The homology of Peiffer products of groups
type: Item
instance of: scholarly article
title: The homology of Peiffer products of groups
zbMATH Open document ID: 0964.20029
author: N. D. Gilbert
author: William A. Bogley
published in: The New York Journal of Mathematics
publication date: 2000-03-02T00:00:00Z
full work available at URL: https://eudml.org/doc/121056
full work available at URL: 6-3nf.htm
review text: The Peiffer product of groups \(G\bowtie H\) first arose in a topological setting in the work of \textit{J.~H.~C.~Whitehead} [Ann. Math., II. Ser. 42, 409-428 (1941; Zbl 0027.26404)] on the structure of relative homotopy groups, posing the following question about asphericity: Are subgroups of aspherical two-complexes themselves aspherical? which is closely related with the embedding question: when are the natural maps \(G\to G\bowtie H\leftarrow H\) injective? In this paper the authors prove that the Peiffer product of perfect groups is perfect, describe the second integral homology \(H_2(G\bowtie H)\) of a Peiffer product \(G\bowtie H\) in terms of information about the factors \(G\) and \(H\), obtaining that a Peiffer product of superperfect (\(H_1=H_2=0\)) groups is superperfect. They also introduce a double wreath product of cyclic groups as a Peiffer product and determine when a Peiffer product of cyclic groups has trivial second homology.
reviewed by: M. Ladra
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1409628
zbMATH Keywords: homology
zbMATH Keywords: Peiffer products
zbMATH Keywords: asphericity
zbMATH Keywords: perfect groups
zbMATH Keywords: two-complexes
zbMATH Keywords: double wreath products
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1409628