Nielsen coincidence theory for $(n,1)$-valued pairs

TL;DR

This paper extends Nielsen coincidence theory to pairs of multivalued and single-valued maps between manifolds, providing new formulas and a Wecken theorem for this generalized setting.

Contribution

It introduces a generalized Nielsen theory for $(n,1)$-valued pairs, including explicit formulas for invariants on infra-nilmanifolds and a Wecken theorem.

Findings

Proves a Wecken theorem for $(n,1)$-valued pairs.

Derives formulas for Nielsen, Lefschetz, and Reidemeister numbers.

Provides explicit formulas for infra-nilmanifolds based on fundamental group morphisms.

Abstract

We generalise Nielsen theory to coincidences of pairs $(f,g)$ where $f:X\multimap Y$ is $n$-valued multimap and $g:X\to Y$ is a single-valued map, for $X$ and $Y$ closed oriented triangulable manifolds of equal dimension. We prove a Wecken theorem in this setting, and formulas for the Nielsen, Lefschetz and Reidemeister numbers in terms of the analogous invariants for single-valued maps. If $X$ and $Y$ are orientable infra-nilmanifolds, we derive explicit formulas in terms of the fundamental group morphisms of $f$ and $g$.