A limit approach to group homology

Abstract

In this paper, we consider for any free presentation $G=F/R$ of a group $G$ the coinvariance $H_0(G,R_{ab}^{\otimes n})$ of the $n$-th tensor power of the relation module $R_{ab}$ and show that the homology group $H_{2n}(G,\mathbb{Z})$ may be identified with the limit of the groups $H_0(G,R_{ab}^{\otimes n})$, where the limit is taken over the category of these presentations of $G$. We also consider the free Lie ring generated by the relation module $R_{ab}$, in order to relate the limit of the groups ${\gamma }_{n}R/[{\gamma }_{n}R,F]$ to the $n$-torsion subgroup of $H_{2n}(G,\mathbb{Z})$.