Analysis of Floer Homology, its Natural Ring Structure and its S1-equivariant version, in Relation with the Free Loop Space and Symplectic Invariants

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The target of the proposed project is the theory of Floer homology. We are interested in its version for the free loop space of the underlying symplectic manifold generated by fixed points of Hamiltonian diffeomorphisms. Originally, Floer homology had been developed for closed symplectic manifold in order to prove Arnold's conjecture, exploiting that it is always isomorphic to the ordinary homology. Subsequent further developments showed also essential symplectic invariants encoded in it. Of particular interest is the natural pair-of-pants ring structure. The proposed project consists of different parts. One is concerned with the S1-equivariant Floer homology theory with respect to the action by reparametrizing loops. There also, the ring structure has still to be established and analyzed. It is planned to find more essential relations with the topological structure of the free loop space. This requires to consider classes of non-compact symplectic manifolds, and also a localized version of Floer homology together with its ring structure will be considered. Moreover, the project also asks for a precise relation between S1-equivariant Floer homology and contact homology, in particular for the case of cotangent/unit cotangent bundles with the respective symplectic/contact structure.