A limit approach to group homology

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 $H_{2n}(G,{\mathbb Z})$.

💡 Research Summary

The paper investigates a novel categorical‑limit description of even‑dimensional group homology. Starting with an arbitrary group (G) and a free presentation (G=F/R) (where (F) is a free group and (R) a normal subgroup), the authors focus on the relation module (R_{ab}=R/