Coincidence free pairs of maps

TL;DR

This paper investigates topological coincidence theory by measuring how close pairs of maps are to being coincidence free, and describes the structure of loose pairs within homotopy classes, with explicit calculations for spheres and projective spaces.

Contribution

It introduces a framework for quantifying the 'distance' from coincidence free maps and explores the set of loose pairs in homotopy classes, especially in homotopy groups.

Findings

Nielsen and minimum numbers help measure proximity to coincidence free maps.

A natural filtration of homotopy sets is described.

Explicit calculations for maps into spheres and projective spaces are provided.

Abstract

This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations are carried out for maps into spheres and projective spaces.