Miguel Abreu

Jean Gutt
email: gutt@math.uni-koeln.de

AND Jungsoo Kang
email: jungsoo.kang@snu.ac.kr

Leonardo Macarini
email: leomacarini@gmail.com

TWO CLOSED ORBITS FOR NON-DEGENERATE REEB FLOWS

Keywords: 2010 Mathematics Subject Classification. 53D40, 53D25, 37J10, 37J55 Closed orbits, Conley-Zehnder index, Reeb flows, equivariant symplectic homology

published or not. The documents may come

Introduction

A contact manifold is an odd-dimensional manifold M 2n`1 endowed with a contact structure, i.e. a codimension one distribution ξ having a maximal non-integrability property. If we write locally the distribution as the kernel of a 1-form, ξ " Ker α, the condition is that α ^pdαq n is nowhere vanishing; such a 1-form α is called a contact form. Throughout this paper we will tacitly assume that a contact structure is co-oriented, that is, α is defined globally. Associated to a contact form α we have the corresponding Reeb vector field R α uniquely characterized by the equations ι Rα dα " 0 and αpR α q " 1.

A central question in contact geometry is the Weinstein conjecture which states that every contact form on a closed manifold carries at least one closed Reeb orbit. The Weinstein conjecture was proved in dimension three by Taubes in 2007 [START_REF] Taubes | The Seiberg-Witten equations and the Weinstein conjecture[END_REF]. Taubes' result was later improved by Cristofaro-Gardiner and Hutchings [START_REF] Cristofaro-Gardiner | From one Reeb orbit to two[END_REF] who proved that every contact form on a closed manifold of dimension three carries at least two simple closed Reeb orbits (this result for the tight sphere S 3 was proved independently in [START_REF] Ginzburg | Closed Reeb orbits on the sphere and symplectically degenerate maxima[END_REF]; see also [START_REF] Liu | The existence of two closed characteristics on every compact star-shaped hypersurface in R 4[END_REF] for another alternative proof using a result from [START_REF] Ginzburg | Closed Reeb orbits on the sphere and symplectically degenerate maxima[END_REF]). Unfortunately, these results hold only in dimension three and the situation in higher dimensions is wide open.

Suppose that M admits a strong symplectic filling pW, ωq such that c 1 pT W q " 0. For the sake of simplicity, assume, for a moment, that W is a Liouville domain, that is, ω " dλ is exact and the dual vector field of λ with respect to ω points outwards along BW . Then, one can associate to W its positive S 1 -equivariant symplectic homology HC ˚pW q with an integer grading. It was introduced by Viterbo [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF] and developed by Bourgeois and Oancea [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF][START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF][START_REF] Bourgeois | The Gysin exact sequence for S1-equivariant symplectic homology[END_REF][START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]. (For the sake of brevity and to emphasize the analogy with contact homology, we denote this homology by HC ˚pW q rather than the usual notation SH `,S 1 ˚pW q used in [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF][START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF][START_REF] Bourgeois | The Gysin exact sequence for S1-equivariant symplectic homology[END_REF][START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF].) The positive equivariant symplectic homology has a filtration given by the free homotopy classes of loops in W and given a set Γ of free homotopy classes of loops we denote by HC Γ ˚pW q the corresponding homology. Given a non-degenerate contact form α on M , it turns out that HC Γ ˚pW q can be obtained as the homology of a chain complex generated by the good closed Reeb orbits of α with free homotopy class in Γ; see [START_REF] Ginzburg | Lusternik-Schnirelmann theory and closed Reeb orbits[END_REF][START_REF] Ginzburg | Multiplicity of Closed Reeb Orbits on Prequantization Bundles[END_REF][START_REF] Gutt | The positive equivariant symplectic homology as an invariant for some contact manifolds[END_REF][START_REF] Gutt | Symplectically knotted codimension-zero embeddings of domains in R 4[END_REF]. In particular, the Weinstein conjecture holds whenever HC ˚pW q does not vanish.

Assuming that HC ˚pW q does not vanish, it is natural to ask if we can get more than one closed orbit. This is the multiplicity problem for closed Reeb orbits and the fundamental difficulty here is how to distinguish simple orbits from iterated ones. This difficulty already manifests itself in the classical problem of the multiplicity of closed geodesics on Riemannian manifolds; c.f. [START_REF] Klingenberg | Lectures on closed geodesics[END_REF].

If HC ˚pW q is asymptotically unbounded, that is, there exists a sequence k i Ñ 8 such that lim iÑ8 dim HC k i pW q " 8 then it was proved in [START_REF] Hryniewicz | Local contact homology and applications[END_REF][START_REF] Mclean | Local Floer homology and infinitely many simple Reeb orbits[END_REF] that every Reeb flow on M carries infinitely many simple closed orbits. (These results were inspired by the corresponding result for geodesic flows due to Gromoll and Meyer [START_REF] Gromoll | Periodic geodesics on compact riemannian manifolds[END_REF].) Thus, it is natural to ask what happens when HC ˚pW q is not asymptotically unbounded.

In this case, the situation is more delicate since we do have several examples of contact forms with finitely many simple closed orbits. Some works provide lower bounds for the number of simple closed orbits (some of these estimates are sharp), but, in general, these bounds request certain index conditions on the closed Reeb orbits; see [START_REF] Abreu | Multiplicity of periodic orbits for dynamically convex contact forms[END_REF][START_REF] Duan | Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in R 2n[END_REF][START_REF] Ginzburg | Lusternik-Schnirelmann theory and closed Reeb orbits[END_REF][START_REF] Ginzburg | Multiplicity of Closed Reeb Orbits on Prequantization Bundles[END_REF][START_REF] Gutt | On the minimal number of periodic orbits on some hypersurfaces in R 2n[END_REF] and references therein.

The main result in this work establishes that a contact form α on M has at least two simple closed orbits provided that HC ˚pW q satisfies a certain mild condition and assuming only the non-degeneracy of α. More precisely, we have the following statement. Throughout this paper, we will use the convention that the natural numbers are given by the positive integers. Moreover, the grading of HC ˚pW q is given by the Conley-Zehnder index. Under the assumption that c 1 pT W q " 0, this grading is well defined up to a choice of the homotopy class of a non-vanishing section of the determinant line bundle Λ n`1 C T W ; see [START_REF] Abreu | On the mean Euler characteristic of Gorenstein toric contact manifolds[END_REF]Section 3], [START_REF] Espina | On the mean Euler characteristic of contact manifolds[END_REF], [START_REF] Mclean | Reeb orbits and the minimal discrepancy of an isolated singularity[END_REF] and [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]Appendix C]. For a shorter notation, we will omit this dependence.

Theorem 1.1. Let pM 2n`1 , ξq be a closed contact manifold admitting a strong symplectic filling W such that c 1 pT W q " 0. Let Γ be a set of free homotopy classes of loops in W closed under iterations and assume that there exist K P N and a non-vanishing section σ of the determinant line bundle Λ n`1

C T W such that dim HC Γ n pW q ă dim HC Γ n`jK pW q or dim HC Γ ´npW q ă dim HC Γ ´n´jK pW q for every j P N, where the grading in HC Γ ˚pW q is taken with respect to the homotopy class of σ. Then every non-degenerate Reeb flow on M carries either infinitely many geometrically distinct closed Reeb orbits or at least two geometrically distinct closed Reeb orbits γ 1 and γ 2 such that their Conley-Zehnder indices satisfy µpγ k 1 q ‰ µpγ k 2 q for some k P N. Moreover, all these orbits have free homotopy class in Γ.

Remark 1.2. When Γ " t0u, i.e. when looking at contractible orbits, it is enough to assume that c 1 pT W q| π 2 pW q " 0. Moreover, in this case the grading of HC Γ ˚pW q does not depend on the choice of the homotopy class of σ.

Remark 1.3. Note that we are not assuming that W is a Liouville domain. When ω is not exact, HC Γ ˚pW q is a vector space over the universal Novikov field and we have to use an action filtration introduced by McLean and Ritter in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]; see Section 2. Using this action filtration we also have that, given a non-degenerate contact form α on M , HC Γ ˚pW q is the homology of a chain complex generated by the good closed Reeb orbits of α with free homotopy class in Γ.

Let M τ Ý Ñ Ď
M be a finite covering. Suppose that M carries a contact structure ξ preserved by the deck transformations so that Ď M has an induced contact structure s ξ such that τ ˚s ξ " ξ. We say in this case that Ď M is a contact finite quotient of M . Assume that pM, ξq satisfies the hypotheses of Theorem 1.1 and that H 1 pM, Rq " 0. Given a non-degenerate contact form s α on Ď M we can consider its lift to M and applying Theorem 1.1 we can deduce that s α has at least two simple closed orbits; see Section 4. Thus, we have the following corollary.

Corollary 1.4. Let pM, ξq be a contact manifold satisfying the hypotheses of Theorem 1.1 and H 1 pM, Rq " 0. Then every non-degenerate Reeb flow on a contact finite quotient of M carries at least two simple closed orbits. Moreover, the closed lifts of iterates of these orbits to M have free homotopy class in W contained in Γ.

Remark 1.5. The hypothesis that H 1 pM, Rq " 0 can be dropped if the first Chern class of the contact structure on the quotient vanishes; see Remark 4.1.

Remark 1.6. When Γ " t0u, the hypothesis that H 1 pM, Rq " 0 can be replaced by the assumption that the map π 1 pM q Ñ π 1 pW q induced by the inclusion is injective.

Remark 1.7. Since the deck transformations do not necessarily extend to the filling W , it is not clear that when Γ " t0u the assumption that c 1 pT W q| π 2 pW q " 0 is enough to conclude the previous corollary, unless the map π 1 pM q Ñ π 1 pW q induced by the inclusion is injective; c.f. Remark 1.

2.

In what follows, we will provide several examples of contact manifolds satisfying the hypotheses of Theorem 1.1. The simplest one is the standard contact sphere S 2n`1 with its obvious filling W given by the ball D 2n`2 whose positive equivariant symplectic homology is

HC j pW q - # Q if j " n `2k for some k P N 0 otherwise.
The existence of at least two simple closed orbits for every non-degenerate contact form on the standard contact sphere is well known; see, for instance, [START_REF] Gurel | Perfect Reeb flows and action-index relations[END_REF][START_REF] Kang | Equivariant symplectic homology and multiple closed Reeb orbits[END_REF]. We will furnish a very large class of examples, for which the sphere is a very particular case.

To the best of our knowledge, all the examples known so far of closed contact manifolds admitting contact forms with finitely many simple closed orbits are prequantization circle bundles over symplectic orbifolds admitting Hamiltonian circle actions with isolated fixed points. Our list below covers several examples of such manifolds; see Corollaries 1.17, 1.19 and 1.26. This raises the following question.

Question: Let M be a prequantization circle bundle over a closed symplectic orbifold admitting a Hamiltonian circle action with isolated fixed points. Suppose that M admits a strong symplectic filling W such that c 1 pT W q " 0. Is it true that W satisfies the hypothesis of Theorem 1.1 for some Γ and σ?

Note that there are examples of contact manifolds admitting fillings with vanishing first Chern class that do not satisfy the hypothesis of Theorem 1.1, at least for a naturally chosen non-vanishing section of the determinant line bundle. For instance, consider the cosphere bundle S ˚N of an orientable closed manifold admitting a Riemannian metric g with negative sectional curvature and take the symplectic filling D ˚N given by the unit disk bundle. We have that c 1 pT D ˚N q " 0 because N is orientable. Moreover, the choice of a volume form in the base naturally induces a non-vanishing section of Λ n`1 C T D ˚N so that the Conley-Zehnder index of every non-degenerate closed geodesic coincides with its Morse index; see [START_REF] Abouzaid | Symplectic cohomology and Viterbo's theorem[END_REF]Section 1.4.5]. Since the geodesic flow of g is non-degenerate and the index of every closed geodesic vanishes, we have that HC ˚pD ˚N q is non-trivial only in degree zero and therefore it does not satisfy the hypothesis of Theorem 1.1. However, every contact form on S ˚N supporting the standard contact structure has infinitely many simple closed orbits; see [START_REF] Macarini | Equivariant symplectic homology of Anosov contact structures[END_REF].

Displaceable contact manifolds.

Given a contact manifold pM, ξq and an exact symplectic manifold pX, dλq, an embedding M ãÑ X is called an exact contact embedding if it is bounding and there exists a contact form α supporting ξ such that α ´λ| M is exact. Here bounding means that M separates X into two connected components, with one of them relatively compact. This embedding is displaceable if M can be displaced from itself by a Hamiltonian diffeomorphism with compact support on X. We say that X is convex at infinity if there exists an exhaustion X " Y k X k by compact subsets X k Ă X k`1 with smooth boundaries such that λ| BX k is a contact form for every k. A big class of contact manifolds admitting displaceable exact contact embeddings in exact symplectic manifolds that are convex at infinity is given in [START_REF] Biran | Lagrangian embeddings into subcritical Stein manifolds[END_REF]: the boundary of every subcritical Stein manifold.

Let pM, ξq be a contact manifold admitting a displaceable exact contact embedding into a convex at infinity exact symplectic manifold X such that c 1 pT Xq| π 2 pXq " 0 and denote by W the compact region in X bounded by M . In Section 5 we show that W satisfies the hypothesis of Theorem 1.1 for Γ " t0u. Hence, we get the following result; see Remarks 1.2 and 1.7.

Corollary 1.8. Let pM, ξq be a contact manifold admitting a displaceable exact contact embedding into a convex at infinity exact symplectic manifold X such that c 1 pT Xq| π 2 pXq " 0 and denote by W the compact region in X bounded by M . Then every non-degenerate Reeb flow on M has at least two simple closed orbits contractible in W . If c 1 pT Xq " 0 and H 1 pM, Rq " 0 then every Reeb flow on a contact finite quotient of M carries at least two simple closed orbits. Moreover, the closed lifts of iterates of these orbits to M are contractible in W .

Cosphere bundles and closed geodesics.

Let N be a closed Riemannian manifold and ΛN its free loop space. There is an isomorphism between the (non-equivariant) symplectic homology of T ˚N and the homology of ΛN twisted by a local system of coefficients. For the S 1 -equivariant version, if N is orientable and spin, we have the isomorphism

(1.1) HC ˚pD ˚N q -H ˚pΛN {S 1 , N ; Qq,
where N Ă ΛN {S 1 indicates the subset of constant loops, D ˚N is the obvious filling of the cosphere bundle S ˚N given by the unit disk bundle and we are taking the grading of HC ˚pD ˚N q given by a non-vanishing section of Λ n`1 C T D ˚N induced from the choice of a volume form in the base so that the Conley-Zehnder index of a non-degenerate closed geodesic coincides with its Morse index, see e.g. [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]. This isomorphism respects the filtration given by the free homotopy classes, that is,

(1.2)
HC Γ ˚pD ˚N q -H ˚pΛ Γ N {S 1 , N ; Qq for every set Γ of free homotopy classes in D ˚N , where Λ Γ N denotes the set of loops in N with free homotopy class in Γ. (Note that, since π 1 pD ˚N q -π 1 pN q, the set of free homotopy classes in D ˚N and N are naturally identified. Moreover, N Ă Λ 0 N {S 1 and therefore if Γ does not contain the trivial free homotopy class then the right hand side of the isomorphism (1.2) has to be understood as H ˚pΛ Γ N {S 1 ; Qq.) For general N , it is expected that the same isomorphism holds with a local system of coefficients as in the non-equivariant case but a rigorous proof has not been written in the literature yet.

It turns out that if N is simply connected and H ˚pΛN {S 1 , N ; Qq is not asymptotically unbounded then it satisfies the assumption in Theorem 1.1. Using this, we can prove the following result. The proof is given in Section 6. Before we state it, let us recall a definition and introduce a notation. A topological space X is k-simple if π 1 pXq acts trivially on π k pXq. If a closed manifold N has dimension bigger than one and π 1 pN q -Z then N is not rationally aspherical, that is, there exists j ą 1 such that π j pN q b Q ‰ 0; see Section 6. Let k be the smallest such j. In what follows, ξ can denotes the canonical contact structure on S ˚N . Corollary 1.9. Let N be a closed oriented spin manifold with dimension bigger than one. Suppose that N satisfies one of the following conditions:

(i) π 1 pN q is finite;

(ii) π 1 pN q -Z, π 2 pN q " 0 and N is k-simple, with k as discussed above;

(iii) π 1 pN q is infinite and there is no a P π 1 pN q such that every non-zero b P π 1 pN q is conjugate to some power of a. In case (i), we have that every non-degenerate contact form on pS ˚N, ξ can q has at least two simple closed orbits. Under hypothesis (ii) or (iii), we have two simple closed orbits for any contact form on pS ˚N, ξ can q, without assuming that it is non-degenerate.

Remark 1.10. The hypothesis that N is oriented spin is used only to have the isomorphism (1.2). Possibly, it can be relaxed once we have this isomorphism with the relative homology of pΛN {S 1 , N q twisted by a local system of coefficients.

Remark 1.11. In case (ii), the hypothesis that N is k-simple can be relaxed in the following way: let a be a generator of π 1 pN q and denote by A the linear map corresponding to the action of a on π k pN q b Q. Then it is enough that kerpA ´Id q ‰ 0; see Remark 6.6. This hypothesis and the assumption that π 2 pN q " 0 are probably just technical but we do not know how to drop them so far. Theorem 1.1 is used to prove Corollary 1.9 only under hypothesis (i). For hypotheses (ii) and (iii), we show the existence of two periodic orbits γ 1 and γ 2 such that no iterate of γ 1 is freely homotopic to γ 2 . This is easy in case (iii) using (1.2) but highly non-trivial in case (ii) where we show that HC 0 ˚pD ˚N q ‰ 0 and HC a ˚pD ˚N q ‰ 0 for some non-trivial homotopy class a. The proof in case (ii) actually shows the following result; see Remark 6.8. It can be considered as a sort of Lusternik-Fet theorem for Reeb flows; see e.g. [START_REF] Asselle | The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles[END_REF].

Theorem 1.12. Let N be a closed not rationally aspherical manifold. Suppose that N is oriented spin, π 1 pN q is abelian, π 2 pN q " 0 and N is k-simple, with k as discussed above.

Then every (possibly degenerate) Reeb flow on S ˚N carries a contractible closed orbit. As a consequence, if, furthermore, π 1 pN q is infinite, then every Reeb flow on S ˚N has at least two simple closed orbits. Remark 1.13. Similarly to Corollary 1.9, the hypothesis that N is k-simple can be weakened; see Remark 6.8.

The hypothesis that N is oriented spin and the second and third conditions in item (ii) can be dropped when we restrict ourselves to Reeb flows given by geodesic flows of Finsler metrics as follows. The proof of item (i) in Theorem 1.1 uses only the fact that, given a non-degenerate contact form α on M , HC Γ ˚pW q is the homology of a chain complex generated by the good periodic orbits of α with free homotopy class in Γ graded by the Conley-Zehnder index; the nature of the differential is absolutely unessential. Let F be a Finsler metric on N . It is well known that the closed geodesics of F are the critical points of the corresponding energy functional defined on the free loop space. We will say that F has only one prime closed geodesic if either the corresponding geodesic flow has only one simple closed orbit or F is reversible (i.e. F px, vq " F px, ´vq for every px, vq P T N ) and its geodesic flow has only two simple periodic orbits (given by the lifts of a closed geodesic γptq and its reversed geodesic γp´tq).

It turns out that if F is bumpy (i.e. its geodesic flow is non-degenerate) and has only one prime closed geodesic then H ˚pΛN {S 1 , N ; Qq is the homology of the chain complex generated by the good periodic orbits of the geodesic flow of F with trivial differential. Using this fact we can prove the following result. In what follows, we say that F has at least two prime closed geodesics if it does not have only one prime closed geodesic in the sense above. (Note that every Finsler metric has at least one prime closed geodesic.) Corollary 1.14. Let N be a closed manifold with dimension bigger than one. Suppose that N satisfies one of the following conditions:

(i) π 1 pN q is finite;

(ii) π 1 pN q -Z;

(iii) π 1 pN q is infinite and there is no a P π 1 pN q such that every non-zero b P π 1 pN q is conjugate to some power of a. In case (i), we have that every bumpy Finsler metric F on N has at least two prime closed geodesics. Under hypothesis (ii) or (iii), we have two prime closed geodesics for any Finsler metric F on N , without assuming that it is bumpy. Remark 1.15. Our contribution in this corollary is that we find two closed geodesics when N has finite fundamental group and F is bumpy. The remaining cases can be covered by classical minimax methods. This is in contrast with Corollary 1.9 for which these minimax methods are not available, making the proof of item (ii) much harder than in the case of geodesic flows.

Remark 1.16. We are not aware of any example of N which is excluded in the statement, see [START_REF] Taimanov | The type numbers of closed geodesics[END_REF]Section 5]. For instance, if π 1 pN q is abelian, N meets the hypothesis in Corollary 1.14.

Good toric contact manifolds.

Toric contact manifolds are the odd dimensional analogues of toric symplectic manifolds. They can be defined as contact manifolds of dimension 2n `1 equipped with an effective Hamiltonian action of a torus of dimension n `1. Good toric contact manifolds of dimension three are pS 3 , ξ st q and its finite quotients. Good toric contact manifolds of dimension greater than three are compact toric contact manifolds whose torus action is not free. These form the most important class of compact toric contact manifolds and can be classified by the associated moment cones, in the same way that Delzant's theorem classifies compact toric symplectic manifolds by the associated moment polytopes. We refer to [START_REF] Lerman | Contact toric manifolds[END_REF] for details.

In [START_REF] Abreu | Contact homology of good toric contact manifolds[END_REF] we showed that on any good toric contact manifold pM 2n`1 , ξq such that c 1 pξq " 0 any non-degenerate toric contact form is even, that is, all contractible closed orbits of its Reeb flow have even contact homology degree, where the contact homology degree of a closed orbit γ is given by µpγq `n ´2. (As proved in [START_REF] Abreu | On the mean Euler characteristic of Gorenstein toric contact manifolds[END_REF], this is also true for the non-contractible closed Reeb orbits.) Suppose that M admits a symplectic filling W with vanishing first Chern class. Then, as showed in [START_REF] Abreu | Contact homology of good toric contact manifolds[END_REF][START_REF] Abreu | On the mean Euler characteristic of Gorenstein toric contact manifolds[END_REF], HC 0 ˚pW q can be computed in a purely combinatorial way in terms of the associated momentum cone. Using this computation, we show in Section 8 that W satisfies the hypothesis of Theorem 1.1 for Γ " t0u and consequently we get the following result. Note that the fundamental group of every good toric contact manifold M is finite and consequently H 1 pM, Rq " 0.

Corollary 1.17. Let pM, ξq be a good toric contact manifold admitting a strong symplectic filling W such that c 1 pT W q " 0. Then every non-degenerate contact form on a contact finite quotient of M carries at least two geometrically distinct contractible closed orbits.

Remark 1.18. It turns out that every good toric contact manifold pM, ξq in dimensions three and five such that c 1 pξq " 0 admits a (toric) filling with vanishing first Chern class [START_REF] Abreu | On the mean Euler characteristic of Gorenstein toric contact manifolds[END_REF].

Prequantization circle bundles over symplectic manifolds.

Let pB 2n , ωq be a closed integral symplectic manifold. Consider the prequantization circle bundle pM, ξq of pB, ωq, that is, the contact manifold given by the total space of a principle circle bundle over B whose first Chern class is ´rωs and with contact structure given by the kernel of a connection form. Suppose that M admits a symplectic filling W with vanishing first Chern class. Then, under some assumptions on B, we can show that W satisfies the hypothesis of Theorem 1.1 with Γ " t0u. More precisely, we have the following result. In what follows, c B :" inftk P N | DS P π 2 pBq with xc 1 pT Bq, Sy " ku denotes the minimal Chern number of B.

Corollary 1.19. Let pM, ξq be a prequantization circle bundle of a closed integral symplectic manifold pB, ωq such that ω| π 2 pBq ‰ 0, c 1 pT Bq| π 2 pBq ‰ 0 and, furthermore, H k pB; Qq " 0 for every odd k or c B ą n. Suppose that M admits a strong symplectic filling W such that c 1 pT W q " 0. Then every non-degenerate Reeb flow on M carries at least two geometrically distinct closed orbits contractible in W . If, additionally, H 1 pM, Rq " 0 then every contact form on a contact finite quotient of M carries at least two geometrically distinct closed orbits. Moreover, the closed lifts of iterates of these orbits to M are contractible in W . Remark 1.20. It follows from the Gysin exact sequence that H 1 pM, Rq " 0 whenever H 1 pB; Qq " 0.

Remark 1.21. When ω| π 2 pBq " 0 and B satisfies some extra few conditions (for instance, when π i pBq " 0 for every i ě 2) it is proved in [START_REF] Ginzburg | On the Conley conjecture for Reeb flows[END_REF] (c.f. [START_REF] Ginzburg | On the Filtered Symplectic Homology of Prequantization Bundles[END_REF]) that every Reeb flow on M (possibly degenerate) carries infinitely many simple closed orbits.

Remark 1.22. We have that H ˚pB; Qq vanishes in odd degrees and c 1 pT Bq| π 2 pBq ‰ 0 whenever B admits a Hamiltonian circle action with isolated fixed points.

The proof of the previous corollary is given in Section 9. Now, note that the prequantization bundle M has a natural symplectic filling W given by the corresponding disk bundle in the complex line bundle L π Ý Ñ B whose first Chern class is ´rωs. Suppose that B is monotone, that is, rωs " λc 1 pT Bq for some λ P R. (We say that B is positive monotone if λ ą 0.) One can check that c 1 pT W q " p1 ´λqπ ˚c1 pT Bq.

Consequently, when λ " 1 we have that c 1 pT W q " 0. Now, suppose that λ is an integer bigger than one and let Ă M be the prequantization bundle of pB, 1 λ ωq. It is easy to see that M is the finite quotient of Ă M by the Z λ -action induced by the obvious S 1 -action on Ă M . Thus, we have the following corollary; see Remark 1.20.

Corollary 1.23. Let pM, ξq be the prequantization circle bundle of a closed integral symplectic manifold pB, ωq such that ω| π 2 pBq ‰ 0, c 1 pT Bq| π 2 pBq ‰ 0 and, furthermore, H k pB; Qq " 0 for every odd k or c B ą n. Suppose that rωs " λc 1 pT Bq for some λ P N and that H 1 pB; Qq " 0. Then every contact form on a contact finite quotient of M carries at least two geometrically distinct closed orbits. Moreover, the closed lifts of iterates of these orbits to M have contractible projections to B.

Remark 1.24. If a closed curve γ in M has contractible projection to B then it must be homotopic to some iterate of the fiber and therefore, since ω| π 2 pBq ‰ 0, its homotopy class rγs P π 1 pM q is nilpotent, i.e. rγs k " 0 for some k.

Remark 1.25. Note that if c B " 1 then, since ω has to be integral, λ must be an integer. Therefore, rωs " λc 1 pT Bq for some λ P N whenever B is positive monotone and c B " 1.

Brieskorn spheres.

Given a " pa 0 , . . . , a n`1 q P N n`2 define Σ a as the intersection of the hypersurface

z a 0 0 `¨¨¨`z a n`1 n`1 " 0 in C n`2
with the unit sphere S 2n`3 Ă C n`2 . It is well known that α a " i 8 ř n`1 j"0 a j pz j dz j źj dz j q defines a contact form on Σ a and pΣ a , ξ a :" ker α a q is called a Brieskorn manifold. When n is even, a 0 " ˘1 mod 8 and a 1 " ¨¨¨" a n`1 " 2 we have that Σ a is diffeomorphic to the sphere S 2n`1 and called a Brieskorn sphere. Brieskorn spheres admit strong symplectic fillings given by Liouville domains W satisfying c 1 pT W q " 0 and it turns out that W satisfies the hypothesis of Theorem 1.1 with Γ " t0u; see Section 10. Therefore, we obtain the following result which is a generalization of [START_REF] Kang | Equivariant symplectic homology and multiple closed Reeb orbits[END_REF]Theorem C].

Corollary 1.26. Let M be a contact finite quotient of a Brieskorn sphere. Then every nondegenerate Reeb flow on M carries at least two geometrically distinct closed orbits.

Connected sums.

Let pW 1 , λ 1 q and pW 2 , λ 2 q be two Liouville domains of dimension 2n `2. The boundary connected sum of them is again a Liouville domain pW 1 #W 2 , λ 1 #λ 2 q and the contact connected sum pBW 1 #BW 2 , ξ 1 #ξ 2 q is the boundary of it. The following result establishes that the main hypothesis of Theorem 1.1 is preserved by boundary connected sums of Liouville domains, furnishing many other examples of contact manifolds satisfying the assumptions of Theorem 1.1.

Theorem 1.27. Let pW 1 , λ 1 q and pW 2 , λ 2 q be Liouville domains of dimension 2n`2 with vanishing first Chern class. Assume that there are non-vanishing sections σ 1 and σ 2 of Λ n`1 C T W 1 and Λ n`1 C T W 2 respectively satisfying the hypothesis of Theorem 1.1 with Γ given by the set of all free homotopy classes. Suppose that c 1 pT pW 1 #W 2 qq " 0 and let σ be a non-vanishing section of Λ n`1 C T pW 1 #W 2 q extending σ 1 and σ 2 . Then W 1 #W 2 satisfies the hypothesis of Theorem 1.1 with the grading of HC ˚pW 1 #W 2 q induced by σ.

Organization of the paper. In Section 2 we discuss equivariant symplectic homology for symplectic fillings with vanishing first Chern class. The proof of Theorem 1.1 is established in Section 3. Corollaries 1.4, 1.8, 1.9, 1.14, 1.17, 1.19 and 1.26 are proved in Sections 4, 5, 6, 7, 8, 9 and 10 respectively. Finally, Section 11 is devoted to the proof of Theorem 1.27.

Acknowledgements. We are grateful to Viktor Ginzburg, Marco Mazzucchelli and Otto van Koert for useful discussions and to Gustavo Granja for suggesting us the proof of Proposition 6.2 and explaining us the Whitehead-Serre theorem for nilpotent spaces.

Equivariant symplectic homology for symplectic fillings with vanishing first Chern class

Let pM 2n`1 , ξq be a closed contact manifold admitting a strong symplectic filling W such that c 1 pT W q " 0. Fix a non-vanishing section of the determinant line bundle Λ n`1 C T W . Then, as we will shortly explain in this section, one can associate to W its positive equivariant symplectic homology HC ˚pW q. This construction, briefly discussed in Section 2.1, is well known when W is a Liouville domain, but without this assumption we have to use coefficients in the universal Novikov field and an action filtration introduced by McLean and Ritter [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]

; c.f. [26, Section 2.2].
In Section 2.2, we discuss Morse-Bott spectral sequences which play a key role in the computation of equivariant symplectic homology. Using these sequences, one can show that given a non-degenerate contact form α on M , HC ˚pW q can be obtained as the homology of a chain complex generated by the good closed Reeb orbits of α. In particular, this homology is a contact invariant whenever pM, ξq admits a non-degenerate contact form such that every periodic orbit has Conley-Zehnder index with the same parity or, more generally, the set of Conley-Zehnder indexes is lacunary, i.e. does not contain any two consecutive integers.

2.1. Equivariant symplectic homology. Suppose that the symplectic filling W is a Liouville domain. The positive equivariant symplectic homology HC ˚pW q of W was introduced by Viterbo [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF] and developed by Bourgeois and Oancea [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF][START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF][START_REF] Bourgeois | The Gysin exact sequence for S1-equivariant symplectic homology[END_REF][START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]. As noticed in the introduction, the equivariant symplectic homology has a filtration given by the free homotopy classes in W and given a set Γ of free homotopy classes we denote by HC Γ ˚pW q the corresponding homology.

It turns out that HC Γ ˚pW q can be obtained as the homology of a chain complex CC Γ ˚pαq with rational coefficients generated by the good closed Reeb orbits of a non-degenerate contact form α on M with free homotopy class in Γ graded by the corresponding Conley-Zehnder indices; see [START_REF] Ginzburg | Lusternik-Schnirelmann theory and closed Reeb orbits[END_REF]Proposition 3.3] and [START_REF] Gutt | Symplectically knotted codimension-zero embeddings of domains in R 4[END_REF]Lemma 2.1]. (Recall that a non-degenerate closed Reeb orbit is good if its index has the same parity of the index of the underlying simple closed orbit; otherwise it is called bad.) In general, the differential in CC Γ ˚pαq depends on the filling W and several extra choices. However, if M admits a contact form such that the Conley-Zehnder indices of every good closed orbit have the same parity then clearly, by degree reasons, the differential must vanish and consequently the positive equivariant symplectic homology is a contact invariant. Now, suppose that pW, ωq is a strong symplectic filling with vanishing first Chern class but not necessarily a Liouville domain (in particular, ω does not need to be exact.). Then we can still associate to W its positive equivariant symplectic homology as we will briefly explain below. The novelty in this case is that HC Γ ˚pW q is a vector space over the universal Novikov field and we have to use an action filtration introduced by McLean and Ritter in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]. We will follow [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]Appendixes D and F] closely.

Let α be a non-degenerate contact form on BW such that dα " ω| BW . Consider the symplectic completion x W of W obtained by attaching to W the symplectic cone pW cone :" r1, 8q ˆBW, dprαqq where r is the coordinate in r1, 8q.

Let Z " rB r be the Liouville vector field defined on W cone and uniquely characterized by the equation ι Z ω " rα. Denote by Y the Reeb vector field on W cone defined by ι Y dα " 0 and αpY q " 1 (clearly, here we are tacitly identifying BW with t1u ˆBW ). In what follows, by Reeb periods we mean the periods of the closed orbits of Y . Given r 0 ě 1, a compatible almost complex structure J on x W is said to be of contact type for r ě r 0 if JZ " Y on rr 0 , 8q ˆBW . Let J be a compatible almost complex structure on x W and H : x W Ñ R a smooth Hamiltonian. We will use the sign convention that its Hamiltonian vector field X H is given by the equation ι X H ω " dH. The pair pH, Jq is called admissible if there exists r 0 ě 1 such that (1) J is of contact type for r ě r 0 , (2) H| rr 0 ,8qˆBW " hprq only depends on the radial coordinate, (3) h 1 pr 0 q ą 0 is smaller than the minimal Reeb period, (4) h 2 pr 0 q ą 0 and h 2 prq ě 0 for r ě r 0 , (5) if h 2 prq " 0 for some r ě r 0 then we require that h 1 prq is not a Reeb period, (6) for large r, h 1 prq is a constant not equal to a Reeb period, [START_REF] Bangert | The existence of two closed geodesics on every Finsler 2-sphere[END_REF] for r ď r 0 , i.e. on W Yr1, r 0 sˆBW , H is Morse and C 2 -small so that all closed 1-orbits of X H in r ď r 0 are the critical points of H.

Note that in the definition of admissible Hamiltonians we are fixing the contact form α used to define the radial coordinate r in W cone . Moreover, by our aforementioned sign convention, the Hamiltonian vector field X H is a negative multiple of the Reeb vector field for r ě r 0 .

With the conditions above, the necessary compactness results for Floer trajectories hold, and this allows us to define the Floer homology HFpH, Jq using coefficients in the universal Novikov field Λ "

" 8 ÿ i"1 n i T a i ; a i P R, a i Ñ 8, n i P K * ,
where K is any given field of characteristic zero. Moreover, this homology does not depend on the choice of J satisfying property [START_REF] Abouzaid | Symplectic cohomology and Viterbo's theorem[END_REF]. (More precisely, we need to pick time-dependent perturbations H t and J t (1-periodic in time) such that H t is non-degenerate and we have transversality for the relevant spaces of Floer trajectories. Moreover, it is required that, for a sufficiently large R, J t is of contact type and H t is of the form ar `b for r ě R, where a is not equal to any Reeb period. We call the corresponding pair pH t , J t q regular.)

Remark 2.1. Due to our sign conventions, the grading of Floer homology is given by minus the Conley-Zehnder index. This is in accordance with the fact that the Hamiltonian vector field is a negative multiple of the Reeb vector field for r ě r 0 .

We define a partial ordering on the set of admissible Hamiltonians in the following way: H 0 ď H 1 if H 0 pxq ď H 1 pxq for every x P x W . (For time-dependent perturbations H i,t (i " 0, 1) we require that H 0,t pxq ď H 1,t pxq for every px, tq P x W ˆS1 .) An admissible homotopy is a family of pairs pH s , J s q satisfying properties (1)-( 7) above for a fixed r 0 such that B s h 1 s ď 0, where h s is the function such that H s | rr 0 ,8qˆBW " h s prq. Given an admissible homotopy pH s , J s q we can define the continuation morphisms HFpH 0 , J 0 q Ñ HFpH 1 , J 1 q. (As before, we actually have to take suitable time-dependent perturbations H s,t and J s,t chosen generically in order to achieve transversality for the relevant spaces of parametrized Floer trajectories.)

Using this, we can define the symplectic homology as SH ˚pW, αq :" lim Ý Ñ HF ˚pH q

where the direct limit is taken over the set of admissible Hamiltonians with the partial ordering and continuation morphisms defined above. W Ñ R and consider a smooth non-decreasing cut-off function φ : R Ñ r0, 1s such that (1) φ " 0 for r ď r 0 and φ ą 0 for r ą r 0 , (2) φ 1 ą 0 for r 0 ă r ă r 1 and h 1 prq is constant for r ě r 1 , (3) φprq " 1 for large r.

Consider the 2-form η on x W given by η "

# dpφprqαq on W cone 0 on W.
Notice that, by property (1) of φ, this form vanishes on W Y r1, r 0 s ˆBW . We associate to η the 1-form Ω η on the free loop space C 8 pS 1 , x W q, with S 1 " R{Z, given by pΩ η q x pXq "

ż 1 0 ηpXptq, 9
xptq ´XH pxptqqq dt for every x P C 8 pS 1 , x W q and X P T x C 8 pS 1 , x W q. Following [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF], we will introduce a primitive for Ω η which will give us our action filtration. Let f : R Ñ r0, 8q be the smooth function given by f prq "

ż r 0 φ 1 pτ qh 1 pτ q dτ.
Note that, by the properties of h and φ, it satisfies (1) f " 0 for r ď r 0 and f ą 0 for r ą r 0 , (2) f is bounded.

The filtration functional F : C 8 pS 1 , x W q Ñ R is defined as

F pxq " ´żS 1 x ˚pφαq ´ż 1 0 f prpxptqqq dt,
where rpxptqq is the r-coordinate of xptq. (Note that φ and f vanish on W and consequently F is well defined for any curve in x W .)

Remark 2.2. Notice that our action functional differs from that in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF] in the sign in the second term. Moreover, we are using the sign convention ι X H ω " dH rather than ι X H ω " ´dH as in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]. The reason is that we are using homological conventions rather than cohomological ones as in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]. It is easy to see that the arguments from [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF] work in this case with the appropriate sign changes.

Given an admissible homotopy pH s , J s q such that h 1 s prq is constant for r ě r 1 and all s (this constant depends on s), define F s as the corresponding filtration functional for H s , that is, F s pxq " ´şS 1 x ˚pφαq ´ş1 0 f s prpxptqqq dt where f s prq " ş r 0 φ 1 pτ qh 1 s pτ q dτ . This action functional has the following nice properties.

Theorem 2.3 (Theorem 6.2 in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]). The filtration functional F satisfies

(1) Exactness: F is a primitive of Ω η .

(2) Negativity: dF pB s uq ď 0 for any Floer trajectory u : R ˆS1 Ñ x W of an admissible pH, Jq.

(3) Separation: F " 0 on all loops in r ď r 0 and F ą 0 on the 1-orbits of H in r ě r 0 . (4) Compatibility: dF s pB s uq ď 0 for any parametrized Floer trajectory u : R ˆS1 Ñ x W of an admissible homotopy pH s , J s q such that h 1 s prq is constant for r ě r 1 and all s. (5) Strictness: F px ´q ą F px `q for any Floer trajectory joining distinct orbits x ´and x ẁith rpx `q ě r 0 . Thus, F determines a filtration on the Floer chain complex of a given admissible pair pH, Jq and this filtration is respected by Floer continuation maps for admissible homotopies pH s , J s q.

Let CF ˚pH, Jq be the Floer complex of pH, Jq, CF 0 ˚pH, Jq the subcomplex generated by the 1-orbits with action F ď 0 and CF `pH, Jq :" CF ˚pH, Jq{CF 0 ˚pH, Jq. The homology of CF `pH, Jq is denoted by HF `pH, Jq. Note that, in principle, this homology depends on the choice of the cut-off function φ but, as proved in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]Corollary 6.4], actually it does not. Moreover, it does not depend on the choice of J and the direct limit SH `pW, αq :" lim Ý Ñ HF `pH q is the positive symplectic homology of pW, αq. As before, it turns out that this homology does not depend on the choice of α (c.f. [START_REF] Ritter | Deformations of symplectic cohomology and exact Lagrangians in ALE spaces[END_REF]Theorem 8]) and therefore will be denoted by SH `pW q. Now, the construction of positive equivariant symplectic homology is analogous to the construction from [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF][START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF][START_REF] Bourgeois | The Gysin exact sequence for S1-equivariant symplectic homology[END_REF][START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF][START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF]. In what follows, we will very briefly explain this; we refer to [START_REF] Bourgeois | Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action[END_REF][START_REF] Bourgeois | The index of Floer moduli problems for parametrized action functionals[END_REF][START_REF] Bourgeois | The Gysin exact sequence for S1-equivariant symplectic homology[END_REF][START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF][START_REF] Kwon | Brieskorn manifolds in contact topology[END_REF] for details. Given N P N, we have to consider S 1 -invariant families of (non-degenerate time-dependent perturbations of) admissible Hamiltonians H :

x W ˆS2N`1 ˆS1 Ñ R, where the S 1 -action is the diagonal one on S 2N `1 ˆS1 , and the space of S 1 -equivariant Floer trajectories, consisting of pairs pu, zq given by maps u : R ˆS1 Ñ x W and z : R Ñ S 2N `1 such that B s u `Jzpsq,t puqpB t u ´XH zpsq,t puqq " 0, pups, tq, zpsqq " pγ `ptq, z `q uniformly in t, where J z,t , with pz, tq P S 2N `1 ˆS1 , is a parametrized family of S 1 -invariant compatible almost complex structures, ∇ z stands for the gradient in the z-coordinate with respect to an S 1 -invariant metric on S 2N `1, z ˘P S 2N `1 and γ ˘are closed 1-orbits of H z ˘,t .

9
Choosing H and J generically, we have the corresponding S 1 -equivariant Floer homology HF S 1 ,N ˚pH q. Using parametrized versions of the previous equations, we can define continuation morphisms HF S 1 ,N ˚pH 0 q Ñ HF S 1 ,N ˚pH 1 q and using these we can define, analogously to the nonequivariant case, SH S 1 ,N ˚pW, αq :" lim Ý Ñ HF S 1 ,N ˚pH q where the direct limit is taken over the set of admissible S 1 -invariant families of Hamiltonians with the partial ordering and continuation morphisms similar to the non-equivariant case. The S 1 -equivariant symplectic homology of W is defined as

SH S 1 ˚pW, αq :" lim Ý Ñ N SH S 1 ,N ˚pW, αq,
where the direct limit is taken with respect to the embeddings S 2N `1 ãÑ S 2N `3, which induce maps SH S 1 ,N ˚pW, αq Ñ SH S 1 ,N `1 ˚pW, αq. As in the non-equivariant case, SH S 1 ˚pW, αq does not depend on the choice of α, since the proof of [START_REF] Ritter | Deformations of symplectic cohomology and exact Lagrangians in ALE spaces[END_REF]Theorem 8] extends to the equivariant setup. Therefore, we will denote it by SH S 1 ˚pW q. Finally, the positive S 1 -equivariant symplectic homology HC ˚pW q :" SH `,S 1 ˚pW q is defined in the same way as in the non-equivariant case, using the action filtration F defined above.

2.2.

Morse-Bott spectral sequence. Assume that the contact form α is non-degenerate or, more generally, Morse-Bott. If α is degenerate, assume, furthermore, that the linearized Reeb flow is complex linear with respect to a unitary trivialization of the contact structure along every periodic orbit. This last condition is satisfied, for instance, when the Reeb flow of α generates a free circle action; see [START_REF] Kwon | Brieskorn manifolds in contact topology[END_REF]Remark 8.8]. Under this assumption, we have a Morse-Bott spectral sequence that is very useful to compute SH S 1 ˚pW q and SH `,S 1 ˚pW q as we will very briefly explain in this section.

Given an admissible Hamiltonian H as in the previous section, note that the non-constant closed 1-orbits of H have radius r 1 ă r 2 ă ¨¨¨ă r pmax where the r p 's are such that h 1 pr p q is a Reeb period of α (by our assumptions on H there are finitely many such radii). Take G : Z Ñ R such that

(1) G is increasing (not necessarily strictly), (2) Gppq " p for p ď 0, (3) Gppq " φpr p qh 1 pr p q ´f pr p q for p P t1, . . . , p max u.

(4) lim pÑ8 Gppq " 8. Note here that φprqh 1 prq ´f prq " 0 for r ď r 0 and pφh 1 ´f q 1 prq " φprqh 2 prq ě 0 for every r and therefore the existence of G is clear. Note also that F pxq " Gppq for every closed 1-orbit x of H such that rpxq " r p and F pxq " 0 for every constant orbit x.

Define ApHq " tF pxq; x is a closed 1-orbit of Hu and choose ą 0 sufficiently small such that (2.2) #pApHq X pGpp ´1q ` , Gppq ` qq ď 1 for every p P Z . Consider the filtration on the Floer complex CF ˚pH q given by r F p CF q pHq " tx P CF q pHq; F pxq ď Gppq ` u.

(As mentioned in the previous section, we actually have to take a generic non-degenerate time-dependent perturbation of H.) This filtration exhausts the complex in finitely many steps due to our assumptions on H and the condition that lim pÑ8 Gppq " 8. Consider the corresponding spectral sequence whose E 0 -page is given by E 0 p,q " r F p CF p`q pHq{ r F p´1 CF p`q pHq.

Using properties (2.1) and (2.2) of our filtration, we can show that the first page of this spectral sequence is given by

E 1 p,q " $ ' &
' % À c HF loc p`q pB p,c , Hq 0 ă p ď p max H q`n`1 pW, BW ; Λq p " 0 0 p ă 0 or p ą p max , where the sum above runs over the connected components B p,c of the Morse-Bott components with F -action p and HF loc ˚pB p,c , Hq stands for the local Floer homology of B p,c ; see [START_REF] Kwon | Brieskorn manifolds in contact topology[END_REF][START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]. In general, this local Floer homology is isomorphic to the singular homology of B p,c twisted by a local system of coefficients, but it turns out that this system is trivial under our assumption on α as the following lemma shows. Lemma 2.4 (Lemma 8.9 in [START_REF] Kwon | Brieskorn manifolds in contact topology[END_REF] and Lemma 7.1 in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]). Suppose that α is Morse-Bott and, if α is degenerate, assume furthermore that the linearized Reeb flow is complex linear with respect to a unitary trivialization of the contact structure along every periodic orbit of α. Then HF loc ˚pB p,c , Hq -H ˚`µpBp,cq pB p,c ; Λq where µpB p,c q is the Robbin-Salamon index of any closed orbit in B p,c . Thus, since our filtration is bounded and exhausting, we obtain convergent spectral sequences E 1 p,q ùñ HF ˚pH q and E 1,p ,q ùñ HF `pH q whose first pages are given by

E 1 p,q " $ ' & ' % À c H p`q`µpBp
,cq pB p,c ; Λq 0 ă p ď p max H q`n`1 pW, BW ; Λq p " 0 0 p ă 0 or p ą p max , and E 1,p ,q " # À c H p`q`µpBp,cq pB p,c ; Λq 0 ă p ď p max 0 p ď 0 or p ą p max .

It turns out that analogous spectral sequences exist in the equivariant setup converging to HF S 1 ˚pH q and HF `,S 1 ˚pH q. Choosing a suitable sequence of Hamiltonians, we have that these spectral sequences pass to the limit, furnishing the following result.

Theorem 2.5 (Corollary 7.2 in [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]). Under our assumption on α, there are convergent spectral sequences E 1 p,q ùñ SH S 1 ˚pW q and E 1,p ,q ùñ SH `,S 1 ˚pW q whose first pages are given by

E 1 p,q " $ ' & ' % À c H S 1
p`q`µpBp,cq pB p,c ; Λq p ą 0 H S 1 q`n`1 pW, BW ; Λq p " 0 0 p ă 0, and E 1,p ,q " # À c H S 1 p`q`µpBp,cq pB p,c ; Λq p ą 0 0 p ď 0,

where the S 1 -action on W is the trivial one so that H S 1 ˚pW, BW ; Λq -À i`j"˚H i pW, BW ; Λqb H j pCP 8 ; Λq.

Clearly, all the constructions above respect the filtration given by the free homotopy classes in W . Using the spectral sequence E 1,p ,q established in the previous proposition, one can show that if α is non-degenerate then, given a set Γ of free homotopy classes in W , HC Γ ˚pW q can be obtained as the homology of a chain complex CC Γ ˚pαq with coefficients in the universal Novikov field generated by the good closed Reeb orbits of α with free homotopy class in Γ graded by the corresponding Conley-Zehnder indices. As a matter of fact, this is proved when W is a Liouville domain in [START_REF] Ginzburg | Lusternik-Schnirelmann theory and closed Reeb orbits[END_REF]Proposition 3.3] and [31, Lemma 2.1] and one can check that the proof extends to our context since it is purely algebraic.

Proof of Theorem 1.1

We will prove the theorem when dim HC Γ n pW q ă dim HC Γ n`jK pW q for some K P N and every j P N, since the argument in the other case is analogous up to some sign changes. Throughout the proof we will use the following theorem taken from [START_REF] Ginzburg | Multiplicity of Closed Reeb Orbits on Prequantization Bundles[END_REF]Theorem 4.1]. This result can also be derived from a refinement of the common index jump theorem proved in [START_REF] Duan | The enhanced common index jump theorem for symplectic paths and nonhyperbolic closed geodesics on Finsler manifolds[END_REF].

Theorem 3.1 ([24]

). Let Φ 1 , . . . , Φ r be strongly non-degenerate paths in Spp2nq starting at the identity with positive mean index. Given N P N and 0 P N there exist two integer vectors pd `, k 1 , . . . , k r q and pd ´, k 1 , . . . , k ŕ q such that d ˘, k 1 , . . . , k ȓ are all divisible by N and µpΦ k ȋ ` i q " d ˘`µpΦ i q, µpΦ k í i q ´d´" ´pµpΦ k ì i q ´d`q , and |µpΦ k ȋ i q ´d˘| ď n for every i P t1, . . . , ru and P Z in the range 1 ď | | ď 0 .

Suppose that α has finitely many simple orbits γ 1 , . . . , γ r with free homotopy class in Γ.

Here, simple means that the closed orbit is not a multiple of a closed orbit with free homotopy class in Γ (but it can be a multiple of an orbit with free homotopy class not in Γ). Note that r ě 1 since HC Γ ˚pW q is not trivial. Arguing by contradiction, suppose that µpγ k i q " µpγ k j q for every i, j P t1, . . . , ru and k P N. Consequently, HC Γ ˚pW q is the homology of a chain complex generated by the good closed orbits of α with trivial differential.

Let γ " γ 1 . By hypothesis, we have that the mean index of γ, denoted by ∆pγq, is positive (otherwise, we would have that HC Γ k pW q " 0 for every k ě n). Since |k∆pγq ´µpγ k q| ă n for every k P N we conclude that µpγ k q ą ´n for every k P N. Lemma 3.2. There exists 0 P N such that µpγ k` q ą µpγ k q `2n for every k P N and ě 0 .

Proof. Taking 0 ě 4n ∆pγq we have µpγ k` q ą pk ` q∆pγq ´n ą µpγ k q ´2n ` ∆pγq

ě µpγ k q `2n
for every ě 0 and k P N.

Applying Theorem 3.1 to the linearized Reeb flow along γ with N " 2K and 0 given by the previous lemma, we find even numbers d and k γ which are multiples of K such that (3.1) µpγ kγ ˘ q " d ˘µpγ q

for every such that 1 ď ď 0 and (3.2) d ´n ď µpγ kγ q ď d.

Lemma 3.3. If µpγ k q " d `n then k " k γ ` for some such that 1 ď ă 0 .

Proof. It follows from Lemma 3.2 and (3.2) that µpγ kγ ´ q ă µpγ kγ q ´2n ď d ´2n and µpγ kγ ` q ą µpγ kγ q `2n ě d `n

for every ě 0 . Since µpγ k q ą ´n for every k P N, we deduce from (3.1) that

µpγ kγ ´ q " d ´µpγ q ă d `n
for every 1 ď ď 0 . Finally, notice that, by (3.2), µpγ kγ q ‰ d `n. Now, it follows from the fact that d is a multiple of K and our hypothesis that HC Γ d`n pW q ‰ 0. This implies that there exists k P N such that µpγ k i q " d `n for every i P t1, . . . , ru. By Lemma 3.3 and our assumptions, k " k γ ` for some such that 1 ď ă 0 . Thus, it follows from (3.1) that µpγ i q " n for every i P t1, . . . , ru. This means that for every contribution to HC Γ d`n pW q, necessarily given by γ kγ ` i with 1 ď ă 0 and i P t1, . . . , ru, we get a contribution to HC Γ n pW q given by γ i with 1 ď ă 0 and i P t1, . . . , ru (notice here that, since k γ is even, γ kγ ` i is good if and only if γ i is good). But this contradicts the assumption that dim HC Γ n pW q ă dim HC Γ n`jK pW q for every j P N.

Proof of Corollary 1.4

Let Ď M be a contact finite quotient of M and τ : M Ñ Ď M the corresponding quotient projection. Given a non-degenerate contact form s α on Ď M , consider its lift α " τ ˚s α. Arguing by contradiction, suppose that s α has only one simple closed orbit. By Theorem 1.1 and our assumptions, α has at least two simple closed orbits γ 1 and γ 2 such that (4.1)

µpγ k 1 q ‰ µpγ k 2 q for some k P N, where the index is computed using a non-vanishing section σ of Λ n`1 C T W . Since s α has only one simple closed orbit, we have that γ 2 " ψ˝γ 1 for some deck transformation ψ.

Let ξ be the contact structure on M and note that T W | M " ξ ' ξ ω , where ξ ω is the trivial bundle given by the symplectic orthogonal of ξ. Choose a compatible almost complex structure J on W satisfying Jpξq " ξ and Jpξ ω q " ξ ω so that

Λ n`1 C T W | M " Λ n C ξ b ξ ω .
Then σ induces a non-vanishing section σ ξ of Λ n C ξ and it turns out that the grading of HC ˚pW q induced by σ (viewing HC ˚pW q as the homology of a chain complex generated by the good closed orbits of a non-degenerate contact form) is given by the Conley-Zehnder index of the closed Reeb orbits defined using a trivialization of the contact structure (along each closed orbit) induced by σ ξ . Given a closed orbit γ, denote by µpγ, σ ξ q the corresponding Conley-Zehnder index.

The deck transformation ψ induces an isomorphism Ψ : Λ n C ξ Ñ Λ n C ξ, where we are identifying the top exterior power of ξ with respect to the complex structures J| ξ and ψ ˚pJ | ξ q. It turns out that µpγ j 1 , Ψ ˚σξ q " µpγ j 2 , σ ξ q for every j P N. But, since H 1 pM, Rq " 0, we have that Ψ ˚σξ and σ ξ are homotopic; see [START_REF] Mclean | Reeb orbits and the minimal discrepancy of an isolated singularity[END_REF]Lemma 4.3]. Hence, µpγ j 1 , σ ξ q " µpγ j 2 , σ ξ q for every j P N, contradicting (4.1).

Remark 4.1. If the first Chern class of the contact structure s ξ on Ď M vanishes, we do not need that Ψ ˚σξ and σ ξ are homotopic to obtain a contradiction. Indeed, choose a section σs ξ of Λ n C s ξ and consider its lift r σ ξ to Λ n C ξ. Since r σ ξ is invariant by deck transformations, we have that µpγ j 1 , r σ ξ q " µpγ j 1 , Ψ ˚r σ ξ q " µpγ j 2 , r σ ξ q for every j P N. However, note that r σ ξ is not necessarily induced by the section σ of Λ n`1 C T W . In order to fix this, we claim that the equality µpγ j 1 , r σ ξ q " µpγ j 2 , r σ ξ q implies that µpγ j 1 , σ ξ q " µpγ j 2 , σ ξ q for any section σ ξ of Λ n C ξ. As a matter of fact, given a periodic orbit γ we have that the difference µpγ; r σ ξ q ´µpγ; σ ξ q depends only on the homology class rγs P H 1 pM ; Rq; see [START_REF] Mclean | Reeb orbits and the minimal discrepancy of an isolated singularity[END_REF]Lemma 4.3] and [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]Remark 5.1]. Hence, it is enough to have that γ j 1 and γ j 2 represent the same homology class in H 1 pM ; Rq. But the deck transformations act trivially in H 1 pM ; Rq and consequently rγ j 1 s " rγ j 2 s for every j P N.

Proof of Corollary 1.8

In this section we prove Corollary 1.8 showing that the compact domain W Ă X bounded by M satisfies the hypothesis of Theorem 1.1. More precisely, we have the following result.

(Note that π 1 pS ˚r N q is finite because r N is simply connected. Therefore, H 1 pS ˚r N , Rq " 0.) So, assume from now on that N is simply connected. A result due to Vigué-Poirrier and Sullivan [START_REF] Vigué-Poirrier | The homology theory of the closed geodesic problem[END_REF] establishes that the dimension of H ˚pΛN ; Qq is asymptotically bounded exactly when

H ˚pN ; Qq -T d,h`1 pxq
for some d and h, where T d,h`1 pxq is the truncated polynomial algebra with generator x of degree d and height h `1. In this case, Rademacher [START_REF] Rademacher | On the average indices of closed geodesics[END_REF] computed H ˚pΛN {S 1 , N ; Qq. First of all, m " dh. When d is even, [START_REF] Rademacher | On the average indices of closed geodesics[END_REF]Theorem 2.4] tells us that the Poincaré series of pΛN {S 1 , N q (for homology with rational coefficients) is given by

P pΛN {S 1 , N ; Qqptq " t d´1 ˆ1 1 ´t2 `tdph`1q´2 1 ´tdph`1q´2
˙1 ´tdh 1 ´td .

Therefore, we deduce from this that dim H dh´1 pΛN {S 1 , N ; Qq " h and dim H dh´1`jpdph`1q´2q pΛN {S 1 , N ; Qq " h `1

for every j P N. When d is odd, h has to be equal to one and N is rationally homotopic to the sphere S d . The corresponding Poincaré series is given by (see [START_REF] Rademacher | On the average indices of closed geodesics[END_REF]Remark 2.5])

P pΛN {S 1 , N ; Qqptq " t d´1 ˆ1 1 ´t2 `td´1 1 ´td´1 ˙.
Consequently, dim H d´1 pΛN {S 1 , N ; Qq " 1 and dim H d´1`jpd´1q pΛN {S 1 , N ; Qq " 2 for every j P N.

(ii) N satisfies the second hypothesis.

We will show that, under hypothesis (ii), HC 0 ˚pD ˚N q ‰ 0 and HC a ˚pD ˚N q ‰ 0 for any non-trivial homotopy class a. This readily implies the result in this case.

The non-triviality of HC a ˚pD ˚N q easily follows from (1.2) and the difficult part is to show that HC 0 ˚pD ˚N q ‰ 0. In order to prove this, we first need the following well known result for which we give a proof for the convenience of the reader. For a shorter notation, we will suppress the superscript zero in Λ 0 N . Proposition 6.1. We have that π j`1 pN q -π j pΛN, N q for every j ě 1.

Proof. Let ΩN be the based loop space of contractible loops. The canonical fibration ΩN ãÑ ΛN ev Ý Ñ N admits a section N i ãÝ Ñ ΛN given by the inclusion of constant loops, and this implies that the associated homotopy long exact sequence reduces to the split short exact sequence 0 π ˚pΩN q π ˚pΛN q π ˚pN q 0 α β i so that β ˝i " Id , where, for a shorter notation, we are using i to denote both the inclusion N i ãÝ Ñ ΛN and its map induced on the homotopy groups. In particular, i is injective. Since the sequence splits, we have that π j pΛN q -π j pΩN q ' π j pN q for every j ě 2 and π 1 pΛN q -π 1 pΩN q ¸π1 pN q. However, using the well known isomorphism π ˚pΩN q -π ˚`1 pN q and our assumptions that π 1 pN q -Z and π 2 pN q " 0, we have that π j pΛN q -π j pΩN q ' π j pN q -π j`1 pN q ' π j pN q for every j ě 1 and consequently the fundamental group of ΛN is abelian.

Thus, we have an inverse on the left of α, p : π ˚pΛN q Ñ π ˚pΩN q, given by (6.1)

ppbq " α ´1pb ´ipβpbqqq and the commutative diagram (6.2)

0 π ˚pΩN q π ˚pΛN q π ˚pN q 0 π ˚pΩN q ' π ˚pN q α β ψ τ 1 τ 2
where ψpbq " pppbq, βpbqq is an isomorphism and τ 1 and τ 2 are the projections onto the first and second factors respectively. On the other hand, the homotopy long exact sequence associated to the pair pΛN, N q furnishes the commutative diagram (6.3)

¨¨¨π k`1 pΛN, N q π k pN q π k pΛN q π k pΛN, N q ¨¨π k pΩN q ' π k pN q π k`1 pN q ' π k pN q 0 i ϕ j ψ 0 -
where ϕ :" ψ ˝i and we are using the isomorphism π k pΩN q -π k`1 pN q. Note that the map π ˚pΛN, N q Ñ π ˚´1 pN q vanishes because i is injective. We claim that ϕpcq " p0, cq. In order to prove this, let us first prove that τ 1 ˝ϕpcq " 0.

for every c P π k pN q. Indeed, we have that τ 1 ˝ϕpcq " τ 1 ˝ψ ˝ipcq " ppipcqq " α ´1pipcq ´i ˝β ˝ipcqq " α ´1pipcq ´ipcqq " 0

where the second equality follows from the fact that ψpipcqq " pppipcqq, βpipcqqq, the third equality holds because of (6.1) and the fourth equality follows from the relation β ˝i " Id . Now, let us show that τ 2 ˝ϕpcq " c. for every c P π k pN q. This is a consequence of the relation τ 2 ˝ϕ " τ 2 ˝ψ ˝i " β ˝i " Id where the second equality follows from the commutative diagram (6.2).

Finally, we conclude from the claim and the diagram (6.3) that j induces an isomorphism between π k`1 pN q and π k pΛN, N q for every k ě 1, as desired. Now, we need the following result which is a rational relative Hurewicz theorem for the pair pΛN, N q. Note that, by Proposition 6.1, we have that π j pΛN, N q is an abelian group for every j ě 1. Proposition 6.2. Suppose that there exists m ě 1 such that π j pΛN, N q b Q " 0 for every j ă m and π m pΛN, N q b Q ‰ 0. Then H j pΛN, N ; Qq " 0 for every j ă m and H m pΛN, N ; Qq -π 1 m pΛN, N q b Q where π 1 m pΛN, N q is obtained from π m pΛN, N q by factoring out the action of π 1 pN q. Proof. Recall that given a continuous map f : X Ñ Y between topological spaces X and Y one can associate to it a fibration F Ñ X 1 Ñ Y whose total space X 1 is homotopically equivalent to X and with fiber F called the homotopy fiber. Consider such a fibration F Ñ N 1 Ñ ΛN associated to the map i : N Ñ ΛN . Lemma 6.3. The homotopy fiber F is homotopy equivalent to Ω 2 N . In particular, we have that π j pF q -π j`2 pN q -π j`1 pΛN, N q for every j ě 0.

Proof. Consider the commutative diagram

N ΛN N N. i ev ev Id
This diagram has a fibre extension (up to homotopy) given by a 3 ˆ3 diagram such that all the rows and columns are fibration sequences (c.f. [45, Section 3.2])

Ω 2 N ˚ΩN F N ΛN ˚N N.
where ˚denotes a contractible space. From the first column we conclude that F is homotopy equivalent to Ω 2 N . Consequently, π j pF q -π j pΩ 2 N q -π j`2 pN q for every j ě 0. The isomorphism π j pF qπ j`1 pΛN, N q follows from Proposition 6.1 (c.f. Remark 6.4). Remark 6.4. Since F is the homotopy fiber of the inclusion N ãÑ ΛN , it is well known that π ˚pF q -π ˚`1 pΛN, N q (see, for instance, [START_REF] Davis | Lecture notes in algebraic topology[END_REF]Page 155]). Consequently, one could also deduce Proposition 6.1 from the isomorphism π ˚pF q -π ˚`2 pN q established in the previous lemma.

Therefore, we conclude from the previous lemma and our assumptions that m ě 2, F is connected and its fundamental group is abelian. Moreover, we have that π m´1 pF q b Q ‰ 0 and, if m ą 2, π j pF q b Q " 0 for every 1 ď j ď m ´2. The next lemma is a rational Hurewicz theorem for F . Lemma 6.5. We have that H j pF ; Qq -π j pF q b Q for every 1 ď j ď m ´1.

Proof. Suppose initially that G :" π 1 pF q is torsion and let r F τ Ý Ñ F be the universal covering. By the rational Hurewicz theorem for simply connected spaces, (6.4) H q p r F ; Qq -π q p r F q b Q for every 1 ď q ď m ´1; see, for instance, [START_REF] Klaus | A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres[END_REF]. Let BG be the classifying space of G and ϕ : F Ñ BG be the classifying map of τ . The fibration associated to ϕ is

r F Ñ F 1 Ñ BG
with homotopy fiber r F and total space F 1 homotopically equivalent to F . Consider the corresponding Serre spectral sequence E r p,q so that (6.5) E 2 p,q -H p pBG; H q p r F ; Qqq, where H q p r F ; Qq is a local system with coefficients H q p r F ; Qq. We claim that H p pBG; Qq " 0 for all p ě 1. Indeed, since G is a finite abelian group, we can write G " ' r i"1 Z k i for positive integers k 1 , . . . , k r . Thus, we have the homotopy equivalence

BG - r ź i"1 BZ k i .
But clearly H p pBZ k i ; Qq " 0 for every p ą 0 and therefore the claim follows from Kunneth theorem.

Hence, we have from (6.5) that E 2 p,q " 0 for all p ě 1 and q ě 0 which implies that H q pF ; Qq -E 8 0,q -E 2 0,q -H 0 pBG; H q p r F ; Qqq.

But H 0 pBG; H q p r F ; Qqq is isomorphic to H q p r F ; Qq factored out by the action of π 1 pBGq -G acting as the deck transformations of r F τ Ý Ñ F . By Lemma 6.3, F is homotopy equivalent to Ω 2 N and therefore is an H-space. (Recall that a topological space X is an H-space if there is a continuous multiplication map µ : X ˆX Ñ X and an identity element e P X such that the maps µp¨, eq and µpe, ¨q are homotopic to the identity through maps pX, eq Ñ pX, eq.) Therefore, G acts trivially on H ˚p r F ; Qq. As a matter of fact, note that r F is also an H-space and τ is an H-map (i.e. if µ and μ are the multiplications on F and r F respectively then the maps τ ˝μ and µ ˝pτ ˆτ q are homotopic). Let G e Ă r F be the fiber of τ over the identity element in F so that there is a natural identification G -G e . Then the deck transformation of τ corresponding to g P G is given by μpg, ¨q where g P G e corresponds to g via the identification G -G e . Since r F is path connected, such map must be homotopic to the identity and therefore G acts trivially on H ˚p r F ; Qq. Thus, by (6.4), H 0 pBG; H q p r F ; Qqq -H q p r F ; Qq -π q p r F q b Q for every 1 ď q ď m ´1. But π j p r F q -π j pF q for every j ě 2 and π 1 p r F q b Q " π 1 pF q b Q " 0 and therefore

π ˚p r F q b Q -π ˚pF q b Q
implying that H q pF ; Qq -π q pF q b Q for every 1 ď q ď m ´1, proving the lemma when π 1 pF q is torsion.

Finally, when π 1 pF q is not torsion, we have by Hurewicz theorem and Lemma 6.3 that H 1 pF ; Zq -π 1 pF q -π 2 pΛN, N q (note that π 1 pF q is abelian) and hence m " 2 and H 1 pF ; Qq -π 1 pF q b Q, as desired. Now, let E r p,q be the Serre spectral sequence associated to F Ñ N 1 Ñ ΛN so that E 2 p,q -H p pΛN ; H q pF ; Qqq, where H q pF ; Qq is a local system with coefficients in H q pF ; Qq such that the action π 1 pΛN q Ĥq pF ; Qq Ñ H q pF ; Qq fits in the commutative diagram (6.6) π 1 pΛN q ˆHq pF ; Qq H q pF ; Qq

π 1 pN q ˆπq`1 pΛN, N q b Q π q`1 pΛN, N q b Q - -
for all 1 ď q ď m ´1, where in the second line we have the usual action of π 1 pN q on π q`1 pΛN, N q b Q and the vertical isomorphisms follow from Lemma 6.5 and the isomorphism π q pF q -π q`1 pΛN, N q. Since F is connected, H 0 pF ; Qq is a trivial system and hence (6.7) E 2 p,0 -H p pΛN ; H 0 pF ; Qqq -H p pΛN ; Qq for every p ě 0. By Lemmas 6.3 and 6.5,

H m´1 pF ; Qq -π m´1 pF q b Q -π m pΛN, N q b Q ‰ 0 and, if m ą 2,
H q pF ; Qq " π q pF q b Q " π q`1 pΛN, N q b Q " 0 for every 1 ď q ď m ´2.

Using this we deduce that (6.8) E 2 0,m´1 -π 1 m pΛN, N q b Q and, if m ą 2, (6.9) E 2 p,q " 0 for every p ě 0 and 1 ď q ď m ´2, where in (6.8) we are using the commutative diagram (6.6) (c.f. [START_REF] Whitehead | Elements of homotopy theory[END_REF]Theorem VI.3.2]).

The spectral sequence converges to (6.10) E 8 p,q -# 0 for every p and q ą 0 H p pN ; Qq for every p and q " 0. Indeed, by the construction of the spectral sequence, E p,q above the line zero and consequently we obtain the isomorphism (6.10).

From (6.7), (6.8), (6.9) and (6.10) we easily conclude that, for every 0 ď j ď m, the differential d j j,0 from E j j,0 -E 2 j,0 -H j pΛN ; Qq to E j 0,j´1 -E 2 0,j´1 -π 1 j pΛN, N q b Q induces an isomorphism (6.11) π 1 j pΛN, N q b Q -H j pΛN ; Qq{i ˚pH j pN ; Qqq.

The evaluation map ev : ΛN Ñ N and i : N Ñ ΛN satisfy ev ˝i " Id and consequently i ˚: H ˚pN ; Qq Ñ H ˚pΛN ; Qq is injective. Therefore, the homology long exact sequence associated to the pair pΛN, N q readily implies that H ˚pΛN, N ; Qq -H ˚pΛN ; Qq{i ˚pH ˚pN ; Qqq and the proposition follows from (6.11). Now, we claim that there exists j ą 1 such that π j pN q b Q ‰ 0. Arguing by contradiction, suppose that π j pN q b Q " 0 for every j ą 1. Recall that a topological space X is nilpotent if π 1 pXq is nilpotent (e.g. if π 1 pXq is abelian) and the action of π 1 pXq on π j pXq is nilpotent (e.g. if π j pXq is finite) for every j ą 1; see [START_REF] Hilton | Localization of nilpotent groups and spaces[END_REF][START_REF] May | More concise algebraic topology[END_REF] for precise definitions. Then, by our assumptions, N is a nilpotent space. Let m ą 1 be the dimension of N and f : S 1 Ñ N be a closed curve whose homotopy class is a generator of π 1 pN q -Z. Then f induces an isomorphism f ˚: π ˚pS 1 qbQ Ñ π ˚pN qbQ. But by the Whitehead-Serre theorem for nilpotent spaces (c.f. [41, Theorems 3.3.8 and 5.3.2]), this implies that f also induces an isomorphism H ˚pS 1 ; Qq Ñ H ˚pN ; Qq, contradicting the fact that H m pN ; Qq -Q. Now, let k be the smallest integer bigger than one such that π k pN q b Q ‰ 0. It follows from Proposition 6.1 that π j pΛN, N q b Q " 0 for every j ă k ´1 and π k´1 pΛN, N q b Q ‰ 0. The action π 1 pN q ˆπk´1 pΛN, N q Ñ π k´1 pΛN, N q fits in the commutative diagram (6.12) π 1 pN q ˆπk´1 pΛN, N q π k´1 pΛN, N q π 1 pN q ˆπk pN q π k pN q, --

where in the second line we have the usual action of π 1 pN q on π k pN q and the vertical isomorphisms follow from Proposition 6.1. Thus, since N is k-simple, we have that π k´1 pΛN, N q -π 1 k´1 pΛN, N q and hence, by Proposition 6.2, (6.13) H k´1 pΛN, N ; Qq ‰ 0.

Remark 6.6. Notice that to obtain this inequality we actually only need the following weaker assumption: let a be a generator of π 1 pN q and denote by A the linear map corresponding to the action of a on π k pN q b Q -π k´1 pΛN, N q b Q. Since π 1 k´1 pΛN, N q b Q is the quotient of π k´1 pΛN, N q b Q by the subgroup tA n g ´g; n P Z, g P π k´1 pΛN, N q b Qu, it is clearly enough that kerpA ´Id q ‰ 0.

We claim that this implies that H ˚pΛN {S 1 , N ; Qq is not trivial, that is, H j pΛN {S 1 , N ; Qq ‰ 0 for some degree j. As a matter of fact, this is a consequence of the following proposition. Proposition 6.7. If the total homology H ˚pΛN {S 1 , N ; Qq is trivial then so is H ˚pΛN, N ; Qq.

Proof. Applying the relative Gysin sequence to the S 1 -bundle ΛN ˆES 1 Ñ ΛN ˆS1 ES 1 we obtain the exact triangle

H ˚pΛN ˆES 1 , N ˆES 1 ; Qq H ˚pΛN ˆS1 ES 1 , N ˆS1 ES 1 ; Qq H ˚´2 pΛN ˆS1 ES 1 , N ˆS1 ES 1 ; Qq r`1s
which yields

H ˚pΛN, N ; Qq H S 1 ˚pΛN, N ; Qq H S 1 ˚´2 pΛN, N ; Qq. r`1s
Since H ˚pΛN {S 1 , N ; Qq -H S 1 ˚pΛN, N ; Qq we immediately conclude the result. Thus, we conclude from (1.2), (6.13) and the previous proposition that HC 0 ˚pD ˚N q ‰ 0 and consequently α carries a contractible periodic orbit γ 1 . Indeed, if α has no contractible closed orbit then we would have that HC 0 ˚pD ˚N q " 0, furnishing a contradiction. Note that we do not need the non-degeneracy of α here. Remark 6.8. The proof above shows that if π k pN q b Q ‰ 0 for some k ą 1 then α carries a contractible periodic orbit assuming only that N is oriented spin, π 1 pN q is abelian, π 2 pN q " 0 and N is k-simple. The hypothesis that N is k-simple can be relaxed by the assumption that the subgroup tag ´g; a P π 1 pN q, g P π k pNq b Qu is strictly contained in π k pN q b Q. When π 1 pN q -Z this is equivalent to ask that kerpA ´Id q ‰ 0, where A is the linear map corresponding to the action of a generator of π 1 pN q on π k pN q b Q; see Remark 6.6.

To find a second closed orbit, take a non-trivial homotopy class a P π 1 pN q -π 1 pD ˚N q. Clearly, we have that

H ˚pΛ a N {S 1 ; Qq ‰ 0.
This, together with (1.2), implies that α has a periodic orbit γ 2 with homotopy class a. Since π 1 pD ˚N q -π 1 pN q -Z, a is not nilpotent (i.e. there is no k P N such that a k " 0) and consequently we have that γ 1 and γ 2 are geometrically distinct.

(iii) N satisfies the third hypothesis.

By hypothesis, given a non-trivial free homotopy class a in N there exists another nontrivial free homotopy class b such that b ‰ a n for every n P Z. Thus, we deduce from (1.2) that (notice that we can naturally identify the free homotopy classes of N and D ˚N ) HC a ˚pD ˚N q -H ˚pΛ a M {S 1 ; Qq ‰ 0 and HC b ˚pD ˚N q -H ˚pΛ b M {S 1 ; Qq ‰ 0. From this we get two closed orbits γ 1 and γ 2 with free homotopy classes a and b respectively. From our choice of a and b we conclude that these orbits must be geometrically distinct.

Proof of Corollary 1.14

As in the proof of Corollary 1.9, we will consider each hypothesis separately. Throughout this section, we will tacitly identify a closed geodesic γ with the corresponding closed orbit of the geodesic flow restricted to the unit sphere bundle. As in the previous section, if N is orientable we will take a non-vanishing section of Λ n`1 C T D ˚N induced from a volume form on N so that the Conley-Zehnder index of a non-degenerate closed geodesic coincides with its Morse index.

(i) N satisfies the first hypothesis.

Arguing as in the proof of Corollary 1.9, it is enough to show that if N is simply connected then every bumpy Finsler metric F has either infinitely many geometrically distinct closed geodesics or at least two geometrically distinct closed geodesics γ 1 and γ 2 such that µpγ k 1 q ‰ µpγ k 2 q for some k P N. If H ˚pΛN {S 1 , N ; Qq is asymptotically unbounded then the Gromoll-Meyer theorem [START_REF] Gromoll | Periodic geodesics on compact riemannian manifolds[END_REF] implies that F has infinitely many geometrically distinct closed geodesics. (Note here that the proof the Gromoll-Meyer theorem works for Finsler metrics.) When H ˚pΛN {S 1 , N ; Qq is asymptotically bounded, then, as explained in the proof of Corollary 1.9, there exists K P N such that dim H m´1 pΛN {S 1 , N ; Qq ă dim H m´1`jK pΛN {S 1 , N ; Qq for every j P N where m is the dimension of N .

Arguing by contradiction, suppose that F has only one prime closed geodesic. As explained in the introduction, the proof of Theorem 1.1 uses only the fact that given a non-degenerate contact form α on M then HC Γ ˚pW q is the homology of a chain complex generated by the good periodic orbits of α with free homotopy class in Γ graded by the Conley-Zehnder index. Thus, it is enough to show that H ˚pΛN {S 1 , N ; Qq is the homology of a chain complex generated by the good periodic orbits of the geodesic flow of F . (Here, the geodesic flow of F means the geodesic flow of F restricted to the unit sphere bundle.)

In order to do this, consider the energy functional E : ΛN Ñ R induced by F . Let γ 1 , . . . , γ r be the simple closed orbits of the geodesic flow of F . By our assumption, r " 1 if F is not reversible or r " 2 and γ 2 " ´γ1 if F is reversible, where ´γ1 denotes the corresponding reversed geodesic. In what follows, we will use well known facts about closed geodesics; see e.g. [START_REF] Bangert | The existence of two closed geodesics on every Finsler 2-sphere[END_REF][START_REF] Klingenberg | Lectures on closed geodesics[END_REF][START_REF] Rademacher | Morse Theorie und geschlossene Geodatische[END_REF]. Given a closed geodesic γ, define Λpγq " E ´1pr0, Epγqqq. The (equivariant) Morse type numbers are defined as

m k " ÿ iPt1,...,ru jPN dim H k ppΛpγ j i q Y S 1 ¨γj i q{S 1 , Λpγ j i q{S 1 ; Qq
where S 1 ¨γj i is the orbit of γ j i in ΛN induced by the (obvious) S 1 -action. Since We have the Morse inequalities

H k ppΛpγq Y S 1 ¨γq{S 1 , Λpγq{S 1 ; Qq - # Q if k " µpγq
m k ´mk´1 `mk´2 ´¨¨¨`p´1q k m 0 ě b k ´bk´1 `bk´2 ´¨¨¨`p´1q k b 0
for every k ě 0. Since F has only one prime closed geodesic, the index of every good closed orbit has the same parity p. Hence, m k " 0 for every k ‰ p (mod 2).

This fact, together with the Morse inequalities, implies that b k " m k for every k.

Thus, we conclude that H ˚pΛN {S 1 , N ; Qq is the homology of the chain complex with rational coefficients and trivial differential generated by the good periodic orbits of the geodesic flow of F graded by the Conley-Zehnder index, as desired.

(ii) N satisfies the second hypothesis.

Since N is not diffeomorphic to the the circle and π 1 pN q -Z, there is j ě 2 such that π j pN q ‰ 0. Let k be the smallest integer bigger than one such that π k pN q ‰ 0. Then, by Proposition 6.1, we have that π k´1 pΛ 0 N, N q -π k pN q ‰ 0. Then, applying classical minimax arguments to a non-trivial homotopy class in π k´1 pΛ 0 N, N q (see e.g. [START_REF] Klingenberg | Lectures on closed geodesics[END_REF]) we obtain a contractible closed geodesic γ 1 . Now, taking a non-trivial homotopy class a P π 1 pN q one can take a minimum of E| Λ a N which furnishes a non-contractible closed geodesic γ 2 . Since π 1 pN q -Z we conclude that γ 1 and γ 2 must be geometrically distinct.

(iii) N satisfies the third hypothesis.

By hypothesis, given a non-trivial free homotopy class a in N there exists another non-trivial free homotopy class b such that b ‰ a n for every n P Z. Minimizing the energy functional on Λ a N and Λ b N we obtain two closed geodesics γ 1 and γ 2 with free homotopy classes a and b which have to be geometrically distinct due to our choice of a and b.

Proof of Corollary 1.17

Let pM 2n`1 , ξq be a Gorenstein toric contact manifold (i.e. a toric contact manifold with vanishing first Chern class) determined by a toric diagram D Ă R n (see [START_REF] Abreu | On the mean Euler characteristic of Gorenstein toric contact manifolds[END_REF]). Let F denote one of the facets of D, necessarily a simplex, and assume without loss of generality that its vertices are v 1 , . . . , v n P Z n . Let η P Z n be such that tν 1 " pv 1 , 1q, . . . , ν n " pv n , 1q, pη, 1qu is a Z-basis of Z n`1 .

Let R denote a rational toric Reeb vector, which can be uniquely written as

R " n ÿ j"1 b j ν j `bpη, 1q , with b 1 , . . . , b n , b P Q and b " 1 ´n ÿ j"1 b j ‰ 0 . Write b j |b| "
p j q j with p j P Z , q j P N and gcdpp j , q j q " 1 .

Let (8.1) q :" lcmpq 1 , . . . , q n q so that qb j {|b| P Z for all j " 1, . . . , n. We then have that

(8.2) b `n ÿ j"1 b j " 1 ñ N :" q |b| P N .
We will consider irrational perturbations R δ of R that can be uniquely written as

R δ " n ÿ j"1 b δ,j ν j `bδ pη, 1q , with b δ,1 , . . . , b δ,n , b δ P R , b δ " 1 ´n ÿ j"1 b δ,j ‰ 0 and b δ,j |b δ | " b j |b|
`δj for some δ j P Rzt0u , j " 1, . . . , n .

We will say that R δ is -small for some ą 0 whenever 0 ă |δ j | ă for all j " 1, . . . , n.

We will denote by γ the simple closed R-orbit associated with F and by γ δ the simple closed R δ -orbit associated with F . The Conley-Zehnder index of γ k δ , for any k P N, is given by (see [2, Section 5] and [5, Section 3])

(8.3) µpγ k δ q " 2 n ÿ j"1 Z k ˆbj |b| `δj ˙^`2k b |b| `n .
This implies that the Conley-Zehnder index of γ k 1 `k2 q δ , for any k 1 , k 2 P N and q P N defined by (8.1), is given by

µpγ k 1 `k2 q δ q " 2 n ÿ j"1 ˆZk 1 b j |b| `pk 1 `k2 qqδ j ^`k 2 q b j |b| ˙`2pk 1 `k2 qq b |b| `n " 2 n ÿ j"1 Z k 1 b j |b| `pk 1 `k2 qqδ j ^`2k 2 q |b| `2k 1 b |b| `n. (8.4)
Proposition 8.1. For any L P N there exists ą 0 such that, for any -small R δ and any k 1 , k 2 P N with k 1 , k 2 ď L, we have µpγ k 1 `k2 q δ q " µpγ k 1 δ q `2k 2 N , where q, N P N are defined by (8.1) and (8.2). Moreover, denoting by s the number of negative δ j 's, j " 1, . . . , n, of the -small R δ , we also have that µpγ k 2 q δ q " n ´2s `2k 2 N . Proof. Using (8.3) and (8.4), it suffices to choose ą 0 small enough so that Theorem 8.2. Let pM 2n`1 , ξq be a good toric contact manifold admitting a strong symplectic filling W such that c 1 pT W q " 0. There exists K P N such that dim HC 0 n´2s pW q ă dim HC 0 n´2s`dK pW q for all s " 0, 1, . . . , n , d P N .

Proof. Let D Ă R n be the toric diagram of pM 2n`1 , ξq. Pick a rational toric Reeb vector R, denote by γ 1 , . . . , γ m the simple closed R-orbits, associated with the m facets of D, and by N 1 , . . . , N m the corresponding natural numbers given by (8.2). Define K :" 2 lcmpN 1 , . . . , N m , pq , where p :" |π 1 pM q| is the order of the fundamental of group pf M . It follows from the first part of Proposition 8.1, by considering arbitrarily large L P N and k 1 , k 2 multiples of p, that dim HC 0 d 1 pW q ď dim HC 0 d 1 `d2 K pW q for all d 1 , d 2 P N . On the other hand, one can choose the rational toric Reeb vector R so that one of the above simple closed orbits, say γ 1 , has an arbitrarily high mean index, in particular greater than 2n. This implies that µ CZ pγ k 1,δ q ą n for any -small R δ , with small enough, and any k P N. Hence, there is no contribution of any iterate of γ 1,δ to HC 0 n´2s pW q, for any s " 0, 1, . . . , n, but it follows from the second part of Proposition 8.1 that, given s " 0, 1, . . . , n, we can choose such an R δ so that contractible iterates of γ 1,δ do contribute to HC 0 n´2s`dK pW q for all d P N up to an arbitrarily large L P N . Hence, we can indeed conclude that dim HC 0 n´2s pW q ă dim HC 0 n´2s`dK pW q for all s " 0, 1, . . . , n , d P N .

Corollary 1.17 readily follows from this theorem (with s " 0) and Corollary 1.4. Proposition 9.1. Let pM 2n`1 , ξq be a prequantization circle bundle of a closed integral symplectic manifold pB, ωq such that ω| π 2 pBq ‰ 0, c 1 pT Bq| π 2 pBq ‰ 0 and, furthermore, H k pB; Qq " 0 for every odd k or c B ą n. Suppose that M admits a strong symplectic filling W such that c 1 pT W q " 0. Then dim HC 0 n pW q ă dim HC 0 n`jc B pW q or dim HC 0 ´npW q ă dim HC 0 ´n´jc B pW q for every j P N.

Proof. Since c 1 pT W q " 0, we have that c 1 pξq " 0 and this implies that c 1 pT Bq " ρrωs for some ρ P R. The constant ρ must be different from zero by our assumption on c 1 pT Bq. Suppose initially that ρ ą 0. Arguing as in the proof of [24, Theorem 3.1(a)] we conclude that (9.1) dim HC 0 k pW q " ÿ mPN dim H k´2mc B `npB; Λq for every k P Z. Indeed, this is proved in [24, Theorem 3.1(a)] under the assumption that the symplectic form on the filling W is aspherical. Without this assumption, the argument in the proof of [24, Theorem 3.1(a)] applies word-for-word except with the nuance that in HC ˚pW q we have to use coefficients in the universal Novikov field Λ and the action filtration introduced by McLean and Ritter [START_REF] Mclean | The McKay correspondence via Floer theory[END_REF]; see Section 2. (Note here that every periodic orbit of a connection form on M has a unitary trivialization of the contact structure under which the linearized Reeb flow is complex linear [START_REF] Kwon | Brieskorn manifolds in contact topology[END_REF]Remark 8.8] Given p " 1 mod 8 and an even natural number n let Σ p :" Σ pp,2,...,2q be the corresponding Brieskorn sphere of dimension 2n `1. It is proved in [START_REF] Ustilovsky | Infinitely Many Contact Structures on S 4m`1[END_REF] that Σ p admits a non-degenerate contact form α with finitely many simple closed orbits such that the Conley-Zehnder indices of the good periodic orbits have the same parity. It is well known that Σ p admits a symplectic filling given by a Liouville domain W satisfying c 1 pT W q " 0. Therefore, using the fact that HC ˚pW q is the homology of a complex generated by the good closed orbits of α and the computation of the indices of the orbits from [START_REF] Ustilovsky | Infinitely Many Contact Structures on S 4m`1[END_REF]Lemma 4.3] we can conclude that dim HC n pW q " 1 and dim HC n`jp4`2ppn´1qq pW q " 2 for every j P N. (Note here that in [START_REF] Ustilovsky | Infinitely Many Contact Structures on S 4m`1[END_REF] it is used the contact homology degree and the dimension of the Brieskorn sphere is 2n´1.) Therefore, Σ p satisfies the hypotheses of Theorem 1.1 and we conclude the proof of Corollary 1.26.

Proof of Theorem 1.27

The following exact triangle can be found in [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]Theorem 4.4] or [START_REF] Cieliebak | Symplectic homology and the Eilenberg-Steenrod axioms. Appendix written jointly with Peter Albers[END_REF]Corollary 9.26].

Theorem 11.1. Let W be a Liouville domain of dimension 2n `2 and W 1 be obtained from W by attaching a subcritical handle of index k ă n `1. Suppose that c 1 pT W q and c 1 pT W 1 q vanish. Let σ be a non-vanishing section of Λ n`1 C T W and σ 1 a non-vanishing section of Λ n`1 C T W 1 extending σ. We then have an exact triangle HC ˚pD 2pn`1´kq q / / HC ˚pW 1 q

x x HC ˚pW q r´1s h h in which the map HC ˚pW 1 q Ñ HC ˚pW q is the transfer map and the gradings in HC ˚pW q and HC ˚pW 1 q are induced by σ and σ 1 respectively.

The case that k " 1 and W " W 1 \ W 2 corresponds to boundary connected sums. Since HC n pD 2n q and HC n´1 pD 2n q vanish, we have that dim HC n pW 1 #W 2 q " dim HC n pW 1 q `dim HC n pW 2 q. Now, suppose that W 1 and W 2 satisfy the hypothesis of Theorem 1.27, that is, there exist natural numbers K 1 and K 2 such that dim HC n pW 1 q ă dim HC n`jK 1 pW 1 q, dim HC n pW 2 q ă dim HC n`jK 2 pW 2 q for every j P N, where the gradings in HC ˚pW 1 q and HC ˚pW 2 q are induced by σ 1 and σ 2 respectively. Let K " lcmtK 1 , K 2 u. Since the dimension of HC ˚pD 2n q is at most one in any degree, it follows from the exact triangle that dim HC n`jK pW 1 #W 2 q ě dim HC n`jK pW 1 q `dim HC n`jK pW 2 q ´1 ą dim HC n pW 1 q `dim HC n pW 2 q " dim HC n pW 1 #W 2 q for every j P N, where the grading in HC ˚pW 1 #W 2 q is induced by any non-vanishing section σ which extends σ 1 and σ 2 . Hence, under the additional assumption c 1 pT pW 1 #W 2 qq " 0, we conclude that W 1 #W 2 satisfies the hypothesis of Theorem 1.1.

δ `k2 qδ for all k 1

 1 , k 2 ď L, j " 1, . . . , n, and any 0 ă |δ| ă .

9 . 19 Corollary 1 .

 9191 Proof of Corollary 1.19 is a consequence of Theorem 1.1, Corollary 1.4 and the following proposition.

  It turns out that this homology does not depend on the choice of α; see [48, Theorem 8] and [8, Section 2.2]. Now, we will discuss an action filtration introduced by McLean and Ritter [44, Appendix D]. Fix an admissible Hamiltonian H : x

  8 p,0 is a quotient of H p pN ; Qq and therefore we have a map α p : H p pN ; Qq Ñ E 8

			p,0 (c.f. [55, Section 7, Chapter XIII]). The
	composition
		H p pN ; Qq	αp ÝÑ E 8 p,0 ãÑ H p pΛN ; Qq
	is called the edge homomorphism and it turns out that it is the map induced by the projection
	of the fibration (c.f. [55, Theorem XIII.7.2]) which in our case coincides with the map induced
	by the inclusion N	i ãÝ Ñ ΛN . Since this map is injective, we conclude that nothing can survive
	in E 8

  and γ is good 0 otherwise for every closed geodesic γ, we have that m k is the number of good orbits with index k. Consider the Betti numbers b k " dim H k pΛN {S 1 , N ; Qq.

  and this allows us to use Lemma 2.4.) Λq for every j P N. Thus, dim HC 0 n`2jc B pW q ě dim H 2n pB; Λq `ÿ mPN dim H 2n´2mc B pB; Λq ą dim HC 0 n pW q for every j P N. When ρ ă 0 we have the relation Λq for every k P Z. (Notice here that there is a typo in the sign of n in the isomorphism (3.3) of[START_REF] Ginzburg | Multiplicity of Closed Reeb Orbits on Prequantization Bundles[END_REF].) This implies that, for all j P N,

	It follows from (9.1) that
	dim HC 0 n pW q "	ÿ	dim H 2n´2mc B pB; Λq
		mPN
	and
	ÿ mPN dim H 2n`2pj´mqc B pB; (9.2) dim HC 0 n`2jc B pW q " k pW q " dim HC 0 ÿ
	dim HC 0 ´npW q "	ÿ	dim H 2mc B pB; Λq
			mPN
	and
	´n´2jc B pW q " dim HC 0	ÿ
	Hence,
	dim HC 0 ´n´2jc B pW q ě dim H 0 pB; Λq	`ÿ mPN	dim H 2mc B pB; Λq
	ą dim HC 0 ´npW q
	as desired.
	10. Proof of Corollary 1.26

mPN dim H k`2mc B `npB; mPN dim H 2pm´jqc B pB; Λq.

MA and LM were partially funded by FCT/Portugal through UID/MAT/04459/2013 and project PTDC/MAT-PUR/29447/2017. JK was supported by Samsung Science and Technology Foundation (SSTF-BA1801-01). JG is supported by the SFB/TRR 191 "Symplectic structures in Geometry, Algebra and Dynamics". LM was partially supported by CNPq, Brazil.

Proposition 5.1. Let pM 2n`1 , ξq be a contact manifold admitting a displaceable exact contact embedding into an exact symplectic manifold X which is convex at infinity and satisfies c 1 pT Xq| π 2 pXq " 0. Denote by W the compact region in X bounded by M . We have that dim HC 0 n`2k pW q " dim HC 0 n pW q `1 for every k P N.

Proof. Note that, since X is exact, we can take the positive equivariant symplectic homology with rational coefficients. Since M is displaceable, we have that the S 1 -equivariant symplectic homology of W vanishes (see [START_REF] Ritter | Topological quantum field theory structure on symplectic cohomology[END_REF]Theorem 13.4] and [START_REF] Bourgeois | S 1 -equivariant symplectic homology and linearized contact homology[END_REF]Theorem 1.2]). Using this and the Viterbo sequence (see e.g. [START_REF] Frauenfelder | Displaceability and the mean Euler characteristic[END_REF][START_REF] Kang | Equivariant symplectic homology and multiple closed Reeb orbits[END_REF]) we have that

Therefore,

dim H 2i pW, BW ; Qq for every k P N. Since H 2n`2 pW, BW ; Qq -Q, we conclude the desired result.

6. Proof of Corollary 1.9

Let α be a non-degenerate contact form on pS ˚N, ξ can q. We will split the proof according to the three hypotheses. In what follows, we will identify, without fear of ambiguity, N with the subset of constant loops in ΛN {S 1 and ΛN . Moreover, we will take a non-vanishing section of Λ n`1 C T D ˚N induced from the choice of a volume form on N whenever N is orientable.

(i) N satisfies the first hypothesis.

Let r N

τ Ý Ñ N be the universal covering of N . Recall that H ˚pΛ r N {S 1 , r N ; Qq is asymptotically unbounded if there exists a sequence of integers k i Ñ 8 such that dim H k i pΛ r N {S 1 , r N ; Qq Ñ 8 as i Ñ 8; otherwise, we call H ˚pΛ r N {S 1 , r N ; Qq asymptotically bounded. It is not hard to see that the dimension of H ˚pΛ r N {S 1 , r N ; Qq is asymptotically unbounded if, and only if, so is the dimension of H ˚pΛ r N ; Qq. Note that since N is spin so is r N : the corresponding second Stiefel-Whitney classes satisfy w 2 p r N q " τ ˚w2 pN q " 0. By the isomorphism (1.1), if H ˚pΛ r N {S 1 , N ; Qq is asymptotically unbounded so is HC ˚pD ˚r N q. Thus, in this case results from [START_REF] Hryniewicz | Local contact homology and applications[END_REF][START_REF] Mclean | Local Floer homology and infinitely many simple Reeb orbits[END_REF] establish that every Reeb flow on S ˚r N has infinitely many simple closed orbits. This implies that every Reeb flow on S ˚N has infinitely many simple closed orbits since S ˚N is a contact finite quotient of S ˚r N .

Assume then that H ˚pΛ r N {S 1 , r N ; Qq is asymptotically bounded and let m be the dimension of N . By Corollary 1.4, using again the fact that S ˚N is a contact finite quotient of S ˚r N , it is enough to show that there exists K P N such that dim H m´1 pΛ r N {S 1 , r N ; Qq ă dim H m´1`jK pΛ r N {S 1 , r N ; Qq for every j P N.