6.1 Classifying spaces
Let u\colon X\to S be a continuous map. For every sheaf F on S, the sheaf u^*F is endowed with a “descent data” with respect to u, i.e. we have an isomorphism between the two inverse images of u^*F on X\times_S X, and this isomorphism satisfies a cocycle condition. If u admits a section locally on S, then this construction defines an equivalence between the category of sheaves on S and that of sheaves on X endowed with a descent data.
Take X to be a (left) principal homogeneous space for the group G on a space S. Then the G-equivariant sheaves on X are exactly the sheaves endowed with a descent data: every equivariant sheaf on X is, in a unique way (as an equivariant sheaf), the inverse image of a sheaf on S=X/G.
If a topological group G acts on a space X, then G acts on G^{\Delta_n}\times X by g\cdot(g_0,\ldots,g_n,x) = (g_0g^{-1},\ldots,g_ng^{-1},gx). We denote by [X/G]_\bullet the simplicial space [X/G]_\bullet = \big((G^{\Delta_n}\times X)/G\big)_{n\geqslant0}. \tag{6.1.2.1}
If X is a principal homogeneous space for the group G on S=X/G, then the map \begin{aligned} G^{\Delta_n}\times X &\to X^{\Delta_n} \\(g_0,\ldots,g_n,x) &\mapsto (g_0x,\ldots,g_nx) \end{aligned} identifies [X/G]_n with the iterated fibre product (X/S)^{\Delta_n}: \operatorname{cosq}(X\to S) = ([X/G]_\bullet\to S). In particular, we have (by (iii) of (5.3.5)) \operatorname{H}^\bullet([X/G]_\bullet) \simeq \operatorname{H}^\bullet(X/G) \tag{6.1.2.2} (for a principal homogeneous space).
For all n, G^{\Delta_n}\times X is a principal homogeneous space for the group G on [X/G]_n. For every equivariant sheaf F on X, \operatorname{pr}_2^*F is an equivariant sheaf on G^{\Delta_n}\times X; by (6.1.1), the latter is the inverse image of F^n on [X/G]_n. Every equivariant sheaf F on X thus defines a sheaf on [X/G]_\bullet. It is easy to show that we thus obtain an equivalence between the category of equivariant sheaves on X and the category of sheaves F^\bullet on [X/G]_\bullet that satisfy
(*) For every f\colon\Delta_n\to\Delta_m, the structure morphism f^*F^n\to F^m is an isomorphism.
Construction (b) above is natural in (G,X,F). We set \operatorname{H}^\bullet(X,G;F) = \operatorname{H}^\bullet([X/G]_\bullet,F^\bullet) \tag{6.1.2.3} (mixed cohomology of X,G with coefficients in F). Under the hypotheses of (a), if F is the inverse image of F^{-1} on S=X/G, then \operatorname{H}^\bullet(X,G;F) = \operatorname{H}^\bullet(X/G,F^{-1}) \tag{6.1.2.4} (for a principal homogeneous space). This generalises (6.1.2.2) (which is the case F=\underline{\mathbb{Z}}).
Let P^t be the topological space consisting of a single point. We define the simplicial classifying space of G, denoted B_{\bullet G}, to be the simplicial space B_{\bullet G} = [P^t/G]_\bullet. Let P be a principal homogeneous space for the group G on S. The evident morphism \operatorname{cosq}(P\to S) = [P/G]_\bullet \to [P^t/G]_\bullet = B_{\bullet G} defines a composite morphism [P] \colon \operatorname{H}^\bullet(B_{\bullet G}) \to \operatorname{H}^\bullet([P/G]_\bullet) \xleftarrow[(6.1.2.2)]{\sim} \operatorname{H}^\bullet(S). \tag{6.1.3.1}
We will see below that, in good cases, \operatorname{H}^\bullet(B_{\bullet G})=\operatorname{H}^\bullet(B_G), and that the image of [P] consists of the characteristic classes of P.
Let G be a Lie group, B_G a classifying space for G, and a\colon U_G\to B_G the universal principal homogeneous G-space. Let X be a G-space; then X\times U_G is a principal homogeneous G-space over X\times U_G/G such that, for every equivariant sheaf F on X, \operatorname{pr}_1^*F is the inverse image of a sheaf F^G on X\times U_G/G. Since U_G is contractible, and by (6.1.2.2), we have \operatorname{H}^\bullet(X,G;F) \xrightarrow{\sim}\operatorname{H}^\bullet(X\times U_G,G;\operatorname{pr}_1^*F) \xleftarrow{\sim}\operatorname{H}^\bullet(X\times U_G/G,F^G). \tag{6.1.4.1}
In particular, for X=P^t, \operatorname{H}^\bullet(B_{\bullet G}) = \operatorname{H}^\bullet(B_G). \tag{6.1.4.2}
We can see that the isomorphism in (6.1.4.2) is a particular case of (6.1.3.1) where S=B_G and P=U_G.
The spectral sequence in (5.2.3.2) for B_{\bullet G} E_1^{pq} = \operatorname{H}^q(G^{\Delta_p}/G) \Rightarrow\operatorname{H}^{p+q}(B_{\bullet G}) =\operatorname{H}^{p+q}(B_G) is essentially the Eilenberg–Moore spectral sequence. We briefly recall how it allows us to relate the rational cohomologies of G and of B_G, for connected G.
- The algebra \operatorname{H}^\bullet(G,\mathbb{Q}) is a connected graded Hopf algebra of finite dimension over \mathbb{Q}. If P^\bullet(G) is the graded module of its primitive elements, then we have H^\bullet(G,\mathbb{Q}) = \bigwedge P^\bullet(G) and the generators of P^\bullet(G) are of odd degree.
- The simplicial algebra (E_1^{p\bullet})_{p\geqslant0} is E_1^{p\bullet} = \bigwedge\big(P^\bullet(G)^{\Delta_p}/P^\bullet(G)\big) which is the exterior algebra of the suspension of the constant cosimplicial module P^\bullet(G); we thus have (by Quillen (Quillen 1968)) that E_2^{p\bullet}=\operatorname{Sym}^p(P^\bullet(G)). The pages E_2^{pq} are only zero for p+q even; we thus have that E_2^{pq}=E_\infty^{pq}, and, for a suitable filtration, we canonically have that \operatorname{Gr}\operatorname{H}^\bullet(B_G,\mathbb{Q}) \simeq \operatorname{Sym}^\bullet(P^\bullet(G)[-1]) and non-canonically that \operatorname{H}^\bullet(B_G,\mathbb{Q}) \simeq \operatorname{Sym}^\bullet(P^\bullet(G)[-1]).
Let G be a (complex) linear algebraic group.
- If T is a maximal torus of G, with Weyl group W, then \operatorname{H}^\bullet(B_G,\mathbb{Q}) \xrightarrow{\sim}\operatorname{H}^\bullet(B_T,\mathbb{Q})^W. \tag{6.1.6.1}
- If T is a torus with character group X(T), then \operatorname{H}^\bullet(T,\mathbb{Z}) \simeq \bigwedge^\bullet X(T) \tag{6.1.6.2} (an isomorphism of graded Hopf algebras).
We will only use (a) in the following weaker form:
a’. (The splitting principle). The map \operatorname{H}^\bullet(B_G,\mathbb{Q})\to\operatorname{H}^\bullet(B_T,\mathbb{Q}) is injective.
For completion, we recall a proof of (a’).
If B is a Borel subgroup of G, then the bundle U_G/B on B_G is a fibre in flag spaces. By (Deligne 1968, 2.1 and 2.6.3), which is better explained in (Griffiths 1970, Proposition 3.1), or by (Blanchard 1956), the Leray spectral sequence of U_G/B\to B_G degenerates to rational cohomology. We thus have that \operatorname{H}^\bullet(B_G,\mathbb{Q})\hookrightarrow\operatorname{H}^\bullet(U_G/B,\mathbb{Q}). We conclude by noting that U_G/B\sim B_B\sim B_T.