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What are the cohomology of the classifying space of $E_8$ group and $SU(2)$ group, $H^*(BE_8;\mathbb{Z})$ and $H^*(BSU(2);\mathbb{Z})$?

In the paper http://homepages.math.uic.edu/~bshipley/ConMcohomology1.pdf , it was given that $H^∗[BSU(2); \mathbb{Z}_2] = \mathbb{Z}_2[u_4]$. But I like to know the result for integer coefficient.

== added ==

Mike Miller answered the $BSU(2)$ part of the question. There is no torsion in $H^*(BSU(2);\mathbb{Z})$. A motivation for me to ask the above question is to find simple compact and simply connected Lie groups $G$, such that $H^*(BG;\mathbb{Z})$ has torsions at certain dimensions. So $SU(2)$ is out.

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    $\begingroup$ $BSU(2) = \Bbb{HP}^\infty$. You get the result $\Bbb Z[u_4]$ the same way you get the corresponding result for $\Bbb{CP}^\infty$ - a nice cell decomposition, say. $\endgroup$
    – mme
    Commented Feb 16, 2018 at 4:18
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    $\begingroup$ You'll find some (incomplete) information on the cohomology of $BE_8$ in the final chapter of Mimura and Toda's book "Topology of Lie Groups I, II" $\endgroup$
    – Mark Grant
    Commented Feb 16, 2018 at 10:05
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    $\begingroup$ $BE_8$ looks like $K(\mathbb Z,4)$ up to all the dimensions that you probably care about: mathoverflow.net/questions/52286/… The cohomology of $K(\mathbb Z,4)$ is messy, but you'll find some information here: mathoverflow.net/questions/237469/… $\endgroup$
    – André Henriques
    Commented Feb 18, 2018 at 21:22

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I believe Appendix 1. in ``Finite H-spaces and Lie Groups" by Frank Adams shows that BE8 has 2,3 and 5-torsion. The letter from E8 at the end of this paper is also quite amusing:

....Be it therefore known and proclaimed among you, that my K-theory K(E8) and that of my classifying space K(BE8) cannot be criticised in this respect, at least at the prime 5. Their conduct is such as would be blameless and above reproach in the K-theory of a space without 5-torsion in its ordinary cohomology.

Given at our palace, etc, etc, and signed E8

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  • $\begingroup$ Thanks Jeff. But at which dimensions $k$, $H^k(BE_8;\mathbb{Z})=\mathbb{Z}_2,\mathbb{Z}_3$, or $\mathbb{Z}_5$? (I cannot find the paper.) $\endgroup$
    – Xiao-Gang Wen
    Commented Feb 17, 2018 at 7:08
  • $\begingroup$ I sent you a pdf of the Adams paper by email. $\endgroup$
    – Jeff Harvey
    Commented Feb 17, 2018 at 13:38

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