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Rabinowitz Floer homology and symplectic homology. (Homologie de Rabinowitz-Floer et homologie symplectique.) (English. French summary) Zbl 1213.53105

Let \((W,\lambda)\) be a complete convex exact symplectic manifold with symplectic form \(\omega= d\lambda\) and \((M,\xi)\) a contact manifold exact contact embedded in \(W\), i.e., there is a 1-form \(\alpha\) on \(M\) such that \(\text{ker\,}\alpha=\xi\) and \(\alpha-\lambda|_M\) is exact. In the case when \(W\setminus M\) has two connected components, and \(V\) is the bounded component, the first two authors defined Rabinowitz Floer homology groups \(RFH_*(M,W)\) for the Rabinowitz action functional [K. Cieliebak and U. A. Frauenfelder, Pac. J. Math. 239, No. 2, 251–316 (2009; Zbl 1221.53112)]. Under the same assumptions on \(W\), \(V\) and \(M\), there are already defined symplectic (co)homology groups \(SH_*(V)\) and \(SH^*(V)\) [C. Viterbo, Geom. Funct. Anal. 9, No. 5, 985–1033 (1999; Zbl 0954.57015) and P. Seidel, Current developments in mathematics, 2006. Somerville, MA: International Press. 211–253 (2008; Zbl 1165.57020)].
In this paper, it is shown that the groups \(RFH_*(M,W)\) do not depend on \(W\), so we can use the notation \(RFH_*(V)\) (Prop. 3.1). A new version \(S^\vee H_*(V)\) of symplectic homology associated to a \(\vee\)-shaped Hamiltonian is related to the usual version by the long exact sequence
\[ \cdots \to SH^{-*}(V)\to SH_*(V)\to S^\vee H_*(V)\to SH^{-*+1}(V)\to\cdots, \]
(Th. 1.2. proved in §2.7). Then, the main result of this paper is the existence of an isomorphism
\[ RFH_*(V)\cong S^\vee H_*(V), \]
(Th. 1.5. proved in §6).
Most parts of this paper are devoted to the proof of Th. 1.5. Applications of Th. 1.5, such as the Weinstein conjecture in displaceable manifolds (Cor. 1.9), and computations of Rabinowitz Floer homology groups for cotangent bundles (Th.1.10.). The authors remark that, by a discovery of Thomas Krugh, the isomorphism \(SH_*(DT^*L)\cong H_*(\Lambda)\) holds only for \(\mathbb{Z}/2\)-coefficients or if \(L\) is spin. The assumptions used in Th. 1.10, the condition of \(RFH_*(V)\) to be independent of \(V\) (Th. 1.14) and the nondisplaceability and impossibility condition of exact contact embedding (Th. 1.17, Cor. 1.18) are given in §1 (Introduction), together with their proofs.
Definitions of symplectic (co)homology groups and Rabinowitz Floer homology groups are given in §2 and §3. Symplectic homology groups are limits of Floer homology groups defined by Hamiltonians \(H_t: \check V\to\mathbb{R}\), \(t\in S^1\), where \(\check V\) is the symplectic completion of \(V\), with the action
\[ {\mathcal A}_H(x)= \int^1_0 x^*\lambda- \int^1_0 H(t,x(t))\,dt, \]
where \(x: S^1\to V\) is a loop. \(H\) is assumed to be negative on \(V\), having only nondegenerate 1-periodic orbits and the form \(H(r,x)= ar+ b\) for large \(r\) with \(0< a\not\in\text{Spec}(M,\lambda)\) and \(b\in\mathbb{R}\). Here, \(\text{Spec}(M, \lambda)\) is the set of periods of closed Reeb orbits called the action spectrum. By these data, Floer homologies \(FH^{a,b}_k(H)\) are defined. Since there is a homotopy if \(H_1< H_2\), taking limit with respect to \(H\), the symplectic homology groups \(SH^{(a,b)}_k(V)\) are defined (§2.3). \(S^\vee H_*(V)\) is defined similarly, but taking \(H\) to be a \(\vee\)-shaped Hamiltonian, which is non-positive on a tubular neighborhood of \(M\times\{1\}\) and positive elsewhere (§2.7).
Rabinowitz-Floer homology is defined taking a Hamiltonian \(H: W\to\mathbb{R}\) such that \(H^{-1}(0)= M\) and its Hamiltonian vector field \(X_H\) has compact support and agrees with the Reeb vector field along \(M\). Its Rabinowitz action \(A_H:{\mathcal L}\times \mathbb{R}\to \mathbb{R}\). \({\mathcal L}= C^\infty(S^1,W)\) is defined by
\[ A_H(x,\eta)= \int^1_0 x^*\lambda- \eta \int^1_0 H(x(t))\,dt. \]
Then, Rabinowitz-Floer homology groups are defined as limits of truncated Floer homology groups \(FH^{(a, b)}(A_H,J)\) corresponding to action values \((a, b)\) (§3). To show Th.1.5, the perturbed Rabinowitz action
\[ A_{H,b,c}(x,\eta)= \int^1_0 x^*\lambda- b(\eta) \int^1_0 H(x(t))\,dt+ c(\eta) \]
is used. In §4, after hard estimates, two classes of \(h\) provide good Floer homology groups for the proof of Th. 5.1 are extracted. Their interpolations are studied in §5. After these preparations, Th. 1.5 is proved in §6, the last section.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53D35 Global theory of symplectic and contact manifolds
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