The Goresky-Hingston Coproduct on Based Loop Spaces

TL;DR

This paper develops a lifted coproduct structure on based loop spaces, enabling explicit calculations and understanding of its interactions with fundamental group actions and manifold maps, especially for quotient manifolds.

Contribution

It introduces a lift of the Goresky-Hingston coproduct on based loop spaces and derives formulas for its interactions with the π₁-action and manifold maps.

Findings

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

Established formulas for coproduct interaction with π₁-action

Provided methods to analyze coproduct behavior under manifold maps

Abstract

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 $S^n/G$.