Resolutions of generalized 3-manifolds whose singular sets have general position dimension one.
type: Item
instance of: scholarly article
title: Resolutions of generalized 3-manifolds whose singular sets have general position dimension one.
zbMATH Open document ID: 1042.57014
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: A Euclidean neighborhood retract is a generalized 3-manifold if for each point \(p \in X\) \(H_{k}(X, X \setminus \{p\})\) is isomorphic to \(H_{k}({\mathbb R}^{3}, {\mathbb R}^{3} \setminus \{(0,0,0)\})\) for all \(k \in {\mathbb N}\). The manifold set of \(X\) is the set of points that have neighborhoods homeomorphic to \({\mathbb R}^{3}\) and the singular set is the complement of the manifold set of \(X\). The singular subset is of general position dimension one if any map \(f: B^{2} \rightarrow X\) of a 2-ball into \(X\) can be approximated by one that meets the singular set in a \(0\)-dimensional set. In an earlier paper [Proc. Lond.Math. Soc., III. Ser. 55, 378--432 (1987; Zbl 0629.57011)] the author showed that modulo the classical Poincaré conjecture a generalized 3-manifold with singular set of general position dimension zero is resolvable; meaning that it is the cell like image of a closed 3-manifold. In this paper the result is improved to cover the case where the singular set is of general position dimension one. The paper begins with some errata and clarification of the original paper. This is followed by a reworking of the original material to cover the extension. The author, wisely, advises that the paper should be read with the previous paper at hand as there are numerous references to specific pages in the original.
Mathematics Subject Classification ID: 57P99
Mathematics Subject Classification ID: 57M35
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 2056819
zbMATH Keywords: generalized 3-manifold
zbMATH Keywords: resolution
zbMATH Keywords: cell-like mapping
zbMATH Keywords: loop theorem
zbMATH Keywords: general posItion dimension
author: T. L. Thickstun
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
full work available at URL: j.topol.2003.07.002
OpenAlex ID: W1978137971
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DOI: 10.1016/J.TOPOL.2003.07.002
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description: scientific article; zbMATH DE number 2056819