Floer Homology with DG Coefficients. Applications to cotangent bundles

TL;DR

This paper develops Hamiltonian Floer homology with DG local coefficients for symplectically aspherical manifolds, enabling new computations and applications, including a Viterbo isomorphism for cotangent bundles and criteria for periodic orbits.

Contribution

It introduces DG local coefficients into Floer homology, extending the framework and establishing a Viterbo isomorphism with these coefficients for cotangent bundles.

Findings

Defined Floer homology with DG local coefficients.

Proved a Viterbo isomorphism theorem with DG coefficients.

Established criteria for existence of contractible periodic orbits.

Abstract

We define Hamiltonian Floer homology with differential graded (DG) local coefficients for symplectically aspherical manifolds. The differential of the underlying complex involves chain representatives of the fundamental classes of the moduli spaces of Floer trajectories of arbitrary dimension. This setup allows in particular to define and compute Floer homology with coefficients in chains on fibers of fibrations over the free loop space of the underlying symplectic manifold. We develop the DG Floer toolset, including continuation maps and homotopies, and we also define and study symplectic homology groups with DG local coefficients. We define spectral invariants and establish general criteria for almost existence of contractible periodic orbits on regular energy levels of Hamiltonian systems inside Liouville domains. In the case of cotangent bundles, we prove a Viterbo isomorphism theorem with DG local coefficients. This serves as a stepping stone for applications to the almost existence of contractible closed characteristics on closed smooth hypersurfaces. In this context, our methods allow to access for the first time the dichotomy between closed manifolds that are aspherical and those that are not.