Group cohomology of orthogonal groups with integer coefficient

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

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.