The Goresky-Hingston Coproduct in Morse Homology with DG Coefficients

We describe the Goresky-Hingston coproduct on the free loop space with real coefficients via the quasi-isomorphism $C_{\ast }(\mathrm{\Lambda }M)\simeq C_{\ast }(M,C_{\ast }(\mathrm{\Omega }M))$. 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 $M=S^n\mathrm{/}G$. The appendix contains results on inverting morphisms of ${\mathcal{A}}_{\mathrm{\infty }}$-modules that are required for our computations.