# Group cohomology and singular cohomology of classifying space

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?

• 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. – William Mar 1 at 17:43