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

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 $\phi\circ q\colon\thinspace \widehat{X} \multimap X$.
Type de document :
Pré-publication, Document de travail
To appear in Science China Math. 2017

Littérature citée [20 références]

https://hal.archives-ouvertes.fr/hal-01469497
Contributeur : John Guaschi <>
Soumis le : lundi 24 avril 2017 - 11:52:10
Dernière modification le : dimanche 17 décembre 2017 - 06:54:05

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MultVaJohnBoju10Avr-2017.pdf
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### Identifiants

• HAL Id : hal-01469497, version 2
• ARXIV : 1702.05014

### Citation

Daciberg Lima Gonçalves, John Guaschi. Fixed points of n-valued maps on surfaces and the Wecken property – a configuration space approach. To appear in Science China Math. 2017. 〈hal-01469497v2〉

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