On the topological complexity of manifolds with abelian fundamental group

TL;DR

This paper establishes conditions under which the topological complexity of closed manifolds with abelian fundamental groups is not maximal, extending previous results to broader classes of manifolds and nonorientable surfaces.

Contribution

It generalizes existing results on topological complexity and Lusternik-Schnirelmann category to manifolds with abelian and non-abelian fundamental groups, including nonorientable surfaces.

Findings

Conditions for nonmaximal topological complexity in abelian fundamental group manifolds

Examples demonstrating the sharpness of these conditions

Extension of results to nonorientable surfaces with noncommutative fundamental groups

Abstract

We find conditions which ensure that the topological complexity of a closed manifold $M$ with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik-Schnirelmann 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.