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Given f: M N a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace T ₁^st (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.
Kenigsberg et al. (Thu,) studied this question.