The role of the Legendre transform in the study of the Floer complex of cotangent bundles

TL;DR

This paper constructs an explicit chain-level inverse to a known isomorphism between Morse and Floer complexes on cotangent bundles, highlighting the role of the Legendre transform and analyzing orientations and homology.

Contribution

It provides a new explicit construction of a chain-level inverse map that preserves action filtration, using Floer trajectories with boundary conditions related to the Hamilton equations.

Findings

The inverse map Psi is an isomorphism at the chain level.

The construction preserves the action filtration.

The Floer homology is isomorphic to the loop space homology with local coefficients.

Abstract

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Phi from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Psi of Phi. Contrary to other previously defined maps going in the same direction, Psi is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Phi and Psi are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of TM on 2-tori in M.