Why is the Stiefel-Whitney class of a manifold $M^n$, $w_n = 0$ for odd n?

Question

I am reading a paper by Massey about Stiefel-Whitney classes. In this paper he claims that it is a "well known fact" that $w_n = 0$ for an n-dimensional manifold when n is odd. I can't figure out why or get my hands on a reference. Any help is appreciated.

Answer 1

On a closed $n$-manifold $X$, $w_n$ is the mod $2$ Euler class: in particular, evaluating it on a mod $2$ fundamental class (which exists regardless of orientability) gives the Euler characteristic $\chi(X) \bmod 2$. So we want to show that when $n$ is odd the Euler characteristic is even, which follows from Poincare duality (over $\mathbb{Z}_2$; we can also work over $\mathbb{Q}$ and pass to the orientable double cover in the nonorientable).

I don't know what happens off the cuff if $X$ isn't closed.