Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory

TL;DR

This paper investigates the obstructions to homotopy invariance of the loop coproduct in the context of compact manifolds with boundary, using parametrised fixed-point theory and Reidemeister traces.

Contribution

It introduces a new framework connecting Reidemeister traces with the failure of spectral coproducts to be homotopy invariant, extending understanding of loop space structures.

Findings

Reidemeister trace characterizes failure of spectral coproducts to be homotopy invariant.

Spectral coproducts agree when the homotopy equivalence is simple.

Obstructions are linked to Chas-Sullivan multiplication with the Reidemeister trace.

Abstract

Given $f: M \to N$ a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace $[T] \in \pi_1^{st}(\mathcal{L} N, N)$. We realize the Goresky-Hingston coproduct as a map of spectra, and show that the failure of $f$ to entwine the spectral coproducts can be characterized by Chas-Sullivan multiplication with $[T]$. In particular, when $f$ is a simple homotopy equivalence, the spectral coproducts of $M$ and $N$ agree.