Floer theory and reduced cohomology on open manifolds

TL;DR

This paper develops a new framework for Hamiltonian Floer theory on open symplectic manifolds, utilizing reduced cohomology and non-Archimedean analysis to handle lower semi-continuous functions, with applications to periodic orbits.

Contribution

It introduces a novel approach to Floer theory on open manifolds using reduced cohomology and energy confinement techniques, extending the scope of Floer homology to more general Hamiltonians.

Findings

Constructed Floer complexes for lower semi-continuous Hamiltonians.

Established functorial continuation maps and algebraic operations.

Applied the theory to problems on periodic orbits and displaceability.

Abstract

We construct Hamiltonian Floer complexes associated to continuous, and even lower semi-continuous, time dependent exhaustion functions on geometrically bounded symplectic manifolds. We further construct functorial continuation maps associated to monotone homotopies between them, and operations which give rise to a product and unit. The work rests on novel techniques for energy confinement of Floer solutions as well as on methods of Non-Archimedean analysis. The definition for general Hamiltonians utilizes the notion of reduced cohomology familiar from Riemannian geometry, and the continuity properties of Floer cohomology. This gives rise in particular to localized Floer theory. We discuss various functorial properties as well as some applications to existence of periodic orbits and to displaceability.