Hamiltonian perturbations in contact Floer homology

TL;DR

This paper investigates how contact Floer homology, associated with Liouville domains and contact Hamiltonians, behaves under perturbations, establishing invariance conditions and applying results to contactomorphism rigidity.

Contribution

It provides the first analysis of perturbation invariance in contact Floer homology and applies this to prove a rigidity result for positive loops of contactomorphisms.

Findings

Invariance of contact Floer homology under certain Hamiltonian perturbations

Conditions guaranteeing homology invariance

Algebraic proof of rigidity for positive contactomorphism loops

Abstract

We study the contact Floer homology ${\rm HF}_*(W, h)$ introduced by Merry-Uljarevi\'c, which associates a Floer-type homology theory to a Liouville domain $W$ and a contact Hamiltonian $h$ on its boundary. The main results investigate the behavior of ${\rm HF}_*(W, h)$ under the perturbations of the input contact Hamiltonian $h$. In particular, we provide sufficient conditions that guarantee ${\rm HF}_*(W, h)$ to be invariant under the perturbations. This can be regarded as a contact geometry analogue of the continuation and bifurcation maps along the Hamiltonian perturbations of Hamiltonian Floer homology in symplectic geometry. As an application, we give an algebraic proof of a rigidity result concerning the positive loops of contactomorphisms for a wide class of contact manifolds.