# How is the equivariant cohomology of a space related to the cohomology of the corresponding associated bundle

Let $X$ be a manifold with a left $G$-action, and let $\Sigma$ be a Riemann surface. How is the equivariant cohomology $H^*_G(X)$ of $X$ related to the de Rham cohomology of the associated bundle $H^*(P\times_GX)$, where $P$ is a $G$-bundle over $\Sigma$?

The following is my attempt at a solution. We know that $H^*_G(X)=H^*(EG\times_GX)$, where $EG\rightarrow BG$ is the universal or platonic $G$-bundle, with the base space $BG$ (the classifying space).

Also, any $G$-bundle is a pullback of $EG$, e.g., given a map $f:\Sigma \rightarrow BG$, we have $$P= f^*EG$$ for a $G$-bundle $P\rightarrow\Sigma$.

Therefore, we find that $H^*(P\times_GX)= H^*(f^*EG\times_GX)$. If $H^*(f^*EG\times_GX) = f^*H^*(EG\times_GX)$, then we have $$H^*(P\times_GX)= f^*H^*_G(X).$$ But is it true that $H^*(f^*EG\times_GX) = f^*H^*(EG\times_GX)$? References would be appreciated.

I'm not even sure what you mean by $f^*H^*(EG \times_G X)$. The image of this cohomology under the pullback map? Before you make any other guesses, think very carefully about the case where the $G$ action on $X$ is trivial: $H_G(X)=H(BG)\otimes H(X)$, but $P\times_G X\cong \Sigma \times X$. The pullback $f^*$ is obviously not surjective in this case.
Instead, there's a natural fiber bundle $P\times_G X\to X/G$ (don't worry about what the latter means; it's an Artin stack, but that fact is a distraction) with fiber $P$; you have to apply the Leray-Serre spectral sequence to this bundle.
• Thank you for your answer. I am interested in the case where $P\times_G X$ is a fibration over $\Sigma$, does your answer still apply in this case? Also, do you have any references with more details on how the Serre spectral sequence can be used as you indicated? Mar 26, 2017 at 6:59
The de Rham cohomology of the associated bundle $H^∗(P×_GX)$ is indeed the pullback of the equivariant cohomology $H^∗_G(X)$. See the commutative diagram (2.10) in 'Equivariant Cohomology and Gauged Bosonic $\sigma$-models' by Figueora-O'Farrill and Stanciu (https://arxiv.org/pdf/hep-th/9407149.pdf).