For a graph
$\Gamma$
, let
$K(H_{\Gamma },\,1)$
denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group
$H_{\Gamma }$
defined by
$\Gamma$
. We use the relationship between the combinatorics of
$\Gamma$
and the topological complexity of
$K(H_{\Gamma },\,1)$
to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer
$n$
, we construct a graph
$\mathcal {O}_n$
whose TC-generating function has polynomial numerator of degree
$n$
. Additionally, motivated by the fact that
$K(H_{\Gamma },\,1)$
can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.
On the LS-category and topological complexity of projective product spaces
