  We show that in every commensurability class of cusped arithmetic hyperbolic
manifolds of simplest type of dimension $2n+2\geq 6$ there are manifolds $M$
such that the Stiefel-Whitney classes $w_{2j}(M)$ are non-vanishing for all $0
\leq 2j \leq n$. We also show that for the same commensurability classes there
are manifolds (different from the previous ones) that do not admit a
$\text{spin}^\mathbb{C}$ structure.
