Cohomology of Stiefel manifolds

Question

Define the complex Stiefel space $W_{n,k}$ as $U(n)/U(k)$. What is its (co)homology? (Either singular or de Rham).

I've searched through a bunch of classical references but can't seem to find this information anywhere. I'd be over the moon with either a reference, or even just a sketch proof.

Edit: Quite a few papers seem to refer to specific facts without giving any reference. For instance, I have seen statements such as $H_{2k+1}(W_{n,k};\mathbb{Z})\cong\mathbb{Z}$, but with no real explanation as to why this is true. This leads me to believe that the (co)homology of these complex Stiefel spaces is classically well-known, but I can't seem to find where.

Answer 1

Per OP's request:

I don't have it in front of me, but I'm pretty sure this is in Fuchs' Topology II. Also, almost any paper of Bob Stong's would contain this calculation. Maybe try his Notes on Cobordism Theory.

Found in: Fuchs and Viro's Topology II, Theorem III.3.2.A, page 218 (Springer).

Answer 2

See pages 301-302 of Hatcher's Algebraic Topology for some partial results.