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

Daciberg Lima Gonçalves (1, 2) , John Guaschi (3, 4, 5)

1 IME - Instituto de Matemática e Estatística
2 USP - Universidade de São Paulo = University of São Paulo
3 LMNO - Laboratoire de Mathématiques Nicolas Oresme
4 UNICAEN - Université de Caen Normandie
5 NU - Normandie Université

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 $\phi\circ q\colon\thinspace \widehat{X} \multimap X$.

Domaines: Topologie géométrique [math.GT]

Daciberg Lima Gonçalves, John Guaschi. Fixed points of n-valued maps on surfaces and the Wecken property – a configuration space approach. Science China Mathematics, 2017, 60 (9), pp.1561-1574. ⟨10.1007/s11425-017-9080-x⟩. ⟨hal-01469497v2⟩