Hyperbolic manifolds without ${\text{spin}}^{\mathbb{C}}$ structures and non-vanishing higher order Stiefel-Whitney classes

We show that in every commensurability class of cusped arithmetic hyperbolic manifolds of simplest type of dimension $2n+2\ge 6$ there are manifolds $M$ such that the Stiefel-Whitney classes $w_{2j}(M)$ are non-vanishing for all $0\le 2j\le n$. 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.