Quasi embeddings and embeddings of polyhedra in \(\mathbb{R}{}^ m\)
type: Item
instance of: scholarly article
title: Quasi embeddings and embeddings of polyhedra in \(\mathbb{R}{}^ m\)
zbMATH Open document ID: 0768.57012
DOI: 10.1016/0166-8641(92)90009-O
published in: Topology and its Applications
publication date: 1993-01-16T00:00:00Z
review text: Let \(X\) be a metric space and \(\varepsilon>0\). A map \(f: X\to Y\) is said to be an \(\varepsilon\)-map if for each \(y\in f(X)\), \(\text{diam }f^{- 1}(y)<\varepsilon\). If for each \(\varepsilon>0\) there is an \(\varepsilon\)-map of \(X\) into \(Y\) then \(X\) is said to be quasi-embeddable in \(Y\). For each pair \((m,k)\), \(m\geq k\), of positive integers, one can ask the following question: Does every \(k\)-dimensional compact polyhedron which quasi-embeds in \(\mathbb{R}^ m\), also embed in \(\mathbb{R}^ m\)? This problem has a long history. Among the best results so far is a theorem of \textit{C. Weber} [\(\varepsilon\)-applications dans une variété, Comment. Math. Helv. 49, 125-135 (1974; Zbl 0291.54011)] asserting that the question above has an affirmative answer for \(2m\geq 3(k+1)\). Weber's theorem has certain dimension restrictions due to the use of the Freudenthal suspension isomorphism theorem and general position problems. The main purpose of the paper under review is to show that Weber's restrictions are necessary in all but a finite number of cases. This is done by constructing a particular example of a polyedron, using the higher-dimensional analogue of the classical Casson finger moves, which were first introduced by \textit{A. N. Dranishnikov}, \textit{E. V. Shchepin} and the reviewer in [On intersections of compacta in Euclidean space: The metastable case, Tsukuba J. Math. (in print)].
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 57Q55
zbMATH DE Number: 90163
zbMATH Keywords: \(\varepsilon\)-map
zbMATH Keywords: quasi-embeddable
zbMATH Keywords: embed in \(\mathbb{R}^ m\)
zbMATH Keywords: dimension restrictions
zbMATH Keywords: Freudenthal suspension isomorphism
zbMATH Keywords: Casson finger moves
author: Jack Segal
author: Stanisław Spież
reviewed by: Dušan Repovš
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 90163