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

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 $[T]\in \ddot{I}{€}_1^{st}(\mathcal{L}N,N)$. 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.

Accepted version