Abstract

The Lusternik-Schnirelmann (LS) category of a space was introduced to obtain a lower bound on the number of critical points of a C1-function on a given manifold. The topological complexity (TC) of a space is an integer-valued homotopy invariant motivated by the motion planning problem from robotics. Its definition resembles the one of Lusternik-Schnirelmann category and its topological properties are an active topic of research in topology. However, its connections to critical point theory have not been fully explored yet and in this talk, I will give an overview of how to transfer methods from Lusternik-Schnirelmann theory to topological complexity. After presenting the classical Lusternik-Schnirelmann theorem and its method of proof, I will explain how to generalize this result and present some lower bounds on numbers of critical points of functions, expressed in terms of topological complexity and related invariants. This is joint work with Maximilian Stegemeyer (Uni Freiburg).