; λ ` , J ` , r q z HM ˚ L p X,λ, s q (cid:47) (cid:47) (cid:15) (cid:15) z HM ˚ L p Y ´ , s ´ ; λ ´ , J ´ , r q (cid:15) (cid:15) z HM ˚ L p Y ` , s ` ; λ ` , J ` , r q z HM ˚ L p X,λ, s q (cid:47) (cid:47) z HM ˚ L p Y ´ , s ´ ; λ ´ , J ´ , r q (7.11) commutes, where the vertical arrows are induced by inclusions of chain complexes. z HM ˚ L . For certain cobordisms (e.g. those defining the direct system lim ε Ñ z HM ´˚ L p ε q p Y, λ ε , s ξ, Γ q it is enough to use the composition property [HT13, Lem. 3.4 (b)], but we will need to un-derstand cobordism maps on slightly more complex cobordisms as well. The next lemma isa version of [HT13, Prop. 5.4] explaining the composition law for z HM ˚ L in our setting.In the following lemma, we will consider the following composition. Assume ε ă ε ă ε .We consider the exact symplectic cobordism pr ε , ε sˆ Y, p ` s p ˚ H q λ q from p Y, λ ε q to p Y, λ ε q .It is the composition of the exact symplectic cobordism pr ε , ε sˆ Y, p ` s p ˚ H q λ q from p Y, λ ε q to p Y, λ ε q with the exact symplectic cobordism pr ε , ε sˆ Y, p ` s p ˚ H q λ q from p Y, λ ε q to p Y, λ ε q in the sense of [HT13, § λ ε , λ ε , and λ ε are L -nondegenerate. We also assume J, J ,and J are λ ε -, λ ε -, and λ ε -compatible almost complex structures, respectively. Further,we choose a spin-c structure s on r ε , ε s ˆ Y which restricts to spin-c structures s and s on r ε , ε s ˆ Y and r ε , ε s ˆ Y , respectively, where s restricts to s on t ε u ˆ Y , s restrictsto s on t ε u ˆ Y , and both s and s restrict to s on t ε u ˆ Y . Finally we choose abstractperturbations and r large enough to define the chain complexes y CM ˚ L . Lemma 7.5.
The maps of Lemma 7.4 for the above data satisfy z HM ˚ L pr ε , ε sˆ Y, p ` s p ˚ H q λ, s q “ z HM ˚ L pr ε , ε sˆ Y, p ` s p ˚ H q λ, s ` q˝ z HM ˚ L pr ε , ε sˆ Y, p ` s p ˚ H q λ, s ´ q . Note that [HT13, Prop. 5.4] does not discuss the spin-c structures, but since it is provedwith a neck-stretching argument for holomorphic curves whose ends must be homologous, itwill preserve spin-c structures in the case considered in Lemma 7.5, see [HT13, Rmk. 1.10].
ECH ˚ via z HM ˚ L In this section we prove Theorem 7.1 using the machinery from Seiberg-Witten theory re-viewed in the previous sections.
Proof of Theorem 7.1.
Because all λ ε have the same contact structure ξ as λ , we havelim L Ñ8 ECH L ˚ p Y, λ ε , Γ q “ ECH ˚ p Y, ξ, Γ q . However, if ε ą ε p L q then we cannot compute ECH L ˚ p Y, λ ε , Γ q using our methods, becausethe chain complex ECC L ˚ p Y, λ ε , Γ; J q may contain orbits which do not project to criticalpoints of H . If ε is fixed and only L is sent to , then because ε p L q „ L , there will be some L beyond which ε ą ε p L q and we can no longer compute ECH L ˚ p Y, λ ε , Γ q .Instead, we will explain how to obtain ECH ˚ p Y, ξ, Γ q from lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q .Let L p ε q denote the value of L for which ε p L q “ ε . Note that for all L ă L p ε q , the generatorsof ECH L ˚ p Y, λ ε , Γ q all project to critical points of H . In particular, ECH L p ε q˚ p Y, λ ε , Γ q “ ECH L ˚ p Y, λ ε p L q , Γ q and therefore lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q “ lim ε Ñ ECH L p ε q˚ p Y, λ ε , Γ q .
75o prove Theorem 7.1 we will prove the following sequence of isomorphisms:lim ε Ñ ECH L p ε q˚ p Y, λ ε , Γ q – lim ε Ñ z HM ´˚ L p ε q p Y, λ ε , s ξ, Γ q (7.12) – lim ε Ñ lim L Ñ8 z HM ´˚ L p Y, λ ε , s ξ, Γ q (7.13) – lim ε Ñ z HM ´˚ p Y, s ξ, Γ q (7.14) – ECH ˚ p Y, ξ, Γ q . (7.15)Note that the groups z HM ´˚ L p Y, λ ε , s ξ, Γ q on the right hand side of the second equation (7.13)are only defined for L and ε such that λ ε has no Reeb orbit sets of action exactly L . Thisincludes all L ď L p ε q (similarly all ε ă ε p L q ); for a given ε , this is still a full measure set of L because for generic perfect H , the set of actions of orbits of X H is discrete.The direct limit on the right hand side of the first equation (7.12) is defined using eithercomposition in the commutative diagram (7.11) and the exact symplectic cobordisms pr ε , ε sˆ Y, p ` s p ˚ H q λ q discussed in the proof of Proposition 3.2 in § L on the right hand side of the second equation (7.13) is defined usingthe maps induced on homology by the inclusion of chain complexes. The direct limit in ε is defined using the cobordism maps (7.10) and the same exact symplectic cobordisms as inthe above paragraph.To obtain (7.13), we first show that the “obvious” map is well-defined: F : lim ε Ñ z HM ´˚ L p ε q p Y, λ ε , s ξ, Γ q Ñ lim ε Ñ lim L Ñ8 z HM ´˚ L p Y, λ ε , s ξ, Γ q . Note that every element of the direct system on the left hand side appears on the right handside; the map F is given by sending the equivalence class of a P z HM ´˚ L p ε q p Y, λ ε , s ξ, Γ q as amember of the left hand direct limit to its equivalence class as a member of the right handdirect limit. Throughout the proof of (7.13) we will use the notation z HM ´˚ L p ε q : “ z HM ´˚ L p Y, λ ε , s ξ, Γ q . To show that F is well-defined, assume a P z HM ´˚ L p ε q p ε q , b P z HM ´˚ L p ε q p ε q , and a „ b aselements of lim (cid:15) Ñ z HM ´˚ L p ε q p ε q . That means there is some ε for which the image of a underthe map z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q p ε q (7.16)equals the image of b under the map z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q p ε q . Call this shared image c . On the right hand side, we also have F p a q „ F p c q under thecomposition z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q p ε q , (7.17)76here the first map comes from the first direct limit (and is defined because ε ă ε ) and thesecond from the second, because (7.17) is precisely one of the compositions defining (7.16)in the commutative diagram (7.11). Similarly, we have F p b q „ F p c q .Next we show that F is an injection. Let a P z HM ´˚ L p ε q p ε q , b P z HM ´˚ L p ε q p ε q , but we do notassume a „ b on the left hand side. Assume F p a q „ F p b q on the right hand side. We wantto show a „ b . Because F p a q „ F p b q , there are L ě L p ε q , L p ε q and ε ď ε, ε for which theimage of a under z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q Ñ z HM ´˚ L p ε q equals the image of b under z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q Ñ z HM ´˚ L p ε q . Call this shared image d . Let ε “ min t ε , ε p L qu . We have d “ F p c q , where c is the imageof a under z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q p ε q , as well as the image of b under z HM ´˚ L p ε q p ε q Ñ z HM ´˚ L p ε q p ε q , both again by the definition (7.11) of the maps in the direct limit on the left hand side.Therefore a „ b on the left hand side.Finally we show that F is a surjection, essentially because the direct systems on theright hand side of (7.13) are very simple. If d P z HM ´˚ L p ε q with L ă L p ε q , then it iseventually equivalent to some element of z HM ´˚ L p ε q p ε q because there is an inclusion map sending z HM ´˚ L p ε q to z HM ´˚ L p ε q p ε q . If d P z HM ´˚ L p ε q with L ą L p ε q , then it is eventually equivalentto some element of z HM ´˚ L p ε p L qq because there is a cobordism map sending z HM ´˚ L p ε q to z HM ´˚ L p ε p L qq .We have that (7.14) follows from the last equation of [HT13, § § λ , it is true for all λ .We obtain (7.15) by the fact that the groups z HM ´˚ p Y, s ξ, Γ q on the right hand side of(7.14) are all equal and independent of ε , together with the isomorphism (7.6). We split the proof of the first two conclusions of Theorem 1.1 into the case where g ą g “
0. This is because our methods for g ą g “
0: see Remark 4.10. Instead, because a perfectMorse function on S has only elliptic critical points, the differential vanishes by index parity(Theorem 2.9), as explained in § .2.1 Proof of the main theorem when g ą g ą Proof of Theorem 1.1, assuming g ą . By Theorem 7.1 and the discussion in the introduc-tion to §
7, it is enough to show (7.2). By Proposition 3.2, the direct limit on the lefthand side of (7.2) is the homology of the chain complex generated by orbit sets of the form e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` in the class Γ. We will denote this chain complex by p C ˚ , Bq .Recall that we are abusing notation by thinking of Γ as an element of Z ´ e rather thanas an element of the Z ´ e summand of H p Y q . Therefore e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` is in the classΓ precisely when M “ m ´ ` m ` ¨ ¨ ¨ ` m g ` m ` “ Γ ` p´ e q d for some m P Z ě .We claim that the chain complex C ˚ splits into the submodules Λ Γ `p´ e q d H ˚ p Σ g ; Z q onthe right hand side of (7.2). Let E ´ denote the index zero generator of H ˚ p Σ g ; Z q , let H i denote the i th index one generator, and let E ` denote the index two generator. If A is a generator of H ˚ p Σ g ; Z q let A m denote the m -fold wedge product A ^ ¨ ¨ ¨ ^ A , where m “ A in the wedge product. The Z grading on E m ´ ´ ^ H m ^ ¨ ¨ ¨ ^ H m g g ^ E m ` ` is the Z equivalence class of m ´ | E ´ | ` m | H | ` ¨ ¨ ¨ ` m g | H g | ` m ` | E ` | ” ` m ` ¨ ¨ ¨ ` m g ` m ` ” m ` ¨ ¨ ¨ ` m g ” I p e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` , e Γ ´ q (7.18)where (7.18) follows from the Index Parity property of the ECH index, see Theorem 2.9 (iv).Moreover, the exterior product Λ Γ `p´ e q d H ˚ p Σ g ; Z q consists precisely of wedge products ofthe E ˘ and H i of total multiplicity Γ ` p´ e q d , where for all A, B generators of H ˚ p Σ g ; Z q , B ^ A “ p´ q | A |¨| B | A ^ B. Therefore all elements of the exterior product can be rearranged so that all E ´ terms occurfirst and all E ` terms occur last. However, no H i term can occur twice, and because we areusing Z coefficients, H j ^ H i “ p´ q ¨ H i ^ H j “ p´ q H i ^ H j “ H i ^ H j , hence all H i terms can be arranged in the order of ascending index. Therefore the map e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` Ø E m ´ ´ ^ H m ^ ¨ ¨ ¨ ^ H m g g ^ E m ` ` defines a bijection C ˚ Ø à d P Z ě Λ Γ `p´ e q d H ˚ p Σ g ; Z q , which respects the splitting over d in the sense that there is a splitting of C ˚ over d suchthat M “ Γ ` p´ e q d , and respecting the mod two gradings.Finally, we show that the differential vanishes, so that these submodules are in factsubcomplexes, and (7.2) holds. By Corollary 6.2 we can assume that all all ECH indexone curves contributing to B have degree zero and thus are cylinders by Lemma 4.6. ByProposition 4.7, we know that any moduli space of cylinders which could contribute a nonzero78oefficient to B must have the same mod two count as the space of currents consisting oftrivial cylinders together with a single ECH index one cylinder above a gradient flow line of H which contributes to the Morse differential, and that the count must be the same as thecorresponding count for the Morse differential. Specifically, if there is a pseudoholomorphiccurrent contributing to xBp e m ´ ´ h m ¨ ¨ ¨ h m g g e m ` ` q , e m ´ h m ¨ ¨ ¨ h m g g e m ` y then either • There is some h i for which m i “ , m i “
0, and m “ m ´ ` , m “ m ` , and for j ‰ i , m j “ m j . • There is some h i for which m i “ , m i “
1, and m “ m ` ´ , m “ m ´ , and for j ‰ i , m j “ m j .In either case, the mod two count of all such pseudoholomorphic currents must equal thecount of Morse flow lines from p p h i q to p p e ´ q or p p e ` q to p p h i q , respectively. Because H isperfect, both of the latter counts are zero. g “ g “
0. Notethat the manifolds in question are the lens spaces L p´ e, q . Theorem 7.6.
Let p Y, λ q be the prequantization bundle of Euler number e P Z ă over S with contact structure ξ . Let Γ be a class in H p Y ; Z q “ Z ´ e . Then à Γ P H p Y q ECH ˚ p Y, ξ, Γ q – Λ ˚ H ˚ p S ; Z q (7.19) as Z -graded Z -modules, which proves Theorem 1.1 when g “ . Furthermore, ECH ˚ p Y, ξ, Γ q “ Z if ˚ P Z ě else (7.20) Proof.
By Theorem 7.1, it is enough to understand each lim L Ñ8 ECH L ˚ p Y, λ ε p L q , Γ q , which,by Proposition 3.2, is the homology of the chain complex generated by orbit sets of the form e m ´ ´ e m ` ` in the class Γ. We will denote this chain complex by p C ˚ , Bq .Because a perfect Morse function on S has only elliptic critical points, index parity(Theorem 2.9) tells us that B “
0. Therefore
ECH ˚ p Y, ξ, Γ q – C ˚ (7.21)Recall that we are abusing notation by thinking of Γ as an element of Z ´ e rather than as anelement of H p Y q . The generator e m ´ ´ e m ` ` is therefore in the class Γ if m ´ ` m ` “ Γ ` p´ e q d for some d P Z ě .If E ´ , E ` denote the grading zero and two homology classes in H ˚ p S ; Z q , respectively,then Λ δ H ˚ p S ; Z q is the group generated over Z by terms E m ´ ´ E m ` ` with m ´ ` m ` “ δ ,79here E m denotes the m -fold wedge product E ^ ¨ ¨ ¨ ^ E . By a simplification of the proofof the analogous fact in § C ˚ – à d P Z ě Λ Γ `p´ e q d H ˚ p S ; Z q as Z -graded Z -modules. Invoking (7.21) and taking the sum over all Γ P H p Y q proves(7.19).To prove the improvement (7.20), we will prove that the ECH index is a bijection fromthe generators of p C ˚ , Bq to 2 Z ě which sends e Γ ´ to zero. Therefore C ˚ “ Z if ˚ P Z ě e m ´ ´ e m ` ` to 2 Z ě . Our perspective is similar to that of Choi [Ch],but the contact forms are not the same (by choosing a specific perturbation function H wecould make them essentially the same, but do not need to do so).The index bijection factors through a bijection to a lattice in the fourth quadrant in R determined by the vertical axis and the line through the origin of slope ´ e . We willfirst describe the bijection between generators of C ˚ and this lattice, and then describe thebijection between the lattice and the nonnegative even integers.The generators of C ˚ in the class Γ are of the form e m ´ ´ e m ` ` , where m ˘ P Z ě and m ´ ` m ` ” ´ e Γ. See Figure 7.1. The union of all such generators over Γ P Z ´ e are inbijection with the intersection of the lattice spanned by p , q and ` , ´ e ˘ with the fourthquadrant determined by the vertical axis and the line through the origin of slope ´ e , wherethe bijection is given by e m ´ ´ e m ` ` ÞÑ ˆ m ´ , m ´ ´ m ` ´ e ˙ . The image of e m ´ ´ e m ` ` is to the right of the vertical axis, inclusive, because m ´ ě
0, and isbelow the line through the origin of slope ´ e , inclusive, because m ´ ´ m ` ´ e ď m ´ ´ e . The map is a bijection because it has an inverse, which can be computed directly from theformula. Let V p m ´ , m ` q denote ` m ´ , m ´ ´ m ` ´ e ˘ .The lattice splits into ´ e sublattices, each corresponding to the homology class Γ. Eachis spanned by ` , ´ e ˘ and p , q , but they can be differentiated by a translation: they containthe points ` , ´ Γ ´ e ˘ , respectively.Next we explain the bijections between each of these sublattices and the nonnegative evenintegers. The essential idea is the following. Let T p m ´ , m ` q denote the triangle boundedby the axes and the line through V p m ´ , m ` q of slope ´ e . The relative first Chern classrecords the approximate height of T p m ´ , m ` q , the relative self intersection term recordsapproximately twice its area; when this line moves to the right and/or down, both the80 p , q V p , q V p , q V p , q V p , q V p , q Figure 7.1: Depicted is the lattice for e “ ´
3. The thicker solid lines indicate the axeswhile the dashed lines indicate the grid spanned by the standard lattice generated by p , q and p , q . The Γ “ “ “ T p , q “ T p , q “ T p , q “ T p , q “ T p , q .height and area of T p m ´ , m ` q increase, so moving the line to the right and down groupspoints in the lattice into batches of roughly increasing ECH index. The Conley-Zehnderindex differentiates between lattice points on the same line of slope ´ e by increasing theindex by two as m increases and m decreases, and makes the indices of the groups ondifferent lines of slope ´ e match up exactly.Because these correspondences are only approximate when Γ ‰
0, we will explain themin detail to show that the index is a bijection to 2 Z ě .The index difference between e m ´ ´ e m ` ` and e Γ ´ is I p e m ´ ´ e m ` ` , e Γ ´ q “ p m ´ ` m ` ´ Γ q´ e ` p m ´ ` m ` ´ Γ q ´ e ` p m ´ ` m ` ´ Γ q´ e ` m ` ´ m ´ ` Γ . The first term is the relative first Chern class c τ p e m ´ ´ e m ` ` , e Γ ´ q . The triangle has vertices p , q , ` ´ m ´ ` m ` ´ e , ˘ , ` m ´ ` m ` , m ´ ` m ` ´ e ˘ , so its height is2 p m ´ ` m ` q´ e “ c τ p e m ´ ´ e m ` ` , e Γ ´ q ` ´ e ô c τ p e m ´ ´ e m ` ` , e Γ ´ q “ Height p T p m ´ , m ` qq ´ ´ e . The second two terms comprise the relative self intersection number Q τ p e m ´ ´ e m ` ` , e Γ ´ q .Twice the area of the triangle is2Area p T p m ´ , m ` qq “ ˆ m ´ ` m ` ´ e ˙ p m ´ ` m ` q , Q τ p e m ´ ´ e m ` ` , e Γ ´ q “ p m ´ ` m ` ´ Γ q ´ e ` p m ´ ` m ` ´ Γ q´ e “ ´ e ` p m ´ ` m ` q ´ p m ´ ` m ` q ` Γ ` p m ´ ` m ` q ´ ˘ “ p T p m ´ , m ` qq ´ ´ e . Notice that we can split the sublattices into lattices along the lines of slope ´ e through ` ´ M ` Γ ´ e , ˘ , where M P Z ě . Over each such line, the Conley-Zehnder term ranges from ´ M ` Γ to M ` Γ, where M “ m ´ ` m ` , and is strictly increasing in m ` . Since there isexactly one generator with each value of m ` between zero and m ´ ` m ` on this line, novalues of I p e m ´ ´ e m ` ` , e Γ ´ q are repeated on a given line.Along each line, the triangle T p m ´ , m ` q is constant, therefore its height and area areconstant, and thus both c τ p e m ´ ´ e m ` ` , e Γ ´ q and Q τ p e m ´ ´ e m ` ` , e Γ ´ q are constant. They are alsoboth increasing in M . In order to prove the theorem it therefore suffices to show that thesmallest value the index takes on the line corresponding to M ` p´ e q must be two greaterthan the largest value the index takes on the line corresponding to M .The smallest value the index takes on the line corresponding to M `p´ e q is I p e M `p´ e q´ , e Γ ´ q ,while while the largest value the index takes on the line corresponding to M is I p e M ` , e Γ ´ q . Itis a straightforward computation to show that I p e M `p´ e q´ , e Γ ´ q “ I p e M ` , e Γ ´ q ` References [ABW10] P. Albers, B. Bramham, and C. Wendl,
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Jo NelsonRice University email: [email protected]
Morgan WeilerRice University email: [email protected]@rice.edu