Let $\operatorname{Gr}(k, n)$ and $\operatorname{Gr}^+(k, n)$ denote the unoriented and oriented grassmannians respectively.
The $\mathbb{Z}_2$ cohomology of the unoriented grassmannian is
$$H^*(\operatorname{Gr}(k, n); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_1(\gamma), \dots, w_k(\gamma)]/(\overline{w}_{n-k+1}, \dots, \overline{w}_n)$$
where $\gamma$ is the tautological bundle, $\deg\overline{w}_i = i$ and $\overline{w} = 1 + \overline{w}_1 + \dots + \overline{w}_n$ satisfies $w(\gamma)\overline{w} = 1$.
I was under the impression that for $1 < k < n - 1$, the $\mathbb{Z}_2$ cohomology of the oriented grassmannian is
$$H^*(\operatorname{Gr}^+(k, n); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_2(\gamma), \dots, w_k(\gamma)]/(\overline{w}_{n-k+1}, \dots, \overline{w}_n)$$
but this is false, as can be seen by considering $\operatorname{Gr}^+(2, 4) = S^2\times S^2$.
- What is $H^*(\operatorname{Gr}^+(k, n); \mathbb{Z}_2)$? Can it be expressed in terms of $w_i(\gamma)$?
What I would really like to know is the answer to the following question (which might be easier to answer than the first question):
- When is $w_i(\gamma) \in H^*(\operatorname{Gr}^+(k, n); \mathbb{Z}_2)$ non-zero?