Homologies of inverse limits of 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: $$H_n(\underleftarrow{\lim }{G}_i)\to \underleftarrow{\lim }{H}_n(G_i)$$ 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.

12 pages

Cited by in corpus (1)

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