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.
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$\begingroup$ 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/… $\endgroup$– Jason StarrCommented Jan 1, 2017 at 12:18
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$\begingroup$ 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. $\endgroup$– Arun DebrayCommented Jan 1, 2017 at 17:58
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In the book of Mimura-Toda "Topology of Lie groups" you can find a lot of the computations you are interested on.
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$\begingroup$ 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? $\endgroup$ Commented May 12, 2019 at 3:20
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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.
answered Jun 9, 2017 at 4:33
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$\begingroup$ 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 $p=2,3,5$. $\endgroup$ Commented May 12, 2019 at 3:25