An algebraic derivation of a certain exact sequence
type: Item
instance of: scholarly article
title: An algebraic derivation of a certain exact sequence
zbMATH Open document ID: 0687.20043
DOI: 10.1016/0021-8693(89)90282-2
author: Graham Ellis
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: \textit{R. Brown}, \textit{D. L. Johnson} and \textit{E. F. Robertson} [J. Algebra 111, 177-202 (1987; Zbl 0626.20038)] have used topological methods to obtain the exact sequence \[ H_ 3(G)\to \Gamma (G^{ab})\to J_ 2(G)\to H_ 2(G)\to 0. \] Here \(H_ i(G)\) denotes the i-th integral homomology group of G, \(J_ 2(G)\) is the (abelian) kernel of the canonical homomorphism from the nonabelian tensor square \(G\otimes G\) to G, and \(\Gamma (G^{ab})\) is the so called ``universal quadratic construction'' due to J. H. C. Whitehead. In this paper the author proceeds in an algebraic way, using methods of categorical homology theory (due to Barr-Beck, a.o.). He shows that the above sequence is the right hand end of a long exact sequence involving the integral homology and the derived functors of \(\Gamma\) and \(\otimes\).
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 4125694
zbMATH Keywords: group extension
zbMATH Keywords: integral homomology group
zbMATH Keywords: nonabelian tensor square
zbMATH Keywords: universal quadratic construction
zbMATH Keywords: long exact sequence
zbMATH Keywords: derived functors
reviewed by: Urs Stammbach
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90282-2
OpenAlex ID: W2062066825
cites work: Q5566551
cites work: Some computations of non-Abelian tensor products of groups
cites work: Van Kampen theorems for diagrams of spaces
cites work: Non-abelian exterior products of groups and exact sequences in the homology of groups
cites work: The Ganea Map for Nilpotent Groups
cites work: A certain exact sequence
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description: scientific article; zbMATH DE number 4125694