Averaging formulas for the Reidemeister trace, Lefschetz and Nielsen numbers of $n$-valued maps

TL;DR

This paper develops averaging formulas for Reidemeister, Lefschetz, and Nielsen numbers of n-valued maps on manifolds, providing explicit calculations especially for infra-nilmanifolds, advancing fixed point theory.

Contribution

It introduces averaging formulas linking n-valued map invariants to single-valued maps on covering spaces, with explicit results for infra-nilmanifolds.

Findings

Averaging formula for Reidemeister trace in terms of coincidence traces.

Derived formulas for Lefschetz and Nielsen numbers of n-valued maps.

Explicit formulas for these invariants on infra-nilmanifolds.

Abstract

For an $n$-valued self-map $f$ of a closed manifold $X$, we prove an averaging formula for the Reidemeister trace of $f$ in terms of the Reidemeister coincidence traces of single-valued maps between finite orientable covering spaces of $X$. We then derive analogous formulas for the Lefschetz and Nielsen numbers of $f$. In the special case where $X$ is an infra-nilmanifold, we obtain explicit formulas for the Lefschetz and Nielsen numbers of any $n$-valued map on $X$.