Morse homology for the Hamiltonian action in cotangent bundles

TL;DR

This paper constructs a Morse homology for the Hamiltonian action on cotangent bundles, establishing its independence from Hamiltonian choices and its isomorphism with Floer and singular homologies.

Contribution

It introduces a novel Morse complex for the Hamiltonian action on cotangent bundles using a gradient flow approach with new transversality techniques.

Findings

Morse homology is well-defined and independent of Hamiltonian choices.

The Morse homology is isomorphic to Floer homology and singular homology.

New transversality results for infinite-dimensional Hilbert manifolds.

Abstract

In this paper we use the gradient flow equation introduced in [10] to construct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixed regularity space of loops in the cotangent bundle $T^*M$ of a closed manifold $M$. Connections between pairs of critical points are realized as genuine intersections between unstable and stable manifolds, which (despite being infinite dimensional objects) turn out to have finite dimensional intersection with good compactness properties. This follows from the existence of an additional structure, namely a strongly integrable (0)-essential subbundle, which behaves nicely under the negative gradient flow of the Hamiltonian action and which is needed to make comparisons. Transversality is achieved by generically perturbing the negative gradient vector field $-\nabla \mathbb A_H$ of the Hamiltonian action within a class of pseudo-gradient vector fields preserving all good compactness properties of $-\nabla \mathbb A_H$. This follows from an abstract transversality result of independent interest for vector fields on a Hilbert manifold for which stable and unstable manifolds of rest points are infinite dimensional. The resulting Morse homology is independent of the choice of the Hamiltonian (and of all other choices but the choice of the (0)-essential subbundle, which however only changes the Morse-complex by a shift of the indices) and is isomorphic to the Floer homology of $T^*M$ as well as to the singular homology of the free loop space of $M$.