Four-manifolds with \(\pi_1\)-free second homotopy
type: Item
instance of: scholarly article
title: Four-manifolds with \(\pi_1\)-free second homotopy
zbMATH Open document ID: 1032.57019
author: Fulvia Spaggiari
published in: Manuscripta Mathematica
publication date: 2003-08-17T00:00:00Z
review text: The author studies closed connected topological 4-manifolds \(M\) for which \(\pi_2(M)\) is a \(\Lambda\)-free module, where \(\Lambda\) is the integral group ring \({\mathbb Z}[\pi_1(M)]\). The \textit{quadratic 2-type} of \(M\) is defined as the quadruple of data \([\pi_1(M),\pi_2(M),\lambda_{M},k(M)]\), where \(\pi_2\cong H_2(M;\Lambda)\) is regarded as a \(\Lambda\)-module, \(\lambda_{M}\) is the intersection form of \(M\) (over \(\Lambda\)), and \(k(M)\in H^3(\pi_1;\pi_2)\) is the first \(k\)-invariant of \(M\). The main theorem states that if \(M\) satisfies (i.) \(\pi_1(M)\) is infinite and \(\pi_2\) is \(\Lambda\)-free (with both finitely generated), and (ii.)\(H_4(B\pi_1;{\mathbb Z}) \cong H_5(B\pi_1;{\mathbb Z})\cong 0\), then the homotopy type of \(M\) is determined by the isometry class of its quadratic 2-type. Examples of 4-manifolds for which the conditions of the theorem hold are those where \(\pi_1\) is free, or isomorphic to the fundamental group of a closed aspherical manifold of dimension less than or equal to four. One consequence of this paper's main theorem therefore is to give a different, more algebraic proof of the homotopy classification of closed orientable topological 4-manifolds with free fundamental group than that given originally in [\textit{A. Cavicchioli} and \textit{F. Hegenbarth}, Forum Math. 6, 415-429 (1994; Zbl 0822.57015)].
reviewed by: Terry Fuller
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57R80
zbMATH DE Number: 1965507
zbMATH Keywords: topological 4-manifolds
zbMATH Keywords: quadratic 2-type
zbMATH Keywords: homotopy type
zbMATH Keywords: classification
MaRDI profile type: Publication
full work available at URL: s00229-003-0371-7
OpenAlex ID: W248857201
DOI: 10.1007/S00229-003-0371-7
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description: scientific article; zbMATH DE number 1965507