Approximating \(\ell_2\)-Betti numbers of an amenable covering by ordinary Betti numbers
type: Item
instance of: scholarly article
title: Approximating \(\ell_2\)-Betti numbers of an amenable covering by ordinary Betti numbers
zbMATH Open document ID: 0946.57029
author: Beno Eckmann
published in: Commentarii Mathematici Helvetici
publication date: 2000-10-19T00:00:00Z
review text: The author gives a simple homological proof of a recent result by \textit{J. Dodziuk} and \textit{V. Mathai} [J. Funct. Anal. 154, No. 2, 359-378 (1998; Zbl 0936.57018)] on the \(l_2\)-Betti numbers of an amenable covering of a finite cell complex. These \(l_2\)-Betti numbers [\textit{J. Cheeger} and \textit{M. Gromov}, Topology 25, 189-215 (1986; Zbl 0597.57020)], can be approximated by the ordinary Betti numbers \(\frac{\beta_p(Y_j)}{N_j}\) of a Følner exhaustion \(Y_j\) of the covering \(Y\), where \(N_j\) is the number of translates of a fundamental domain of \(Y\) necessary to cover \(Y_j\).
reviewed by: Clemens Berger
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 55N99
zbMATH DE Number: 1278816
zbMATH Keywords: amenable groups
zbMATH Keywords: covering spaces
zbMATH Keywords: \(l_2\)-homology
MaRDI profile type: Publication
full work available at URL: s000140050081
OpenAlex ID: W2095493290
DOI: 10.1007/S000140050081
description: scientific article; zbMATH DE number 1278816