On the higher homotopy groups of a finite CW-complex
type: Item
instance of: scholarly article
title: On the higher homotopy groups of a finite CW-complex
zbMATH Open document ID: 1068.55005
author: Mihai Damian
published in: Topology and its Applications
publication date: 2005-06-01T00:00:00Z
review text: Suppose that \(X\) is a finite CW complex of dimension \(q \geq 2\); this paper gives some results on the higher homotopy groups of \(X\). If the first Betti number \(\beta_{1}(X)\) is large enough, \(>q\) will do, then in the case that \(X\) has an abelian fundamental group, \(\pi_{i}(X)\) is not finitely generated for some \(i = 2, \ldots ,q\). Similar results are obtained when \(\chi(X) \neq 0\) and \(\pi_{1}(X) = {\mathbb Z} \times G\) where \(G\) is finitely generated. In the case where \(\beta_{1}(X)= q \) or \(q-1\) the same result is shown with the proviso that \(X\) does not have the homotopy type of a torus.
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 57Q10
zbMATH DE Number: 2173010
zbMATH Keywords: Betti number
zbMATH Keywords: Novikov homology
zbMATH Keywords: fibrations over a circle
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
full work available at URL: j.topol.2004.10.002
OpenAlex ID: W2016335148
cites work: The Whitehead group of a polynomial extension
cites work: Fibering manifolds over a circle
cites work: On the structure of complete manifolds of nonnegative curvature
cites work: Formes fermées non singulières et propriétés de finitude des groupes
cites work: The obstruction to fibering a manifold over a circle
cites work: Stable pseudoisotopy spaces of compact nonpositively curved manifolds
cites work: Q3247612
cites work: Contributions à la théorie du type simple d'homotopie
cites work: Whitehead torsion
cites work: Q4742016
cites work: Surgery on the Novikov complex
cites work: Q4210473
cites work: Q4003910
cites work: Finite domination and Novikov rings
cites work: Groupes d'homotopie et classes de groupes abéliens
cites work: A total Whitehead torsion obstruction of fibering over the circle
cites work: Q5522742
DOI: 10.1016/J.TOPOL.2004.10.002
description: scientific article; zbMATH DE number 2173010