Topological complexity of a map

Abstract

We present a survey of recent results on different versions of topological complexity associated to a continuous map. For each version we examine separately a strict form based on sections of certain projection maps, and a homotopy invariant version, based on homotopy sections of the respective projection maps. We discuss similarities and differences between alternative versions and relations with other invariants like the Lusternik–Schnirelmann category and the topological complexity of spaces. We also present several applications of topological complexities of maps to manipulation problems in robotics, to detection of the fixed point property of certain spaces and as a new invariant of homomorphisms of discrete groups.