Characterisation of plus-constructive fibrations
type: Item
instance of: scholarly article
title: Characterisation of plus-constructive fibrations
zbMATH Open document ID: 0531.55012
DOI: 10.1016/0001-8708(83)90087-7
author: A. J. Berrick
published in: Advances in Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The author gives a characterisation of a plus-constructive fibration \({\mathcal E}:\) \(F\to E\to^{p}B\) by proving the equivalence of (i) \({\mathcal E}\) is plus-contructive, that is \({\mathcal E}\) induces a fibration \(F^+\to E^+\to B^+\), (ii) the maximal perfect subgroup \({\mathcal P}\pi_ 1(B)\) of \(\pi_ 1(B)\) acts on \(F^+\) by maps (freely) homotopic to the identity, and (iii) \(\pi_ 1(p)\) is \(EP^ 2R\), that is \(\pi_ 1(p)\) preserves perfect radicals, and \({\mathcal P}\pi_ 1(E)\) acts trivially on \(\pi_*(F^+,*)\). As corollaries of this theorem there are two special cases which cover situations frequently occurring in applications of the plus-construction: 1) Let \(F^+\) be a nilpotent space; then \({\mathcal E}\) is plus-contructive iff \({\mathcal P}_{\kappa 1}(B)\) acts trivially on \(H_*(F;Z)\) (Corollary 2). 2) Let \(N\hookrightarrow G\twoheadrightarrow^{\Phi}Q\) be a group extension, and \({\mathcal P}N=1\); then the fibration B\(N\to BG\to BQ\) is plus-constructive iff \(\Phi\) is \(EP^ 2R\) and [N,\({\mathcal P}G]=1\) (Corollary 3).
Mathematics Subject Classification ID: 55R05
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P05
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 3842751
zbMATH Keywords: maximal perfect subgroup of the fundamental group
zbMATH Keywords: fibrations with nilpotent fibres
zbMATH Keywords: classifying spaces of groups
zbMATH Keywords: plus-constructive fibration
zbMATH Keywords: nilpotent space
zbMATH Keywords: group extension
reviewed by: Götz Brunner
MaRDI profile type: Publication
full work available at URL: 0001-8708(83)90087-7
OpenAlex ID: W2077840187
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description: scientific article; zbMATH DE number 3842751