Obstructions to homotopy invariance of loop coproduct via parameterized fixed‐point theory

Given a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace . We realize the Goresky–Hingston coproduct as a map of spectra, and show that the failure of to entwine the spectral coproducts can be characterized by Chas–Sullivan multiplication with . In particular, when is a simple homotopy equivalence, the spectral coproducts of and agree.