Formality of classifying spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:50:18Z http://mathoverflow.net/feeds/question/46521 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46521/formality-of-classifying-spaces Formality of classifying spaces Geordie Williamson 2010-11-18T18:29:21Z 2011-04-05T13:58:57Z <p>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:</p> <blockquote> <p>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?</p> </blockquote> <p>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 $E_8$.)</p> <p>Note that we know* that $\mathcal{A}$ is formal in characteristic 0.</p> <blockquote> <p>Can one then conclude that it is formal in any characteristic in which the cohomology of $\mathcal{A}$ is torsion free?</p> </blockquote> <p>If so I think this would give the above list of primes.</p> <p>*) 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".</p> http://mathoverflow.net/questions/46521/formality-of-classifying-spaces/60680#60680 Answer by Craig Westerland for Formality of classifying spaces Craig Westerland 2011-04-05T12:53:06Z 2011-04-05T12:53:06Z <p>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 </p> <p>$$H^*(BG, \mathbb{F}_p) \cong H^*(BT, \mathbb{F}_p)^W$$</p> <p>Since $W$ is a reflection group, $H^*(BT, \mathbb{F}_p)^W$ is a polynomial algebra, say on $d$ generators. Pick <em>cocycle</em> 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 $p$.</p> <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.</p>