Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ converges to $H^*(E)$. We can consider singular cohomology with coefficients over a field $k$ and the spaces involved are connected $CW$-complexes for simplicity.

As an aside remark, this spectral sequence arises from the filtration $F_pC_n(E)$ generated by those simplices $\sigma \in C_n(E)$ such that $\pi_*\sigma = \tau(i_0,\ldots, i_n)$ for some $\tau \in C_p(BG)$ . There is a chain equivalence between $E_0^{p,q}$ and $C_p(BG; C^q(X))$ which gives the term described above.

On the other hand, $H^*(E)$ can be also computed using group cohomology. Consider $F_*$ a free resolution of $k$ as $k[G]$-module, then $H^*(E) \cong H^*(\text{Tot}(\hom_{k[G]}(F_{*}, H^*(X)))$. Filtering the double complex $\hom_{k[G]}(F_{*}, H^*(X))$ we have an spectral sequence

$E_2^{p,q} = H^p(G;H^q(X))$ converging to $H^*(E)$.

I want to see if these two spectral sequences are canonically isomorphic; in other words, is there a map between the filtered modules $C^*(E)$ and $\hom_{k[G]}(F_{*}, H^*(X))$ which induces an isomorphism on the $E_2$-page of the respective spectral sequences?

  • $\begingroup$ If you construct $BG$ as the classifying space of the category with one object with automorphism group $G$, I think you can get a chain map from the cellular-cohains of $BG$ to a resolution of $G$ which induces the isomorphism $H^*(BG)\cong H^*(G)$ where this left side is cellular cohomology and the right is group cohomology. These might assemble into the map of filtered modules you're looking for, but I'm not sure. I forget what the resolution of $G$ is supposed to be. $\endgroup$ – William Mar 1 at 17:43

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