Coincidence Reidemeister trace and its generalization

Abstract

We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which coincides with the Hurewicz image of Koschorke's stabilized bordism invariant. To define it, we use a kind of shriek maps appearing string topology. As an application, we compute the coincidence Reidemeister trace for the self-coincidence of the projections of $S^1$-bundles on $\mathbb{C}P^n$. We also mention how to relate our construction to the string topology operation called the loop coproduct.