Lifting classes for the fixed point theory of $n$-valued maps

TL;DR

This paper extends fixed point theory concepts like lifting classes and Reidemeister numbers to n-valued maps using orbit configuration spaces, establishing new algebraic and topological tools for analyzing fixed points.

Contribution

It introduces a novel approach to fixed point theory for n-valued maps via orbit configuration spaces, generalizing classical concepts and establishing algebraic equivalences.

Findings

Reidemeister number for n-valued maps defined via orbit configuration spaces.

Fixed point classes correspond to lift-factors and their fixed points.

Algebraic equivalence of orbit space and universal cover approaches for manifolds of dimension ≥3.

Abstract

The theory of lifting classes and the Reidemeister number of single-valued maps of a finite polyhedron $X$ is extended to $n$-valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicot\'encatl. The liftings of an $n$-valued map $f$ split into self-maps of the universal covering space of $X$ that we call lift-factors. An equivalence relation is defined on the lift-factors of $f$ and the number of equivalence classes is the Reidemeister number of $f$. The fixed point classes of $f$ are the projections of the fixed point sets of the lift-factors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of $X$ such that the number of equivalence classes equals the Reidemeister number. We prove that if $X$ is a manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to $n$-valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an $n$-valued map equals its Reidemeister number. If an $n$-valued map splits into $n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of their Reidemeister numbers.