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Let

$$ O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k). $$

What is $$ H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}^{\oplus k});\mathbb{Z}_2)? $$ Let $\mathbb{Z}<i>=\{a+bi\mid a,b\in\mathbb{Z}\}$. Let $$ O(\mathbb{Z<i>}^{\oplus k})=GL_{\mathbb{C}}(\mathbb{Z<i>}^{\oplus k})\cap U(k). $$ What is $$ H^*(BO(\mathbb{Z<i>}^{\oplus k});\mathbb{Z})? $$

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  • $\begingroup$ Is an element of your first group a $k\times k$ matrix with integer entries? If so then I think this is $Gl_k(\mathbb{Z})$, which might help you get started looking for references. $\endgroup$ – Mark Grant Feb 13 '15 at 13:10
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First of all, this question is fairly close to this question on general linear groups. Computational methods are similar (stable via K-theory, unstable via actions on CAT(0)-spaces), but, well, more involved.

Now for concrete calculations: there is of course a choice of orthogonal groups that can be considered because there are various quadratic forms to consider. The cases of split quadratic forms have been studied a bit more, for the anisotropic forms in the question only very little is known.

In the anisotropic case, the only (unstable) computations I know of are discussed in G. Collinet: Homology stability for unitary groups over $S$-arithmetic rings. J. K-theory 8 (2011), 293-322. The results proved there provide some stabilization results, but the paper mentions a result of Henn and Lannes (which seems to be unpublished still) stating that the reduction map $$ H_\bullet(O_n(\mathbb{Z}[1/2]),\mathbb{Z}_{(2)})\to H_\bullet(O_n(\mathbb{F}_3),\mathbb{Z}_{(2)}) $$
is an isomorphism for $n\leq 14$.

As in the case of the general linear groups, stable cohomology (i.e. cohomology of $O_\infty$) with rational coefficients has been computed by Borel.

For the stable cohomology of $O_\infty(\mathbb{Z}[1/2])$ (maybe also $O_\infty(\mathbb{Z})$) with mod 2 coefficients (or more generally with 2-completed coefficients), you can use the following papers:

  • A.J. Berrick and M. Karoubi: Hermitian K-theory of integers. Amer. J. Math. 127 (2005), 785-823.

  • A.J. Berrick, M. Karoubi and P.A.Østvær: Hermitian periodicity and cohomology of infinite orthogonal groups. J. K-theory 12 (2013), 203-211.

What is shown in the first paper is that the corresponding K-theory space (the plus-construction of the classifying space) sits in a specific homotopy-cartesian square with classifying spaces of orthogonal groups over the $\mathbb{R}$, $\mathbb{F}_3$ and $\mathbb{C}$. This homotopy cartesian square can then be used to compute homotopy groups/K-groups via a long exact sequence or (co)homology groups via a spectral sequence. This is close to the computations of Arlettaz for mod 2 cohomology of $GL_\infty(\mathbb{Z})$ in the the question on general linear groups.

For split orthogonal groups over Gaussian integers with 2 inverted you can look at the Master's thesis of K.J. Moi. Again, the paper contains the computation of the 2-completed homotopy type of the K-theory space. There is still some work required to go from there to cohomology computations...

That about sums up all the computations I can think of right now...

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