On the fixed points of a Hamiltonian diffeomorphism in presence of fundamental group

TL;DR

This paper develops a refined Floer homology approach to establish new lower bounds on the number of periodic orbits of Hamiltonian diffeomorphisms, especially relating to the fundamental group's properties.

Contribution

It introduces a refined Floer chain complex for covering spaces and links the number of periodic orbits to the fundamental group's minimal number of generators.

Findings

Number of periodic orbits ≥ minimal number of generators of fundamental group for finite solvable or simple groups.

Existence of 1-periodic orbit with Conley-Zehnder index 1-n for manifolds with infinite fundamental group.

New lower bounds for periodic orbits based on fundamental group properties.

Abstract

Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and derive from it new lower bounds for the number of periodic orbits. We prove in particular that if the fundamental group of M is finite and solvable or simple, then the number of periodic orbits is not less than the minimal number of generators of the fundamental group. For a general closed symplectic manifold with infinite fundamental group, we show the existence of 1-periodic orbit of Conley-Zehnder index 1-n for any non-degenerate 1-periodic Hamiltonian system.