Generalisations of configuration spaces, relations between braid and almost-crystallographic groups, and applications to the study of the Borsuk-Ulam property and multi-valued maps

Abstract

Our project is related to various aspects of the theory of braid groups, configuration spaces and their generalisations. In problem (1), we shall analyse the homotopy type and the fundamental group of certain orbit configuration spaces, notably for free $Z_2$ actions, and of graph configuration spaces. In problem (2), we investigate the connections between braid and almost-crystallographic groups, as well as generalizations involving lower central series and surface braid groups. In the remaining part of the project, we study applications to two problems in which surface braid groups arise naturally. In problem (3), we exploit a formulation of the Borsuk-Ulam property in terms of braid equations, notably for maps between surfaces, the aim being to decide whether given homotopy classes of maps have this property or not. The second application, in problem (4), is to fixed point and coincidence theory of $n$-valued maps, in particular for maps between surfaces. We intend to explore conditions for $n$-valued maps to be deformable to fixed point free (or coincidence free) maps, as well as whether they satisfy the Wecken property. (AU)