Cockcroft-Swan theorem for projective crossed chain complexes
type: Item
instance of: scholarly article
title: Cockcroft-Swan theorem for projective crossed chain complexes
zbMATH Open document ID: 1370.18015
author: N. A. Khmel'nitskii
published in: Ukrainian Mathematical Journal
publication date: 2016-10-04T00:00:00Z
review text: \textit{W. H. Cockcroft} and \textit{R. G. Swan} [Proc. Lond. Math. Soc. (3) 11, 193--202 (1961; Zbl 0096.37603)] proved that if \(f: C \to C'\) is a chain map between two \(n\)-dimensional projective (free) chain complexes that induces isomorphisms \(f_* : H_j(C)\cong H_j (C') \) for all \(j<n\), then by adding elementary projective (free) complexes to \(C\) and \(C'\), the chain map \(f\) can be extended to a chain isomorphism. An elementary complex is a complex \(D\) with \(D_i=0\;(i\neq q, q+1)\) for some \(q\), and \(D_{q+1}=D_q\) with boundary operator \(\partial _{q+1}=\mathrm{id}\). This theorem was proved in [loc. cit.] and the process is called the stabilization of \(f\). In [Functions on manifolds. Algebraic and topological aspects. Translated from the Russian by V. V. Minachin. Translation edited by Simeon Ivanov. Providence, RI: American Mathematical Society (1993; Zbl 0791.57001)], \textit{V. V. Sharko} proved an analog of Cockcroft-Swan theorem for free crossed chain complexes. In the paper under review, an analog of the Cockcroft-Swan theorem is proved for \(n\)-dimensional projective crossed chain complexes. In this setting, the stabilization of a projective crossed chain complex \((P_i, G, \partial _i)\) is done by replacing the group \(G\) with the group \(G\ast F\) for some finitely generated free group \(F\) and by adding elementary projective complexes at higher dimensions.
reviewed by: Ergün Yalçın
Mathematics Subject Classification ID: 18G35
zbMATH DE Number: 6632552
zbMATH Keywords: Cockcroft-Swan theorem
zbMATH Keywords: projective crossed modules
zbMATH Keywords: chain complexes
MaRDI profile type: Publication
full work available at URL: s11253-015-1058-4
OpenAlex ID: W1470771946
cites work: On the Homotopy Type of Certain Two-Dimensional Complexes
cites work: On the chain equivalence of projective chain complexes
cites work: Q4278732
cites work: Nonabelian algebraic topology. Filtered spaces, crossed complexes, cubical homotopy groupoids. With contributions by Christopher D. Wensley and Sergei V. Soloviev
cites work: Combinatorial homotopy. II
cites work: Free and Projective Crossed Modules
cites work: Q3769623
DOI: 10.1007/S11253-015-1058-4
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description: scientific article; zbMATH DE number 6632552