equivariant cohomology

Question

In this question we consider only cohomology with rational coefficients. All groups will be connected Lie groups. All group actions will be smooth.

Let $M$ be a manifold. Let $G$ be a group acting on $M$. Let $H$ be a normal Lie subgroup of $G$. What can be said about the relation between $H^\ast_{G/H}(M)$ and $H^\ast_G(M)$?

When is $H^\ast_{G/H}(M)$ ring-isomorphic to $H^\ast_{G/H}\otimes_{\mathbb{Q}}H^\ast_G(M)$?

Consider the following particular case: $M$ is the complex projective space $\mathbb{P}^{n-1}$, $G=(\mathbb{C}^*)^n$

acting in the standard way, and $H$ is the sub-torus

$\{(t,\ldots,t):t\in\mathbb{C}^*\}$ of $G$.

Then, $$H^*_G=\mathbb{Q}[\alpha_1,\ldots,\alpha_n],$$ while

$$H^*_{G/H}=\mathfrak{\mathbb{Q}[\alpha_1,\ldots,\alpha_n]}{\alpha_1+\ldots+\alpha_n=0},$$ and

the above ring isomorphism holds.

Answer 1

If $G$ is compact and $H$ is a closed normal subgroup of $G$ that acts trivially on $M$, then there is a spectral sequence $$E_2^{i,j}=H_{G/H}^i(M;H^j(H;A)) \Rightarrow H^{i+j}_G(M;A)$$ where $A$ is a $G$-module.

If $H$ is central in $G$ and $A$ is a trivial $G$-module then all coefficients are trivial. Otherwise they are understood to be local coefficients.

This is a result of Duflot and can be found in section 3 of her celebrated paper "Depth and equivariant cohomology".

Added: Concerning your "hope" expressed in the comment above: If $A=k$ is a field (with trivial $G$-action) and $H$ is central, then the $E_2$-term becomes $$E_2^{\ast,\ast}=H^\ast_{G/H}(M;k)\otimes_k H^\ast(H;k).$$

By taking $G$ finite and $M$ a point it's obvious that one can't hope for more in general.