A chain-level model for Chas-Sullivan products in Morse homology with differential graded coefficients

TL;DR

This paper develops a Morse-theoretic, chain-level framework with differential graded coefficients to describe Chas-Sullivan products and related operations on free loop space homology, enhancing algebraic and geometric understanding.

Contribution

It introduces a novel Morse-theoretic, chain-level model for Chas-Sullivan products using differential graded coefficients, with functorial properties and new algebraic constructions.

Findings

Chain-level description of Chas-Sullivan product in Morse homology.

Functorial properties with respect to differential graded coefficients.

Differential graded K{"u}nneth formula and Pontryagin-Thom construction.

Abstract

We use the framework of Morse theory with differential graded coefficients to study certain operations on the total space of a fibration. More particularly, we focus in this paper on a chain-level description of the Chas-Sullivan product on the homology of the free loop space of an oriented, closed and connected manifold. The idea of ''intersecting on the base'' and ''concatenating on the fiber'' are well-adapted to this framework. We also give a Morse theoretical description of other products that follow the same principle. For this purpose, we develop functorial properties with respect to the coefficient in terms of morphisms of A $\infty$ -modules and morphisms of fibrations. We also build a differential graded version of the K{\"u}nneth formula and of the Pontryagin-Thom construction.