# Fixed points of n-valued maps on surfaces and the Wecken property -- a configuration space approach

## TL;DR

This paper investigates the fixed point properties of n-valued maps on surfaces using configuration spaces and braid groups, establishing the Wecken property for certain surfaces and describing Nielsen numbers via covering spaces.

## Contribution

It proves the Wecken property for n-valued maps on the projective plane and spheres (except for some cases), and provides a method to compute Nielsen numbers using covering spaces and lifts.

## Key findings

- The projective plane has the Wecken property for all n-valued maps.
- The 2-sphere has the Wecken property for all n ≥ 3.
- A partial result is obtained for n=2 on the 2-sphere.

## Abstract

In this paper, we explore the fixed point theory of $n$-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp.\ the $2$-sphere ${\mathbb S}^{2}$) has the Wecken property for $n$-valued maps for all $n\in {\mathbb N}$ (resp.\ all $n\geq 3$). In the case $n=2$ and ${\mathbb S}^{2}$, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split $n$-valued map $\phi\colon\thinspace X \multimap X$ of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering $q\colon\thinspace \widehat{X} \to X$ with a subset of the coordinate maps of a lift of the $n$-valued split map $\phi\circ q\colon\thinspace \widehat{X} \multimap X$.