Embeddings of polyhedra in \(\mathbb R^m\) and the deleted product obstruction
type: Item
instance of: scholarly article
title: Embeddings of polyhedra in \(\mathbb R^m\) and the deleted product obstruction
zbMATH Open document ID: 0934.57025
DOI: 10.1016/S0166-8641(97)00157-0
author: Jack Segal
author: A. B. Skopenkov
author: Stanisław Spież
published in: Topology and its Applications
publication date: 1999-07-20T00:00:00Z
review text: Weber has proved that if \(2m>3(n+1)\) then an \(n\)-dimensional polyhedron \(K\) embeds in \(\mathbb R^m\) if and only if there exists an equivariant map from the deleted product \(K^*\) into the sphere \(S^{m-1}\). As a consequence he has obtained that in the same range of dimension an \(n\)-dimensional polyhedron embeds in \(\mathbb R^m\) if and only if it quasi-embeds in \(\mathbb R^m\). The authors prove that for any pair of integers \((m,n)\) such that \(m\geq \max(4,n)\) and \(2m<3(n+1)\) there exists an \(n\)-dimensional polyhedron \(R\) which is quasi embeddable in \(\mathbb R^m\) but which is not embeddable in \(\mathbb R^m\). The only two cases remaining open are for \(m=3\) and \(n=2,3\). We say that \(X\) quasi embeds in \(\mathbb R^m\) if for each \(\varepsilon >0\) there exists an \(\varepsilon\)-map from \(X\) onto some subspace in \(\mathbb R^m\). If \(X\) is a polyhedron this definition is equivalent to saying that for each triangulation of \(X\) there exists an almost-embedding of \(X\) in \(\mathbb R^m\).
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 55S15
zbMATH DE Number: 1308006
zbMATH Keywords: quasi-embedding
zbMATH Keywords: \(\varepsilon\)-map
reviewed by: Joachim Grispolakis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308006