Morse homology with DG coefficients

TL;DR

This paper develops a Morse homology theory with derived local system coefficients, connecting it to DG modules over chains on loop spaces, and explores its algebraic and topological properties.

Contribution

It introduces a new framework for Morse homology with DG local system coefficients, linking it to DG Tor and Ext functors and relating to Floer theory.

Findings

Morse homology with DG coefficients is isomorphic to DG Tor and Ext.

The theory recovers homology of total spaces of fibrations.

It establishes invariance, functoriality, and Poincaré duality properties.

Abstract

We develop a theory of Morse homology and cohomology with coefficients in a derived local system, for manifolds and also more generally for colimits of spaces that have the homotopy type of manifolds, with a view towards Floer theory. The model that we adopt for derived, or differential graded (DG) local systems is that of DG modules over chains on the based loop space of a manifold. These encompass both classical (non DG) local systems and chains on fibers of Hurewicz fibrations. We prove that the Morse homology and cohomology groups that we construct are isomorphic to DG Tor and Ext functors. The key ingredient in the definition is a notion of twisting cocycle obtained by evaluating into based loops a coherent system of representatives for the fundamental classes of the moduli spaces of Morse trajectories of arbitrary dimensions. From this perspective, our construction sits midway between classical Morse homology with twisted coefficients and more refined invariants of Floer homotopical flavor. The construction of the twisting cocycle is originally due to Barraud and Cornea with Z/2-coefficients. We show that the twisting cocycle with integer coefficients is equivalent to Brown's universal twisting cocycle. We prove that Morse homology with coefficients in chains on the fiber of a Hurewicz fibration recovers the homology of the total space of the fibration. We study several structural properties of the theory: invariance, functoriality, and Poincar\'e duality, also in the nonorientable case.