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Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let

$$ O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\} $$

What is $$ H^*(BO(\mathbb{Z}_2^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}_2^{\oplus k});\mathbb{Z}_2)? $$ or for prime $p\geq 3$, $$ H^*(O(\mathbb{Z}_2^{\oplus k});\mathbb{Z}_p)? $$

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    $\begingroup$ It would be better to use $GL_k(\mathbb{F}_2)$ instead of $O(\mathbb{Z}_2^{\oplus k})$. You can find computations in Chapter 1 of Knudson's "Homology of linear groups" or in Chapter VII of Adem-Milgram "Cohomology of finite groups". $\endgroup$
    – Matthias Wendt
    Commented Feb 16, 2015 at 8:29

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More of a long comment.

The paper you may look into is On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field by Daniel Quillen.

He first shows that $$BGL(F_q) \cong F\psi^q$$ where $F_q$ is a field of order $q$, $\psi^q$ are the Adams operations on $BU$ and $F\psi^q$ is the homotopy equalizer of $1$ and $\psi^q$. Then he computes the cohomology rings. See Theorem 1 and Theorem 2 of the paper.

Here $$BGL(F_q) = \bigcup_k BGL_k(F_q).$$ I do not know the answer to cohomology of $BGL_k(F_q)$ for a fixed $k$.

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