Acyclic covers
type: Item
instance of: scholarly article
title: Acyclic covers
zbMATH Open document ID: 0935.57004
author: James A. Schafer
published in: Commentarii Mathematici Helvetici
publication date: 2000-05-03T00:00:00Z
review text: Whitehead's asphericity question, cf. e.g. [\textit{R. Brown} and \textit{J. Huebschmann}, Identities among relations, in `Low-dimensional topology', Lond. Math. Soc. Lect. Note Ser. 48, 153-202 (1982; Zbl 0485.57001)], asks whether any subcomplex of an aspherical 2-complex is itself aspherical. Given a class \(\mathcal P\) of groups containing the trivial group, the acyclic cover questions asks whether, under the assumption that \(X'\) be acyclic, \(\pi_1(X')\) is in \(\mathcal P\) only if \(X'\) is contractible. In the paper under review, it is proved that, given a class of groups \(\mathcal P\), if there exists a subcomplex \(L\) of an aspherical 2-complex \(K\) which is not aspherical, and if the maximal perfect subgroup of \(\pi_1(L)\) is in \(\mathcal P\), there exists a regular, acyclic, infinite cyclic cover \(Y'\) of some 2-complex \(Y\) having \(\pi_1(Y')\) in \(\mathcal P\). Hence a positive answer to the acyclic cover question would limit the maximal perfect subgroups that could appear in any counterexample to the asphericity question. In the paper, some evidence for a positive answer to the acyclic cover question is given for the class of finite groups. More precisely, it is proved that, given a regular infinite cyclic cover \(X'\) of a finite 2-complex \(X\), when \(X'\) is acyclic and has \(\pi_1(X')\) amenable, it is necessarily aspherical; hence when \(\pi_1(X')\) is finite, it is necessarily trivial and \(X'\) is contractible. (A group is amenable if every action thereof on a compact metric space has an invariant Borel measure.).
reviewed by: J. Huebschmann
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 57M20
zbMATH DE Number: 1333318
zbMATH Keywords: 2-complexes
zbMATH Keywords: covering spaces
zbMATH Keywords: amenable groups
zbMATH Keywords: Whitehead's asphericity question
MaRDI profile type: Publication
full work available at URL: s000140050084
OpenAlex ID: W4242982029
DOI: 10.1007/S000140050084
description: scientific article; zbMATH DE number 1333318