On \(L^ 2\)-homology and asphericity
type: Item
instance of: scholarly article
title: On \(L^ 2\)-homology and asphericity
zbMATH Open document ID: 0887.57002
DOI: 10.1007/BF02760686
author: Jonathan A. Hillman
published in: Israel Journal of Mathematics
publication date: 1998-05-25T00:00:00Z
review text: The paper describes the use of \(L^2\)-homology and \(L^2\)-Betti numbers for questions about deficiency and Euler characteristics of finitely presented groups. For instance, if \( G\) is a finitely presented group with deficiency one and can be written as an extension of \(\mathbb{Z} \) by a finitely generated normal subgroup, then the 2-complex associated to any presentation of optimal deficiency of \(G\) is aspherical. If \(X\) is an \(m\)-dimensional CW-complex with fundamental group \(\pi\) whose universal covering is \((m-1)\)-connected, then \(X\) is aspherical if and only if \(\chi (X)=0\). Applications to the Whitehead conjecture, 4-manifolds and 2-knot groups are discussed.
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 1062235
zbMATH Keywords: \(L^ 2\)-cohomology
zbMATH Keywords: aspherical 2-complexes
reviewed by: Wolfgang Lück
MaRDI profile type: Publication
cites work: Q4086472
cites work: Q4129638
cites work: \(L_ 2\)-cohomology and group cohomology
cites work: Ribbon Knots in S 4
cites work: Amenable groups and Euler characteristic
cites work: Manifolds of even dimension with amenable fundamental group
cites work: Ribbon concordance of knots in the 3-sphere
cites work: A Certain Subgroup of the Fundamental Group
cites work: Aspherical four-manifolds and the centres of two-knot groups
cites work: Q3995077
cites work: Q3359410
cites work: On 3-dimensional Poincaré duality complexes and 2-knot groups
cites work: Q4293067
cites work: On 4-Dimensional Mapping Tori and Product Geometries
cites work: Some remarks on a problem of J. H. C. Whitehead
cites work: Zero divisors and group von Neumann algebras
cites work: \(L^ 2\)-Betti numbers of mapping tori and groups
cites work: Solvable groups that are simple connected at \(\infty\)
cites work: Q4086416
cites work: A vanishing theorem for Euler characteristics
cites work: Examples of 3-knots with no minimal Seifert manifolds
cites work: HNN bases and high-dimensional knots
description: scientific article; zbMATH DE number 1062235