Fixed points of n-valued maps, the fixed point property and the case of surfaces-A braid approach

Author(s)

Goncalves, Daciberg Lima ; Guaschi, John

• Document type: Journal article
• Source: INDAGATIONES MATHEMATICAE-NEW SERIES; v. 29, n. 1, p. 34-pg., 2018-02-01.

Abstract

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. Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). (AU)

• FAPESP's process: 12/24454-8 - Algebraic, geometric and differential topology
• Grantee: Daciberg Lima Gonçalves
• Support Opportunities: Research Projects - Thematic Grants