  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.
