Timeline for Homology versus cohomology of Lie groups
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2012 at 16:24 | comment | added | Peter May | Surely an alternative proof is not irrelevant! In the Rothenberg-Steenrod spectral sequence, when H_*(G;k) is an exterior algebra, E_2 is a polynomial algebra, there are no differentials for dimensional reasons (if char k is not 2) and it follows immediately that H^*(BG;k) is a polynomial algebra. In char 0, being an exterior algebra is equivalent to being commutative. McCleary's 6.38 is stated for general (perfect) fields, with the evident hypotheses. The universally transgressive hypothesis ensures that H_*(G;k) is an exterior algebra, which is what connects the proofs. Peace. | |
| Jul 1, 2012 at 15:48 | comment | added | Pierre | while using the Hopf algebra structure a little. It was worth providing complements to what I said. I object to the way it was phrased. PS I should make the following point clearer: the simple (and very nice IMHO) approach in McCleary's book does not work for other fields, unless you throw more hypotheses in. | |
| Jul 1, 2012 at 15:46 | comment | added | Pierre | If we want to be precise (and I think we do), this spectral sequence is indeed irrelevant if one wants to prove the result I mentioned, to the effect that $H^*(BG,\mathbb{Q})$ is a polynomial ring (see theorem 6.38 in McCleary's book as well as th 3.27). I never said one had to restrict to $\mathbb{Q}$. As should be apparent now, I was trying to state something as elementary as possible, while... (to be continued) | |
| Jul 1, 2012 at 13:54 | history | answered | Peter May | CC BY-SA 3.0 |