Surfaces and the second homology of a group (Q1092462)

Let G be a group and K(G,1) an Eilenberg-MacLane space, i.e. \(\pi_ 1(K(G,1))\cong G\), \(\pi_ i(K(G,1))=0\), \(i\neq 1\). We give a purely algebraic proof that the second homology group \(H_ 2(G)=H_ 2(G, {\mathbb{Z}})\cong H_ 2(K(G,1))\) is isomorphic to the group of stable equivalence classes of continuous maps \({\mathcal F}\to K(G,1)\) inducing surjections on fundamental groups (resp. surjections \(F\twoheadrightarrow G\), \(F=\pi_ 1({\mathcal F}))\), where \({\mathcal F}\in \{{\mathcal F}_ g = closed\) orientable surface of genus g, \(g\in {\mathbb{N}}\}\). As a corollary we obtain an algebraic proof of the well-known isomorphism \(H_ 2(G)\cong \Omega_ 2(K(G,1))\) (2-dimensional bordism group).