Each crumpled 4-cube is a closed 4-cell-complement
type: Item
instance of: scholarly article
title: Each crumpled 4-cube is a closed 4-cell-complement
zbMATH Open document ID: 0617.57010
DOI: 10.1016/0166-8641(87)90061-7
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: In studying the wildness of an (n-1)-sphere \(\Sigma\) topologically embedded in the n-sphere \(S^ n\), it can be useful to tackle the pathology one side at a time. Feasability is ensured by the following fairly general theorem: for each crumpled n-cube C in \(S^ n\) (i.e., C is the closure of a component of \(S^ n-\Sigma)\) and each \(\epsilon >0\) there exists an embedding h of C in \(S^ n\) moving points less than \(\epsilon\) for which the closure of \(S^ n-h(C)\) is an n-cell. This was proved for \(n=3\) by \textit{N. Hosay} [Notices Am. Math. Soc. 9, 666 (1963)] and \textit{L. L. Lininger} [Trans. Am. Math. Soc. 118, 534-549 (1965; Zbl 0134.430)], for \(n\geq 5\) by the author [Mich. Math. J. 24, 225-241 (1977; Zbl 0376.57007)], and for \(n=4\) in the paper under review. Instrumental in each of these arguments is the locally flat approximation theorem for embeddings of (n-1)-manifolds in n-manifolds, the 4- dimensional version of which was established recently by \textit{F. D. Ancel} [Topology Appl. 24, 13-40 (1986; Zbl 0607.57011)]. Ancel's paper also surveys the history of these results as well as their (possible) extensions to embeddings in n-manifolds of generalized n-manifolds with boundary X, where \(\partial X\times (-1,1)\) is required to be an n- manifold.
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57N45
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 57N50
zbMATH DE Number: 4001002
zbMATH Keywords: 1-LCC
zbMATH Keywords: embedding dimension
zbMATH Keywords: closed n-cell-complement
zbMATH Keywords: Sierpinski curve
zbMATH Keywords: crumpled 4-cube
zbMATH Keywords: 4-cell
zbMATH Keywords: wildness of an (n-1)-sphere topologically embedded in the n-sphere
zbMATH Keywords: crumpled n-cube
zbMATH Keywords: locally flat approximation theorem for embeddings of (n-1)-manifolds in n-manifolds
author: Robert J. Daverman
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90061-7
OpenAlex ID: W1992264045
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description: scientific article; zbMATH DE number 4001002