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We show that in every commensurability class of cusped arithmetic hyperbolicmanifolds 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 thereare manifolds (different from the previous ones) that do not admit a$\text{spin}^\mathbb{C}$ structure.

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