  We show that if a closed oriented $n$-manifold $M$ has a non-trivial
cohomology class of even degree $k$, whose all pullbacks to products of type
$S^1\times N$ vanish, then the topological complexity $\mathrm{TC}(M)$ is at
least $6$, if $n$ is odd, and at least $7$ or $9$, if $n$ is even. These bounds
extend and improve a result of Mescher and apply for instance to negatively
curved manifolds and to connected sums with at least one such summand. In fact,
better bounds are obtained due to the non-vanishing of the Gromov norm. As a
consequence, in dimension four, we completely determine the topological
complexity of these connected sums, namely we show that it is equal to its
maximum value nine. Furthermore, we discuss realisation of degree two homology
classes by tori, and show how to construct non-realisable classes out of
realisable classes in connected sums. The examples of this paper will quite
often be aspherical manifolds whose fundamental groups have trivial center and
connected sums. We thus discuss the possible relation between the maximum
topological complexity $2n+1$ and the triviality of the center for aspherical
$n$-manifolds and their connected sums.
