Homologies of inverse limits of groups

TL;DR

This paper investigates the kernel and cokernel of the natural map between the homology of inverse limits of groups and the inverse limits of their homologies, revealing new phenomena especially for the second homology group.

Contribution

It extends previous results on the first homology to the second, showing the kernel can be non-cotorsion even for abelian groups, and provides detailed analysis for towers of abelian groups.

Findings

For n=1, the kernel is cotorsion, as shown by Barnea and Shelah.

For n=2, the kernel can be non-cotorsion even for abelian groups.

Detailed study of kernels and cokernels for towers of abelian groups.

Abstract

Let $H_n$ be the $n$-th group homology functor (with integer coeffcients) and let $\{G_i\} _ {i \in \mathbb{N}}$ be any tower of groups such that all maps $G_{i+1} \to G_i$ are surjective. In this work we study kernel and cokernel of the following natural map: $$H_n(\varprojlim G_i) \to \varprojlim H_n(G_i)$$ For $n=1$ Barnea and Shelah [BS] proved that this map is surjective and its kernel is a cotorsion group for any such tower $\{G_i\} _ {i \in \mathbb{N}}$. We show that for $n=2$ the kernel can be non-cotorsion group even in the case when all $G_i$ are abelian and after it we study these kernels and cokernels for towers of abelian groups in more detail.