  We compute the Lusternik-Schnirelmann category and the topological complexity
of no $k$-equal manifolds $M^{(k)}_d(n)$ for certain values of $d$, $k$ and
$n$. This includes instances where $M^{(k)}_d(n)$ is known to be rationally
non-formal. The key ingredient in our computations is the knowledge of the
cohomology ring $H^*(M^{(k)}_d(n))$ as described by Dobrinskaya and Turchin. A
fine tuning comes from the use of obstruction theory techniques.
