On the embedding of zero-dimensional double loops in locally Euclidean double loops
type: Item
instance of: scholarly article
title: On the embedding of zero-dimensional double loops in locally Euclidean double loops
zbMATH Open document ID: 0768.22001
DOI: 10.1007/BF03323113
author: Richard Bödi
published in: Results in Mathematics
publication date: 1993-04-01T00:00:00Z
review text: Let \(D\) be a topological double loop on the topological space \(\mathbb{R}^ n\). Then \(n\in \{1,2,4,8\}\). Sub-double-loops arise naturally in the study of automorphism groups of \(D\). The conjecture that every closed sub-double-loop has positive topological dimension is known to be true for \(n \leq 2\). The author proves that a closed sub-double-loop of dimension zero is always tamely embedded in \(D\) (i.e. the embedding is equivalent to a piecewise linear embedding), and that its complement in \(D\) is simply connected. To prove this, the author establishes three tameness criteria for Cantor sets in \(n\)-spheres. These criteria are variations of results of \textit{R. P. Osborne} [Mich. Math. J. 13, 57-63 (1966; Zbl 0138.189)] and \textit{D. G. Wright} [Houston J. Math. 2, 439-447 (1976; Zbl 0332.57002)].
reviewed by: Theo Grundhöfer
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 57N45
Mathematics Subject Classification ID: 12K99
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 147375
zbMATH Keywords: tame embedding
zbMATH Keywords: topological double loop
zbMATH Keywords: automorphism groups
zbMATH Keywords: closed sub- double-loop
zbMATH Keywords: positive topological dimension
zbMATH Keywords: linear embedding
zbMATH Keywords: tameness criteria
zbMATH Keywords: Cantor sets
MaRDI profile type: Publication
cites work: Some remarks concerning topologically homogeneous spaces
cites work: Deformations of spaces of imbeddings
cites work: Q3972230
cites work: Piecewise Linear Isotopy and Embedding of Elements and Spheres (I)
cites work: Embedding Cantor sets in a manifold
cites work: Topological planes
cites work: Q4098381
description: scientific article; zbMATH DE number 147375