# Fixed points of n-valued maps, the fixed point property and the case of surfaces -- a braid approach

## TL;DR

This paper investigates fixed point properties of n-valued maps on various surfaces using braid groups, providing algebraic criteria and classifications for fixed point free maps, especially on surfaces like spheres and tori.

## Contribution

It introduces an algebraic approach via braid groups to analyze fixed point properties of n-valued maps on surfaces, including criteria and classifications.

## Key findings

- Fixed point property holds for n-valued maps on certain spaces like balls and projective spaces.
- Classified homotopy classes of 2-valued maps on the 2-torus.
- Identified infinite families of fixed point free homotopy classes on the 2-torus.

## Abstract

We study the fixed point theory of n-valued maps of a space X using the fixed point theory of maps between X and its configuration spaces. We give some general results to decide whether an n-valued map can be deformed to a fixed point free n-valued map. In the case of surfaces, we provide an algebraic criterion in terms of the braid groups of X to study this problem. If X is either the k-dimensional ball or an even-dimensional real or complex projective space, we show that the fixed point property holds for n-valued maps for all n $\ge$ 1, and we prove the same result for even-dimensional spheres for all n $\ge$ 2. If X is the 2-torus, we classify the homotopy classes of 2-valued maps in terms of the braid groups of X. We do not currently have a complete characterisation of the homotopy classes of split 2-valued maps of the 2-torus that contain a fixed point free representative, but we give an infinite family of such homotopy classes.