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I would like to know the group cohomology of orthogonal groups $SO(n)$, which is the topological cohomology of the classifying space of the group: $H^*(BSO(n);\mathbb{Z}) = $ ? (for example for $n=10$)

I also like to know $H^*(BPSU(n);\mathbb{Z})$ (say for $n=3$), where $PSU(n)=SU(n)/Z_n$ and $Z_n$ is the center of $SU(n)$.

Thanks!

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    $\begingroup$ Note: there exist more than one thing that one might call "group cohomology", and only one of them has the property that $H^*_{group}(G)=H^*(BG)$ (for compact $G$). $\endgroup$
    – André Henriques
    Commented Feb 17, 2013 at 18:21
  • $\begingroup$ For continuous group, the group cohomology that I am interested in is the Borel group cohomology. $\endgroup$
    – Xiao-Gang Wen
    Commented Feb 18, 2013 at 0:51

3 Answers 3

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For a precise answer to your first question, see Theorem 1.5 of

Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.

For your second question, note that there is an isomorphism $PSU(n)\cong PU(n)$ for each $n$, and that the cohomology $H^\ast(BPU(3);\mathbb{F}_3)$ is worked out in

Kono, Akira; Mimura, Mamoru; Shimada, Nobuo Cohomology of classifying spaces of certain associative H-spaces. J. Math. Kyoto Univ. 15 (1975), no. 3, 607–617.

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  • $\begingroup$ Thanks Mark! Do we have any results for spin group: $H^*(BSpin(n),\mathbb{Z}) = $? (say for $n=10$). $\endgroup$
    – Xiao-Gang Wen
    Commented Feb 18, 2013 at 13:25
  • $\begingroup$ You're welcome! For the Spin case take a look at R. Stong's "Notes on cobordism theory", around page 290. You can certainly piece together the additive structure from what's there, I'm not sure about the multiplicative structure with integral coefficients. $\endgroup$
    – Mark Grant
    Commented Feb 18, 2013 at 13:55
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When $n$ is a prime, the additive structure of $H^*(BPU_n, \mathbb Z)$ has been computed independently in

  • Kameko, Masaki; Yagita, Nobuaki, The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups ${\rm PU}(p)$ and exceptional Lie groups, Trans. Amer. Math. Soc. 360 (2008), no. 5, 2265–2284, doi:10.1090/S0002-9947-07-04425-X

and in

  • Vistoli, Angelo, On the cohomology and the Chow ring of the classifying space of ${\rm PGL}_p$, J. Reine Angew. Math. 610 (2007), 181–227, doi:10.1515/CRELLE.2007.071, arXiv:math/0505052.

For $n = 3$, the second paper contains a computation of the multiplicative structure.

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I am clearly too late for the party, but I would like to mention my recent works which concern the cohomology and Chow ring of $BPGL_n$ for $n$ not necessarily a prime number:

  • Gu, X, On the cohomology of the classifying spaces of projective unitary groups, to appear, Journal of Topology and Analysis, doi:10.1142/S1793525320500211, arXiv:1612.00506,

  • Gu, X, Some torsion classes in the Chow ring and cohomology of $BPGL_n$, J. London Math Soc, 103(1), 2021, doi:10.1112/jlms.12368, arXiv:1901.10090.

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