A limit approach to group homology

TL;DR

This paper establishes a novel connection between group homology and limits of coinvariance groups derived from free presentations, providing new insights into the structure of homology groups via algebraic limits.

Contribution

It introduces a limit approach to relate the homology groups of a group to coinvariance groups from free presentations, linking these to the relation module and free Lie ring structures.

Findings

Homology group H_{2n}(G, Z) can be identified with a limit of coinvariance groups.

The limit of certain quotient groups relates to the n-torsion subgroup of H_{2n}(G, Z).

A new algebraic framework connects group presentations to homological invariants.

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