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Can anyone provide me with the reference for the following fact (idea of the proof will be appreciated too):

Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what the assumptions are: reductive complex algebraic group or maybe complex Lie group G with some restrictions. The cases I'm interested in are $GL_n(\mathbb C)$ and $SL_n(\mathbb C)$) is the exterior algebra on the generators of odd degrees, with the number of generators equal to the rank of $G$.

This fact is attributed to H.Hopf, but I wasn't able to find a reference.

Thanks.

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    $\begingroup$ You may find "Connections, Curvature & Cohomology, vol. II" by Greub, Halerpin and Vanstone useful. They prove that for connected, compact Lie group the de Rham cohomology ring is isomorphic as graded algebras to the free tensor algebra generated by the primitive elements. $\endgroup$
    – Somnath Basu
    Commented Mar 18, 2010 at 23:40
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    $\begingroup$ You might want to check Hermann Weyl's "The classical groups: their invariants and representations". $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 19, 2010 at 9:14

3 Answers 3

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If $G$ is a connected Lie group (or just a connected loop space with finite homology) then $H^*(G,\mathbf{Q})$ is a Hopf algebra. Graded connected Hopf algebras over $\mathbf{Q}$ are always tensor products of exterior algebras in odd degrees with polynomial algebras in even degrees. Since polynomial algebras are infinite, they can't occur. The reference to Hopf is probably H. Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), 493–524. For the classification of Hopf algebras, see also A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207.

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    $\begingroup$ This doesn't yet explain that the number of generators is equal to the rank. For this you can use that $H^*(BG;\mathbf{Q}) \cong H^*(BT;\mathbf{Q})^W$, the invariants of the Weyl group in the cohomology of the maximal torus. This is a reflection group acting by reflections on the generators of a polynomial ring, and it's a fact from algebra that the invariant ring is again polynomial with the same number of generators. Then $H*(G;\mathbf{Q})$ is the exterior algebra on the generators shifted down one degree. I can look up references for these things if you need them. $\endgroup$
    – Tilman
    Commented Mar 19, 2010 at 9:22
  • $\begingroup$ Thanks! Can you please give me a reference for the statement that "H(G;Q) is the exterior algebra on the generators shifted down one degree". $\endgroup$
    – Evgeny Shinder
    Commented Mar 23, 2010 at 19:52
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    $\begingroup$ This follows from the Serre or the Eilenberg-Moore spectral sequence for the path-loop fibration G -> EG -> BG. The EM-ss is maybe the simpler one of the two: it collapses at $E^2 = Tor^{H^*(BG;\mathbf{Q})}_{**}(\mathbf{Q},\mathbf{Q})$. You just have to compute that $Tor^{\mathbf{Q}[x]}(\mathbf{Q},\mathbf{Q}) = \bigwedge (\sigma x)$, an exterior algebra on the class $\sigma x$ which is the "transgression" of x -- this is what I meant by shifted down one degree. You can look this up in any treatment of the Eilenberg-Moore spectral sequence, e.g. McCleary's "User's guide to spectral sequences". $\endgroup$
    – Tilman
    Commented Mar 23, 2010 at 22:01
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For classical groups such as $GL_n(F)$, $SL_n(F)$ for $F = \mathbb{R}, \mathbb{C}$, $SU(n)$, $U(n)$ and $O(n)$, you may find the cohomology ring structure and its proof in M. Mimura and H. Toda, Topology of Lie groups I, translations of mathematical monographs, vol 91.

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For $GL_n(\mathbf{C})$ and $SL_n(\mathbf{C})$ we can use the Leray spectral sequences of the mappings to $\mathbf{C}^n\setminus \{ 0\}$ that take a matrix to its last column. For other compact Lie groups (and $\mathbf{Q}$-coefficients) see e.g. A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. (French) Ann. of Math. (2) 57, (1953). 115--207. The case of $\mathbf{Z}$ coefficients is much harder, it was settled only recently for simply-connected groups and the answer in general is unknown.

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  • $\begingroup$ Could you give a reference for the case of Z coefficients? It's somewhat hard to search for online because the search terms are all so similar to the case of Q coefficients, or to GL(n,Z), etc. $\endgroup$
    – Joshua Grochow
    Commented Feb 13, 2019 at 16:26

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