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Pré-Publication, Document De Travail Année : 2017

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

Résumé

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.
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Dates et versions

hal-01469497 , version 1 (16-02-2017)
hal-01469497 , version 2 (24-04-2017)

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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-01469497v1⟩
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