We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling.2000 Mathematics Subject Classification: 53D40.We denote by P(H) the set of 1-periodic orbits of X θ H and by P a (H) the set of 1-periodic orbits in a given free homotopy class a in W .Let J denote the set of admissible almost complex structureswhich are compatible with ω and have the following standard form for t large enough:An almost complex structure J ∈ J is called regular for u ∈ M A (γ, γ; H, J) if D u is surjective, and it is called regular if D u is surjective for all γ, γ ∈ P(H), A ∈ H 2 (W ; Z) and u ∈ M A (γ, γ; H, J). It is proved in [15] that the space J reg (H) of regular almost complex structures is of the second category in J . For every J ∈ J reg (H) the space M A (γ, γ; H, J) is a smooth manifold of dimension µ(γ) − µ(γ) + 2 c 1 (T W ), A . From now on we fix some J ∈ J reg (H). According to Floer [12] we have ∂ 2 = 0. We define the symplectic homology groups of the pair (H, J) by SH a * (H, J) := H * (SC a * (H), ∂).Remark 2.3. In view of condition (1) the Novikov ring Λ ω can be replaced by Z[H 2 (W ; Z)], or even by Z at the price of losing the grading. Indeed, the energy of a Floer trajectory depends only on its endpoints, hence the moduli spaces M(γ, γ; H, J) := A M A (γ, γ; H, J) are compact. Therefore the sum (14) involves only a finite number of classes A.By a standard argument [12] the groups SH a * (H, J) do not depend on J ∈ J reg (H). Nevertheless, they do depend on H and, in order to obtain an invariant of (W, ω), we need an additional algebraic limit construction. We define an

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Abstract. A symplectic manifold W with contact type boundary M = ∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M ). We establish a Gysin-type exact sequence in which the symplectic homology SH(W ) of W maps to HC(M ), which in turn maps to HC(M ), by a map of degree −2, which then maps to SH(W ). Furthermore, we give a description of the degree −2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M . ContentsSuch an X is called a Liouville vector field. The 1-form λ := (ι X ω)| M is a contact form on M . We denote by ξ the contact distribution defined by λ and we call (W, ω) a filling of (M, ξ).We assume throughout the paper that (W, ω) satisfies the conditionT 2 f * ω = 0 for all smooth f :where T 2 is the 2-torus. This condition guarantees that the energy of a Floer trajectory (for a definition, see for example [7, Section 2]) does not depend on its homology class, but only on its endpoints. Our main class of examples is provided by exact symplectic forms.Theorem 1 ties together the symplectic homology groups of (W, ω) and the linearized contact homology groups of (M, ξ). Both these invariants encode algebraically the dynamics of the same vector field, the Reeb vector field R λ defined by ker ω| M = R λ and λ(R λ ) = 1. But their natures are quite different: the former belongs to the realm of Floer theory [17,32], whereas the latter belongs to the realm of symplectic field theory (SFT) [16]. Our result can be read as a way to make symplectic homology fit into SFT.Let us introduce some relevant notation. Given a free homotopy class a of loops in W we denote by SH a * (W, ω) the symplectic homology groups of (W, ω) in the homotopy class a. The free homotopy class of the constant loop will be denoted by 0. We also denote by SH + * (W, ω) the symplectic homology groups in the trivial homotopy class truncated at a small positive value of the action functional. We refer to Section 2 for the definitions.Let i : M ֒→ W be the inclusion. Given a free homotopy class a of loops in W we denote by i −1 (a) the set of free homotopy classes in M which are mapped to a via i, and we use the convention i −1 (+) := i −1 (0). We denote by HC i −1 (a) * (M, ξ) the linearized contact homology groups of (M, ξ) based on closed Reeb orbits whose free homotopy class belongs to i −1 (a). We refer to Section 3.1 for the definition.Both SH a * (W, ω) and HC i −1 (a) * (M, ξ) are defined over the Novikov ring Λ ω with Q-coefficients consisting of formal combinations λ := A∈H2(W ;Z) λ A e A , λ A ∈ Q such that #{A|λ A = 0, ω(A) ≤ c} < ∞ for all c > 0. The multiplication in Λ ω is given by the convolution product.We assume the existence of an almost complex structure J such that linearized contact homology is defined. This means that J needs to be regular for rigid holomorphic planes in the symplectic completion of W , as well as for rational holomorphic curves with one positive puncture in the symplectization of M satisfyin...

Abstract. The first two authors have recently defined RabinowitzFloer homology groups RF H * (M, W ) associated to an exact embedding of a contact manifold (M, ξ) into a symplectic manifold (W, ω). These depend only on the bounded component V of W \ M . We construct a long exact sequence in which symplectic cohomology of V maps to symplectic homology of V , which in turn maps to Rabinowitz-Floer homology RF H * (M, W ), which then maps to symplectic cohomology of V . We compute RF H * (ST * L, T * L), where ST * L is the unit cosphere bundle of a closed manifold L. As an application, we prove that the image of an exact contact embedding of ST * L (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided dim L ≥ 4 and the embedding induces an injection on π1. In particular, ST * L does not admit an exact contact embedding into a subcritical Stein manifold if L is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings. IntroductionLet (W, λ) be a complete convex exact symplectic manifold, with symplectic form ω = dλ (see Section 3 for the precise definition). An embedding ι : M ֒→ W of a contact manifold (M, ξ) is called exact contact embedding if there exists a 1-form α on M such that such that ker α = ξ and α − λ| M is exact. We identify M with its image ι(M ). We assume that W \ M consists of two connected components and denote the bounded component of W \ M by V . One can classically [25] associate to such an exact contact embedding the symplectic (co)homology groups SH * (V ) and SH * (V ). We The first two authors have recently defined for such an exact contact embedding Floer homology groups RF H * (M, W ) for the Rabinowitz action functional [9]. We refer to Section 3 for a recap of the definition and of some useful properties. We will show in particular that these groups do not depend on W , but only on V (the same holds for SH * (V ) and SH * (V )). We shall use in this paper the notation RF H * (V ) and call them Rabinowitz Floer homology groups. Remark 1.1. All (co)homology groups are taken with field coefficients. Without any further hypotheses on the first Chern class c 1 (V ) of the tangent bundle, the symplectic (co)homology and Rabinowitz Floer homology groups are Z 2 -graded. If c 1 (V ) = 0 they are Z-graded, and if c 1 (V ) vanishes on π 2 (V ) the part constructed from contractible loops is Z-graded. This Zgrading on Rabinowitz Floer homology differs from the one in [9] (which takes values in 1 2 + Z) by a shift of 1/2 (see Remark 3.2).Our purpose is to relate these two constructions. The relevant object is a new version of symplectic homology, denotedŠH * (V ), associated to " -shaped" Hamiltonians like the one in Figure 1 on page 20 below. This version of symplectic homology is related to the usual ones via the long exact sequence in the next theorem. Theorem 1.2. There is a long exact sequence (1). . .The long exact sequence (1) can be seen ...

We present three equivalent definitions of S 1 -equivariant symplectic homology. We show that, using rational coefficients, the positive part of S 1equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduce a rigorously defined substitute for cylindrical/linearized contact homology based on an S 1 -equivariant construction.Date: June 9, 2014.[S 1 ] * (W ) captures the interaction between the Reeb dynamics on M and the differential topology of W , represented by critical points of a Morse function (these have action close to zero).Remark 1.1 (comparison of notation with [9]). We use the following notational convention. We do not specify in our notation the free homotopy class of loops. Through the whole of Section 2 the notation SH [S 1 ] * and SH +,[S 1 ] * stands for the invariant built on all free homotopy classes simultaneously. Thus, what we denote in section 2 by SH + * coincides with SH + * ⊕ c =0 SH c * in the notation of [9]. Through the whole of Section 3 we need to restrict to a collection C of free homotopy classes for which the transversality assumptions of §3.1 are satisfied. Thus, what we denote in this paper by CH lin * coincides with c∈C HC i −1 (c) * +n−3 in the notation of [9]. Implicitly, in all statements relating symplectic homology and linearized contact homology, we understand that we restrict to such a collection C of free homotopy classes in the notation SH [S 1 ] * and SH +,[S 1 ] * as well.From the perspective of the above remark, it is perhaps useful to remark that the exact triangle (1.1) only carries information related to contractible orbits: the maps having H [S 1 ] * +n (W, M ) as source or target land in, respectively are defined on, the summand of the symplectic homology groups determined by contractible orbits.In order to better situate the arguments of this paper, let us consider the example of S 1 -spaces, i.e. topological spaces endowed with a topological S 1 -action.Each of the above geometric pictures sheds its own light on X Borel . The first point of view gives rise to the Gysin exact triangle in Theorem 1.3(i) and motivates Viterbo's definition of S 1 -equivariant symplectic homology (see [60,12] and §2.1), in which Morse theory on X Borel is seen as Morse theory modulo S 1 on X × ES 1 , where the S 1 -action is free. The second point of view gives rise to the spectral sequence in Theorem 1.3(ii) and motivates the more algebraic approach of Seidel and Smith (see [55,57] and §2.2-2.3), in which Morse theory on X Borel is seen as Morse theoryà la Hutchings [42] for a family of Morse functions on X, parametrized by CP ∞ . The third point of view leads to the isomorphism in Theorem 1.4, and its topological counterpart is the following. Lemma 1.2. Assume Stab(x) is finite at every point x ∈ X. Then the canonical projection X Borel → X/S 1 induces an isomorphism in homology with Q-coefficients.This Lemma has an algebraic counterpart in cyclic homology as the isomorph...