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

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Abstract

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S2) has the Wecken property for n-valued maps for all n ∈ ℕ (resp. all n ≥ 3). In the case n = 2 and S2, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q: X̂ → X with a subset of the coordinate maps of a lift of the n-valued split map .

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Acknowledgements

The first author was supported by Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP), Projeto Temático Topologia Algébrica, Geométrica e Diferencial (Grant No. 2012/24454-8). The second author was supported by the same project as well as the Centre National de la Recherche Scientifique (CNRS)/Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP) Projet de Recherche Conjoint (PRC) project (Grant No. 275209) during his visit to the Instituto de Matemática e Estatística, Universidade de São Paulo, from the 4th to the 22nd of February 2017.

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Authors and Affiliations

  1. Departamento de Matemática & IME, Universidade de São Paulo, São Paulo, 05508-090, Brazil

    Daciberg Lima Gonçalves

  2. Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, UNICAEN, Normandie Université, CS 14032, 14032, Caen Cedex 5, France

    John Guaschi

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  1. Daciberg Lima Gonçalves
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  2. John Guaschi
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Correspondence to Daciberg Lima Gonçalves.

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Dedicated to Professor Boju Jiang on the Occasion of His 80th Birthday

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Gonçalves, D.L., Guaschi, J. Fixed points of n-valued maps on surfaces and the Wecken property—a configuration space approach. Sci. China Math. 60, 1561–1574 (2017). https://doi.org/10.1007/s11425-017-9080-x

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  • DOI: https://doi.org/10.1007/s11425-017-9080-x

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