Equivariant cohomology of finite group actions and invariant cohomology classes

Question

Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certainly not an isomorphism with $\mathbb{Z}$-coefficients (take $X = E_W$, the total space of a universal bundle). But is it true with $\mathbb{Q}$-coefficients? Perhaps if we invert the order of the group?

This is a follow up to my prior question about equivariant cohomology. Again, I am referring to the notes http://www-fourier.ujf-grenoble.fr/~mbrion/notesmontreal.pdf on equivariant cohomology by Michel Brion.

I am interested in understanding the proof of Proposition 1 on pages 6 and 7. Let $G$ be a compact Lie group, let $T$ be a maximal torus, let $N$ be the normalizer of $T$ in $G$, and let $W = N/T$ denote the Weyl group. In part (i), we have the $W$-bundle $G/T \rightarrow G/N$, from which the author claims $H^* (G/N) = H^* (G/T)^W$ when using $\mathbb{Q}$-coefficients.

A few sentences later, a similar statement is made for a more arbitrary $W$ bundle. So it seems like the above statement about $W$-invariants of cohomology is true in some generality. Could someone explain why this is true or give a reference? Or perhaps I am mistaken in interpreting this argument.

Answer 1

These results follow from the Cartan-Leray spectral sequence, which for a regular covering map $X\to X/W$ and a commutative ring $k$ of coefficients has $$ E_2^{p,q}=H^p(W,H^q(X;k)) $$ (cohomology of the group $W$ with coefficients in the $kW$-module $H^\ast(X,k)$) and converges to a graded group associated to $H^\ast(X/W)$. A reference is Ken Brown's "Cohomology of groups", section VII.7.

In case the group $W$ is finite, if $|W|$ is invertible in $k$ then $H^p(W;H^q(X;k))=0$ for all $q$ and all $p>0$ (see Brown, Corollary III.10.2). In particular this is true if $k=\mathbb{Q}$. Thus the spectral sequence is concentrated in the $0$ column and therefore collapses, giving $H^\ast(X/W)\cong H^0(W;H^\ast(X;k))$. Since $H^0(W;M)=M^W$ for any group $W$ and any $W$-module $M$, this gives the stated results. So you were exactly right in your first paragraph!

More generally, the same collapse happens for the Serre spectral sequence of the fibration $X\to X_W\to BW$, which has $$ E_2^{p,q}=H^p(BW;H^q(X))\cong H^p(W;H^q(X)), $$ giving the isomorphism $H^\ast_W(X)\cong H^\ast(X)^W$ you mentioned.