Fixed points of n-valued maps, the fixed point property and the case of surfaces – a braid 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 NU - Normandie Université
4 UNICAEN - Université de Caen Normandie
5 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We study the fixed point theory of n-valued maps of a space X using the fixed point theory of maps between X and its configuration spaces. We give some general results to decide whether an n-valued map can be deformed to a fixed point free n-valued map. In the case of surfaces, we provide an algebraic criterion in terms of the braid groups of X to study this problem. If X is either the k-dimensional ball or an even-dimensional real or complex projective space, we show that the fixed point property holds for n-valued maps for all n ≥ 1, and we prove the same result for even-dimensional spheres for all n ≥ 2. If X is the 2-torus, we classify the homotopy classes of 2-valued maps in terms of the braid groups of X. We do not currently have a complete characterisation of the homotopy classes of split 2-valued maps of the 2-torus that contain a fixed point free representative, but we give an infinite family of such homotopy classes.

Domaines: Topologie géométrique [math.GT]

Daciberg Lima Gonçalves, John Guaschi. Fixed points of n-valued maps, the fixed point property and the case of surfaces – a braid approach. Indagationes Mathematicae, 2018, Virtual special issue - L.E.J. Brouwer after 50 years, 29 (1), pp.91-124. ⟨10.1016/j.indag.2017.03.003⟩. ⟨hal-01469449⟩