Floer Homology and Rabinowitz-Floer Homology

Abstract

Let (M,ω) be a closed symplectic 2n-dimensional manifold that is symplectically aspherical with vanishing first Chern class. The (weak) Arnold conjecture states that the number of contractible periodic orbits P 0 (H) of the Hamiltonian vector field X H associ- ated to a Hamiltonian H is bounded from below by the Betti numbers of the manifold with Z_2 coefficients.

These contractible orbits can also be viewed as the fixed points of a Hamiltonian diffeomor- phism. The first goal of this thesis is to prove the Arnold conjecture using Floer homology. Floer homology is an infinite-dimensional type of Morse homology where the peri- odic orbits are described as critical points of the symplectic action functional A_H on loop space. We prove that this homology is well-defined and does not depend on the choice of the Hamiltonian H and almost complex structure J used to define it. To prove the Arnold conjecture, we show that the Floer homology HF_∗ (M) is isomorphic to the Morse homology HM_∗ (M) of M.

In the second part we explore several recent papers on Rabinowitz-Floer homology, a Floer homology associated to the Rabinowitz action functional. The Rabinowitz-Floer homology RFH_∗(Σ,W) is defined for an exact embedding of a contact manifold (Σ,ξ) into a symplectic manifold (W,ω). We look at two applications.

The first one is the existence of leaf-wise fixed points. These are generalizations of fixed points, associated to a coisotropic submanifold. We prove an existence result for leaf-wise fixed points for a hypersurface of contact type (Σ,ξ) in a symplectic manifold.

The second application is orderability of contact manifolds. A contact manifold is or- derable when there exists a partial order on the universal cover of the group of contactomorphisms Cont_0(Σ,ξ). We establish conditions in terms of the Rabinowitz-Floer homology RFH_∗ (Σ,W) under which a Liouville fillable closed coorientable contact manifold (Σ,ξ) with Liouville filling (W,ω) is orderable.