It is well-known that the singular homology of the classifying space of a group $G$ is isomorphic to the group homology of $G$ with coefficients in the trivial $G$-module $\mathbb{Z}$, i.e. $H_*(BG,\mathbb{Z})\cong H_*(G,\mathbb{Z})$. The question is whether there exists a chain level map (naturally defined) $C_*(G) \rightarrow C_*(BG)$ that lifts the above isomorphism in homology?
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3$\begingroup$ This is commented on in Ken Brown's bible on group cohomology, and attributed to Eilenberg-MacLane's 1945 paper "Relations between homology and homotopy groups of spaces". $\endgroup$– Chris GerigCommented Apr 2, 2021 at 1:16
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8$\begingroup$ Depending on which chain complex you use to compute the homology of G and which model of BG you’re talking about, these might literally be the same. They are for my favourite choices here... $\endgroup$– Andy PutmanCommented Apr 2, 2021 at 1:38
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$\begingroup$ To make clear what is already implicitly stated in the previous two comments: the original 1943 paper by Eilenberg and MacLane (Relations between Homology and Homotopy Groups, Proceedings of the National Academy of Sciences 29:5 (1943), 155–158) defined group cohomology as the cohomology of the classifying space. $\endgroup$– Dmitri PavlovCommented Apr 5, 2021 at 16:36
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Indeed, the answer is given in Eilenberg-MacLane's 1945 paper "Relations between homology and homotopy groups of spaces", as pointed out by Chris Gerig.