# Cohomology of $BE_8$ and $BSU(2)$

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.

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.

• $BSU(2) = \Bbb{HP}^\infty$. 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. – Mike Miller Feb 16 '18 at 4:18
• 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" – Mark Grant Feb 16 '18 at 10:05
• $BE_8$ 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/… – André Henriques Feb 18 '18 at 21:22

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:
• 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.) – Xiao-Gang Wen Feb 17 '18 at 7:08