Coincidence Reidemeister trace and its generalization

TL;DR

This paper introduces a homotopy invariant construction of the Reidemeister trace for coincidences of maps between closed manifolds, generalizing previous concepts and connecting to string topology.

Contribution

It provides a new homotopy invariant of the Reidemeister trace for coincidences, realized via homology classes and shriek maps, extending previous invariants to more general settings.

Findings

Computed the coincidence Reidemeister trace for self-coincidences of $S^1$-bundle projections.

Connected the construction to Koschorke's stabilized bordism invariant.

Related the invariant to string topology operations like the loop coproduct.

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.