Simple homotopy type of the Hamiltonian Floer complex

TL;DR

This paper establishes that for certain symplectic manifolds, the Floer chain complex's continuation maps are simple homotopy equivalences, leading to bounds on Hamiltonian orbits, thus connecting Floer theory with algebraic topology.

Contribution

It introduces a new perspective on Floer complexes by proving continuation maps are simple homotopy equivalences for specific manifolds.

Findings

Continuation maps are simple homotopy equivalences for non-degenerate Hamiltonians of the same slope.

The number of contractible Hamiltonian orbits of period 1 can be bounded from below.

The results apply to aspherical symplectic manifolds with vanishing first Chern class.

Abstract

For an aspherical symplectic manifold, closed or with convex contact boundary, and with vanishing first Chern class, a Floer chain complex is defined for Hamiltonians linear at infinity with coefficients in the group ring of the fundamental group. For two non-degenerate Hamiltonians of the same slope continuation maps are shown to be simple homotopy equivalences. As a corollary the number of contractible Hamiltonian orbits of period 1 can be bounded from below.