5
$\begingroup$

I want to do some computation which need the mod p cohomologies of classifying spaces of connected compact Lie groups as input. I need the table for both the simply connected case and the central quotient case($G/\Gamma$, $\Gamma\subset Z(G)$). Are there a complete table for all these results.

$\endgroup$
2
  • $\begingroup$ Not for homology, but for homotopy groups, I always end up at the tables of the deputy director of the SCGP: felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/… $\endgroup$ Commented Jan 1, 2017 at 12:18
  • $\begingroup$ If $p$ is coprime to $\lvert W\rvert$, where $W$ is the Weyl group, then you can compute $H^*(BG; \mathbb F_p)$ in terms of a maximal torus, as described in this MO answer. $\endgroup$ Commented Jan 1, 2017 at 17:58

2 Answers 2

3
$\begingroup$

In the book of Mimura-Toda "Topology of Lie groups" you can find a lot of the computations you are interested on.

$\endgroup$
1
  • $\begingroup$ Thank you! Arun Debray. The book really contains some examples like $BG_2$ for $Z_2$ and $BF_4$ for $Z_2$. But I want more: for example $BE_8$ for $p=2,3,5$. Are there a complete table for all these results? $\endgroup$
    – Zhao Xu-an
    Commented May 12, 2019 at 3:20
3
$\begingroup$

Most of the time, H*(BG,𝔽p) is a polynomial algebra, generated in the degrees of the Weyl group.

The exceptional cases are known as torsion primes. If G is simply connected and simple, there are no such primes in type A or C, they are 2 in types B (except B2), D and G, 2 and 3 in types E6 and E7, and 2, 3 and 5 in E8. A precise definition is in Definition 2.43 of "Parity Sheaves", by Juteau, Mautner and Williamson.

To get an idea of the cohomology in one of the unusual cases, there is $$H^*(BSO_3;\mathbb{F}_2)\cong \mathbb{F}_2[b,w] $$ where the generators b and w are in degrees 2 and 3. This is not actually hard to prove via a spectral sequence once you have the guts to believe that the cohomology ring is this big.

For a general reference for computing this cohomology ring, look at "Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups" by Kac.

$\endgroup$
1
  • $\begingroup$ Thank you, Peter. Are there a table for all these results as complete as possible. For example it contains the result for $BE_8$ with $p=2,3,5$. $\endgroup$
    – Zhao Xu-an
    Commented May 12, 2019 at 3:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.