Rabinowitz Floer Homology and Symplectic Homology

The first two authors have recently defined RabinowitzFloer homology groups RFH∗(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 RFH∗(M,W ), which then maps to symplectic cohomology of V . We compute RFH∗(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.