Lucile Vandembroucq (11/19/20): Topological complexity for manifolds with abelian fundamental group

Title: On the nonmaximality of the topological complexity for manifolds with abelian fundamental group

Abstract: We will give sufficient conditions for the (normalized) topological complexity of a closed manifold M with abelian fundamental group to be nonmaximal, that is to have TC(M) less than 2dim(M), and see through examples that our conditions are sharp. This generalizes for manifolds some results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the condition of commutativity of the fundamental group, we also generalize Dranishnikov's results on the LS-category of the cofibre of the diagonal map $\Delta: M \to M \times M$ for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds. Joint work with Dan Cohen.