Group (co)homology and classyfing spaces

Question

I would like to ask where I can find in the literature the proof of the following fact: the group cohomology of the group $G$ is naturally isomorphic with the ordinary (say singular) cohomology of the classyfing space $BG$ of $G$.

Answer 1

This is explained in Chater 8 of Weibel's "Introduction to Homological Algebra", specifically Example 8.2.3.

The idea is to revise the construction of $BG$ as the geometric realization of a certain simplicial set — the nerve $N G$ of $G$ viewed as a category: $$BG = |NG|.$$ For any simplicial set $X$ its simplicial homology coincides with the cellular (and hence singular) homology of its geometric realization: $$H_\bullet^\text{Simpl.} (X;\mathbb{Z}) \cong H_\bullet^\text{Sing.} (|X|;\mathbb{Z}).$$ On the other hand, the complex for calculating the simplicial homology of $NG$ coincides with the standard complex for calculating group homology (the one that comes from the bar-resolution), hence $$H_\bullet (G,\mathbb{Z}) \cong H_\bullet^\text{Simpl.} (NG; \mathbb{Z}) \cong H_\bullet^\text{Sing.} (BG; \mathbb{Z}).$$

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Another reference: Section I.6 in Brown's "Cohomology of Groups" (GTM 87). This is more or less the same argument. The author checks that $C_\bullet (EG)$ is a free resolution of $\mathbb{Z}$ by $\mathbb{Z}G$-modules, which coincides with the standard bar-resolution. It only remains to note that if you take the complex of fixed points $C_\bullet (EG)_G$, then you calculate both the group homology $H_\bullet (G,\mathbb{Z})$ and $H_\bullet (BG;\mathbb{Z})$.

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Actually, this is the right way to see where the bar-resolution comes from.