Timeline for Formality of classifying spaces
Current License: CC BY-SA 2.5
6 events
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| May 25, 2020 at 18:12 | comment | added | HenrikRüping | Let me do a bit of gravedigging here. Let $G$ be the compact lie group $\mathbb{Z}/3$, then $C^*(BG;\mathbb{F}_3)$ is not formal, and especially it is not even quasiisomorphic to a graded commutative dga. If $x$ is a generator of $H^1(BG;\mathbb{F}_3)$, then the Massey product $\langle x,x,x\rangle\in H^2(BG;\mathbb{F}_3)$ is non-zero. I believe for arbitrary $p>2$, the same works with $p$-fold Massey-products. | |
| Apr 8, 2011 at 5:54 | comment | added | Tyler Lawson |
@Craig: If we were talking honest commutativity, then I'd construct maps of associative algebras $F_p[y_i] \to R$ and then use a composite $\otimes F_p[y_i] \to \otimes R \to R$; this uses that associative algebras are closed under tensor, and for commutative objects the product map is a map of commutative algebras. The same is true in this case, you just have to $E_\infty$-up everything or work with some kind of honest commutative or associative replacements and deal with the model category issues.
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| Apr 7, 2011 at 6:27 | comment | added | Geordie Williamson | Yes, this was also my concern. For tori I think one can make this work, but I don't see a priori how it works in this case. (The reason that Bernstein and Lunts develop the whole machinery of the de Rham complex on the classifying space is in order to make a choice of representatives so that they commute.) | |
| Apr 7, 2011 at 2:30 | comment | added | Craig Westerland | That's a good point. Can you say more about how to get the map from the tensor product into the range? | |
| Apr 5, 2011 at 14:54 | comment | added | Tyler Lawson |
What worries me here is that cochains with coefficients in $\mathbb{F}_p$ is only an $E_\infty$-algebra, and so I'm not sure that you can make the generators honestly commute. I think you'd need to try something like mapping the generators in separately by maps of associative algebras (where a polynomial algebra on one generator is free) and then use the fact that the range is $E_\infty$ to construct a map from the tensor product.
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| Apr 5, 2011 at 12:53 | history | answered | Craig Westerland | CC BY-SA 2.5 |