If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean?
More generally, does the Stiefel-Whitney height of $M$, i.e., the largest $k$ for which $\\w_1^k(M)\neq 0$, have a geometric meaning? (e.g. in terms of orientations)