Formality of classifying spaces

Question

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".

Answer 1

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."

Answer 2

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.

Answer 3

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.