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 $\operatorname{Gr}^+(2, 4) = S^2\times S^2$.



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> 1. What is $H^\*(\operatorname{Gr}^+(k, n); \mathbb{Z}\_2)$? Can it be expressed in terms of $w\_i(\gamma)$?
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What I would really like to know is the answer to the following question (which might be easier to answer than the first question):



> 
> 2. When is $w\_i(\gamma) \in H^\*(\operatorname{Gr}^+(k, n); \mathbb{Z}\_2)$ non-zero?
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What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?
I was surprised to learn that the ring structure of $H^\*({\rm Gr}^+(k,n);\mathbb{Z}\_2)$ seems to be unknown, in general. The ring structure in the case $k=2$ is given in


*Korbaš, Július; Rusin, Tomáš*, [**A note on the $\mathbb Z\_2$-cohomology algebra of oriented Grassmann manifolds**](http://dx.doi.org/10.1007/s12215-016-0249-7), Rend. Circ. Mat. Palermo (2) 65, No. 3, 507-517 (2016). [ZBL1357.57065](https://zbmath.org/?q=an:1357.57065),


and the recent preprint <https://arxiv.org/abs/1712.00284> by Basu and Chakraborty gives partial results in the case $k=3$.


The main tool seems to be the Gysin sequence of the double cover $\pi:{\rm Gr}^+(k,n)\to {\rm Gr}(k,n)$, which (omitting $\mathbb{Z}\_2$ coefficients) looks like
$$ \cdots \to H^i({\rm Gr}(k,n)) \stackrel{\cup w\_1}{\to} H^{i+1}({\rm Gr}(k,n))\stackrel{\pi^\*}{\to} H^{i+1}({\rm Gr}^+(k,n)) \to H^{i+1}({\rm Gr}(k,n))\to \cdots 
$$ 
and implies that $w\_i(\gamma)$ is nonzero in $H^i({\rm Gr}^+(k,n))$ if and only if it's not in the ideal generated by $w\_1(\gamma)$. But I guess determining this is not always straightforward. At least it seems from the above sources that $w\_2$ is always nonzero when $k=2$ and that $w\_2$ and $w\_3$ are usually nonzero when $k=3$. 


