Timeline for Cohomology rings of $ GL_n(C)$, $SL_n(C)$
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 26, 2010 at 17:00 | vote | accept | Evgeny Shinder | ||
| Mar 23, 2010 at 22:01 | comment | added | Tilman | 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". | |
| Mar 23, 2010 at 19:52 | comment | added | Evgeny Shinder | 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". | |
| Mar 19, 2010 at 9:22 | comment | added | Tilman | 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. | |
| Mar 19, 2010 at 9:14 | history | answered | Tilman | CC BY-SA 2.5 |