Nielsen equalizer theory

TL;DR

This paper extends Nielsen theory to equalizer sets for multiple maps, providing a lower bound for their minimal number and applying it to maps between tori, especially in higher dimensions.

Contribution

It introduces the Nielsen equalizer number for multiple maps, generalizing coincidence theory, and computes it explicitly for maps between tori.

Findings

Nielsen equalizer number bounds minimal equalizer points

Equality holds for non-surface domain manifolds

Complete Nielsen number computation for torus maps

Abstract

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k-1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface. As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.