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Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on $BG$ with coefficients in a field or characteristic $p$ (or if you prefer the dg algebra of endomorphisms of the constant sheaf). My question is:

Is it known for which primes $p$ the dg algebra $\mathcal{A}$ is formal, that is, quasi-isomorphic to a dg algebra with trivial differential?

I assume / hope that the answer is that this is true if $p$ is not a torsion prime for $G$ (i.e. $p$ arbitrary in types $A$ and $C$, $p \ne 2$ in types $B$, $D$ and $G_2$, $p \ne 2, 3$ in types $F_4$, $E_6$ and $E_7$, and $p \ne 2,3,5$ in type $E_8$.)

Note that we know* that $\mathcal{A}$ is formal in characteristic 0.

Can one then conclude that it is formal in any characteristic in which the cohomology of $\mathcal{A}$ is torsion free?

If so I think this would give the above list of primes.

*) for example because $H(BG, \mathbb{Q})$ is a poynomial algebra, and $\mathcal{A}$ admits a graded commutative model using the de Rham complex -- see Bernstein-Lunts "Equivariant sheaves and functors".

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3 Answers 3

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This is an old question. But sometimes old questions get answered!

Benson, Greenlees, Formality of cochains on BG

Here is the abstract:

Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(T)/T|$ is invertible in the field $k$ then the algebra of cochains $C_*(BG;k)$ is formal as an $A_\infty$ algebra, or equivalently as a DG algebra.

(Note that this is not the same condition as I postulate in the question. I still wonder if it is true under the "$p$ not a torsion prime" hypothesis. However in the meantime, I should understand the above paper...)

Update: latest version of paper completely answers question! From Section 5:

"...if $G$ is a connected, simply connected compact Lie group, $k$ is a field of characteristic $p$, and $p$ is not a torsion prime then $H^∗(BG; k)$ is a polynomial ring on even classes, and $C^∗(BG; k)$ is formal."

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  • $\begingroup$ What is the complete answer? $\endgroup$
    – LSpice
    Commented May 24, 2022 at 2:16
  • $\begingroup$ @LSpice: I added the final quote, which answers the question I asked completely. $\endgroup$ Commented May 24, 2022 at 22:09
  • $\begingroup$ Re, thanks! $\endgroup$
    – LSpice
    Commented May 24, 2022 at 22:27
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I've only just seen this rather old thread. I've recently been computing with cochains on $BG$ for $G$ a finite group in characteristic $p$, and have some rather surprising conclusions. If $G$ has either semidihedral or generalised quaternion Sylow $2$-subgroups and no normal subgroup of index two, then the cochains on $BG$ with mod two coefficients is formal. It would be interesting to have a classification of when this occurs for finite, or compact Lie groups. I don't think the answer is at all obvious.

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    $\begingroup$ Happy MO birthday Dave! $\endgroup$
    – Mark Grant
    Commented Nov 13, 2021 at 14:45
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I think that you outlined the proof. In more detail, let $W$ be the Weyl group of $G$, and $T$ its maximal torus. Pick $p$ coprime to $|W|$; this allows to ignore higher $W$ group cohomology in the computation

$$H^*(BG, \mathbb{F}_p) \cong H^*(BT, \mathbb{F}_p)^W$$

Since $W$ is a reflection group, $H^*(BT, \mathbb{F}_p)^W$ is a polynomial algebra, say on $d$ generators. Pick cocycle representatives $x_1, \dots, x_d \in C^*(BG, \mathbb{F}_p)$. Now let $R = \mathbb{F}_p[y_1, \dots, y_d]$ be the free graded commutative algebra on generators $y_i$ in the same degree as $x_i$, and equip $R$ with the $0$ differential. By freeness (and the fact that $d(x_i) = 0$), you get a map $R \to C^*(BG, \mathbb{F}_p)$ of DGA's which sends $y_i$ to $x_i$. You know (because you constructed it that way) that it induces an isomorphism in cohomology, and so $BG$ is formal at the prime $p$.

If you have some other mechanism for ensuring that $H^*(BG, \mathbb{F}_p)$ is a polynomial algebra (e.g., the statement is known integrally, as for $G = U(n)$, $Sp(n)$), the same argument works.

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    $\begingroup$ 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. $\endgroup$ Commented Apr 5, 2011 at 14:54
  • $\begingroup$ That's a good point. Can you say more about how to get the map from the tensor product into the range? $\endgroup$ Commented Apr 7, 2011 at 2:30
  • $\begingroup$ 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.) $\endgroup$ Commented Apr 7, 2011 at 6:27
  • $\begingroup$ @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. $\endgroup$ Commented Apr 8, 2011 at 5:54
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    $\begingroup$ 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. $\endgroup$ Commented May 25, 2020 at 18:12

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