The Goresky-Hingston Coproduct in Morse Homology with DG Coefficients

TL;DR

This paper provides a new description of the Goresky-Hingston coproduct in Morse homology with real coefficients, linking it to the Leray-Serre spectral sequence and enabling explicit computations for certain manifolds.

Contribution

It introduces a novel approach to describe the coproduct using a quasi-isomorphism and spectral sequence techniques, extending the understanding of loop space homology.

Findings

Computed the coproduct for manifolds of the form S^n/G

Linked the coproduct to the diagonal on the manifold and the coproduct on the based loop space

Provided results on inverting morphisms of A_infinity-modules

Abstract

We describe the Goresky-Hingston coproduct on the free loop space with real coefficients via the quasi-isomorphism $C_*(\Lambda M)\simeq C_*(M,C_*(\Omega M))$. This lets us describe the coproduct on the Leray-Serre spectral sequence as the diagonal on the manifold and the coproduct on the based loop space. With this description, we compute the coproduct with real coefficients for manifolds of the form $M=S^n/G$. The appendix contains results on inverting morphisms of $\mathcal{A}_\infty$-modules that are required for our computations.