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Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes called a projective classical group. What is known about the integral cohomology $H^*(G/Z(G);\mathbb{Z})$? I am particularly interested in the integral cohomology of the projective special unitary group $PSU(n)$. I would appreciate any and all references/suggestions.

Thanks!

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Ancient history! I don't remember exactly what is in it, but I think the paper by Paul Baum and William Browder "The cohomology of quotients of classical groups" Topology 3 1965 305–336, considers the cohomology of these groups. Of course, even back then, calculations were made in mod $p$ cohomology first, relying on the Bockstein spectral sequence to bootstrap up to integral information. I don't know of any more recent work that is directly relevant.

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