Neighborhoods of compacta in 4-manifolds
type: Item
instance of: scholarly article
title: Neighborhoods of compacta in 4-manifolds
zbMATH Open document ID: 0669.57005
DOI: 10.1016/0166-8641(89)90100-4
author: G. A. Venema
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let X be a compact subset of the interior of a PL 4-manifold M. Suppose that X satisfies the inessential loops condition (ILC), i.e. for every neighborhood \(U\subset M\) of X there exists a neighborhood \(V\subset U\) of X such that each loop in V-X which is homotopically inessential in V is also inessential in U-X. The main result of the paper asserts that under these conditions, (i) X has arbitrarily close neighborhoods whose components are 4-balls provided \(Fd(X)=0\); or (ii) X has arbitrarily close neighborhoods whose spines are homeomorphic to \(S^ 1\) provided \(Sh(X)=Sh(S^ 1)\). Note that this is a generalization of M. H. Freedman's 4-dimensional cellularity criterion (see the reviewer's paper: ``A criterion for cellularity in a topological 4-manifold'' [Proc. Am. Mat. Soc. 100, 564-566 (1987; Zbl 0623.57009)] for a correct statement and proof). Corollaries include two complement theorems and a weak flatness theorem for compacta in \(S^ 4\). The paper is remarkably well written.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57N25
Mathematics Subject Classification ID: 57N60
zbMATH DE Number: 4096386
zbMATH Keywords: shape theory
zbMATH Keywords: fundamental dimension
zbMATH Keywords: compact subset of interior of a PL 4-manifold
zbMATH Keywords: inessential loops condition
zbMATH Keywords: spines
zbMATH Keywords: 4-dimensional cellularity criterion
zbMATH Keywords: complement theorems
zbMATH Keywords: weak flatness
zbMATH Keywords: compacta in \(S^ 4\)
reviewed by: Dušan Repovš
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90100-4
OpenAlex ID: W1989260646
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description: scientific article; zbMATH DE number 4096386