7
$\begingroup$

What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?

I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{R}/\mathbb{Z})$, for many of $d$.

But I could not show them myself. I find a useful Ref here.

Any systematic results will be highly appreciated.

We may view this from (i) the topological cohomology of the Lie group manifold $E_8$, or from (ii) the group cohomology of $E_8$. Either results/answers are very welcome!

Improve this question
$\endgroup$
2
  • $\begingroup$ You mean the compact form of $E_8$? Or some other real form, or the complex form? $\endgroup$
    – Ben McKay
    Commented Apr 28, 2017 at 21:09
  • $\begingroup$ Concerning (ii): By a well-known lemma, the group cohomology of a compact Lie group $G$ (with any coefficients) is trivial in higher degrees. This is because the group cohomology is a derived functor of the functor of $G$-invariants, and $G$ being compact implies that the latter is exact. $\endgroup$
    – Victor Protsak
    Commented Apr 29, 2017 at 3:53

0

Reset to default

You must log in to answer this question.