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?



Group homology and singular homology
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.


