On the topological complexity of manifolds with abelian fundamental group

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.