The Path-product in Morse Homology with differential graded coefficients

TL;DR

This paper provides a Morse-theoretic framework for a string topology product on path space homology with differential graded coefficients, extending previous singular homology results.

Contribution

It introduces a Morse-theoretic description of the path-product in Morse homology with differential graded coefficients, building on recent tools.

Findings

Defines a Morse-theoretic string topology product

Establishes a module structure over the Chas-Sullivan ring

Extends singular homology results to Morse homology

Abstract

We will use the tools developed in [Rie24] to give a Morse-theoretic description of a string topology product on the homology of the space of paths in a manifold Y with endpoints in a submanifold X and a module structure on this homology over the Chas-Sullivan ring of Y . These operations have been defined and studied by Stegemeyer in [Ste25] in singular homology.