An averaging formula for the coincidence Reidemeister trace

TL;DR

This paper extends an averaging formula to the coincidence Reidemeister trace for maps between compact orientable manifolds, generalizing previous fixed point results and providing two independent proofs.

Contribution

It introduces a new averaging formula for the coincidence Reidemeister trace, broadening the understanding of fixed point and coincidence theory in topology.

Findings

Proves an averaging formula for the coincidence Reidemeister trace.

Provides two independent proofs of the main theorem.

Includes examples and discusses open questions for nonorientable cases.

Abstract

In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for the local Reidemeister trace. We also give some examples and state some open questions for the nonorientable case.