$S^1$-equivariant Rabinowitz-Floer homology

Abstract

We define the $S^1$-equivariant Rabinowitz-Floer homology of a bounding contact hypersurface $\Sigma$ in an exact symplectic manifold, and show by a geometric argument that it vanishes if $\Sigma$ is displaceable. In the appendix we describe an approach to transversality for Floer homologies for which the moduli space ${\hat{M}}_J$ of all gradient flow lines is compact for some almost complex structure $J$. This approach uses a large set of perturbations, namely vector fields on the loop space, and selects from the possibly non-compact perturbed moduli spaces a part near ${\hat{M}}_J$ that turns out to be compact for small enough perturbations.

44 pages, 4 figures, Main body of the paper to appear in the Hokkaido Mathematical Journal. The new appendix describes an approach to transversality for Floer homologies for which one has compactness a priori. A more detailed account of this appendix is under construction