TITLE: What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?
QUESTION [17 upvotes]: 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?

REPLY [3 votes]: This is a partial answer to the second question. 
In the question, I intended $\gamma$ to denote the tautological bundle over $\operatorname{Gr}^+(k, n)$, whereas in Mark Grant's answer, he uses $\gamma$ to denote the tautological bundle over $\operatorname{Gr}(k, n)$. To make things absolutely clear, let $\gamma^+$ denote the tautological bundle over $\operatorname{Gr}^+(k, n)$ instead. As $\pi^*\gamma \cong \gamma^+$, where $\pi : \operatorname{Gr}^+(k, n) \to \operatorname{Gr}(k, n)$ is the natural double cover, we see that $w_i(\gamma^+) = w_i(\pi^*\gamma) = \pi^*w_i(\gamma)$ is non-zero if and only if $w_i(\gamma)$ is not in the ideal generated by $w_1(\gamma)$. 
As the ideal $(\overline{w}_{n-k+1}, \dots, \overline{w}_n)$ is generated by elements of degree at least $n - k + 1$, the elements $w_{i_1}(\gamma)\dots w_{i_t}(\gamma)$ with $i_1 + \dots + i_t \leq n - k$ are independent. In particular, for $1 < i \leq \min\{k, n-k\}$, $w_i(\gamma)$ does not belong to the ideal generated by $w_1(\gamma)$ and hence $w_i(\gamma^+) \neq 0$. Therefore, if $k \leq n - k$, then $w_2(\gamma^+), \dots, w_k(\gamma^+) \in H^*(\operatorname{Gr}^+(k, n); \mathbb{Z}_2)$ are all non-zero. 
This explains the observation at the end of Mark Grant's answer: $w_2(\gamma^+)$ is non-zero in $H^*(\operatorname{Gr}^+(2, n); \mathbb{Z}_2)$ for $n \geq 4$, and $w_2(\gamma^+), w_3(\gamma^+)$ are non-zero in $H^*(\operatorname{Gr}^+(3, n); \mathbb{Z}_2)$ for $n \geq 6$; in addition, $w_2(\gamma^+)$ is non-zero in $H^*(\operatorname{Gr}^+(3, 5); \mathbb{Z}_2)$.