Localization of group rings and applications to 2-complexes
type: Item
instance of: scholarly article
title: Localization of group rings and applications to 2-complexes
zbMATH Open document ID: 0626.57001
DOI: 10.1007/BF02564435
author: Michael N. Dyer
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140074
review text: \textit{S. Rosset} [Math. Z. 185, 211-215 (1984; Zbl 0549.57010)] has shown that for a group G which has a nontrivial normal abelian torsion free subgroup (``G is an R-group'') any finite aspherical complex X with \(\pi_ 1(X)\approx G\) fulfills \(\chi (X)=0\). Rosset's technique concerns localization of group rings, and in the present paper it is applied to problems of 2-complexes. The main result generalizes Rosset's theorem to an exact sequence \(L\hookrightarrow G\twoheadrightarrow H\) with H an R-group, G finitely presented, \(H_ 1(L)\) finitely generated as an abelian group and the minimal number of elements, whose normal closure in G is L, being finite: If \(X^ 2\) is a 2-complex with \(\pi_ 1(G)\approx G\), then \(\chi (X)=0\), and \(\chi (X)=0\) holds if and only if (a) \(H_ 2(L)=0\) and (b) the image of \(\pi_ 2(X)\) in \(H_ 2\) of the covering of X corresponding to L is trivial. After the description of Rosset's method the paper gives an extension of Kaplansky's invariant to localized group rings, and then several beautiful applications to Whitehead's asphericity problem and the deficiency of groups.
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 4017913
zbMATH Keywords: Rosset group
zbMATH Keywords: R-group
zbMATH Keywords: 2-complexes
zbMATH Keywords: Kaplansky's invariant
zbMATH Keywords: localized group rings
zbMATH Keywords: Whitehead's asphericity problem
zbMATH Keywords: deficiency of groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4017913