The Goresky-Hingston Coproduct on Based Loop Spaces

We construct a lift of the Goresky-Hingston coproduct on the based loop space. Using this lift, we produce formulas for the interaction of the coproduct with the ${\pi }_1$-action and with maps of manifolds. These results lets us compute the coproduct on the based loop space for manifolds of the form $S^n\mathrm{/}G$.