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 S^2) has the Wecken property for n-valued maps for all n ∈ N (resp. all n ≥ 3). In the case n = 2 and S^2 , we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map φ : X ⊸ X of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q : X^ −→ X with a subset of the coordinate maps of a lift of the n-valued split map φ • q : X^ ⊸ X. In the final part of the paper, we classify the homotopy classes of non-split 2-valued maps of the 2-torus, and we give explicit formulae for the Nielsen number of such 2-valued maps using our description of it.
Type de document :
Pré-publication, Document de travail
2017


https://hal.archives-ouvertes.fr/hal-01469497
Contributeur : John Guaschi <>
Soumis le : jeudi 16 février 2017 - 15:01:01
Dernière modification le : vendredi 17 février 2017 - 01:07:48

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  • HAL Id : hal-01469497, version 1
  • ARXIV : 1702.05014

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Daciberg Lima Gonçalves, John Guaschi. Fixed points of n-valued maps on surfaces and the Wecken property – a configuration space approach. 2017. <hal-01469497>

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