On the deleted product criterion for embeddability of manifolds in \(\mathbb{R}^m\)
type: Item
instance of: scholarly article
title: On the deleted product criterion for embeddability of manifolds in \(\mathbb{R}^m\)
zbMATH Open document ID: 0905.57015
published in: Commentarii Mathematici Helvetici
publication date: 1999-01-18T00:00:00Z
review text: For any space \(N\), let \(\widetilde N\) denote the deleted square of \(N\), i.e. the set of all \((x,y)\) in \(N\times N\) with \(x\neq y\). For every embedding \(f: N\to{\mathbb R}^m\) we can define a map \(\widetilde f :\widetilde N\to S^{m-1}\) by \(\widetilde f(x,y)=\bigl(f(x)-f(y)\bigr)\big/\big\| f(x)-f(y)\big\| \), and this map is equivariant with respect to the coordinate switching involution of \(\widetilde N\) and the antipodal involution of \(S^{m-1}\). Similarly, if \(\{f_t: N\to{\mathbb R}^m\mid 0\leq t\leq 1\}\) is an isotopy of embeddings, then \(\{\widetilde f_t\mid 0\leq t\leq 1\}\) is an equivariant homotopy. Under suitable restrictions one can prove a converse: if there exists an equivariant map \(\widetilde N\to S^{m-1}\), then \(N\) embeds in \({\mathbb R}^m\), and if \(f,g: N\to{\mathbb R}^m\) are embeddings such that \(\widetilde f\) and \(\widetilde g\) are equivariantly homotopic, then \(f\) and \(g\) are isotopic. A theorem of this kind was first proved by \textit{A. Haefliger} [ibid. 37, 155-176 (1962; Zbl 0186.27302)] for the smooth category, i.e. for \(N\) a compact smooth manifold and for smooth embeddings and isotopies, and later by \textit{C. Weber} [ibid. 42, 1-27 (1967; Zbl 0152.22402)] for the PL category. Their common dimensional restriction was \(m\geq 3(n+1)/2\) for the embedding part and \(m > 3(n+1)/2\) for the isotopy part, where \(n=\dim N\). The bound \(3(n+1)/2\) is sharp in general, but in the present paper the author shows that it can be relaxed in presence of connectivity conditions on \(N\): he proves that if \(N\) is a closed, \(d\)-connected PL \(n\)-manifold with \(d=0,1,2\), then the embedding part of Weber's theorem remains true if \(m\geq(3n+2-d)/2\) and the isotopy part remains true if \(m\geq(3n+3-d)/2\) and \(m\geq 6+d\).
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 57Q37
zbMATH DE Number: 1129397
zbMATH Keywords: PL manifold
zbMATH Keywords: PL embedding into Euclidean space
zbMATH Keywords: PL isotopy of embeddings
zbMATH Keywords: deleted product
zbMATH Keywords: quasi-embeddability
zbMATH Keywords: normal Stiefel-Whitney classes
zbMATH Keywords: classification of embeddings
zbMATH Keywords: deleted square
author: A. B. Skopenkov
reviewed by: Joze Vrabec
MaRDI profile type: Publication
full work available at URL: s000140050033
OpenAlex ID: W2170551139
DOI: 10.1007/S000140050033
Recommended article: Embeddings of homology equivalent manifolds with boundary
Recommended article: Isotopies vis-à-vis level-preserving embeddings
Recommended article: A deleted product criterion for approximability of maps by embeddings
Recommended article: On incompleteness of the deleted product obstruction for embeddability
Recommended article: On the Haefliger-Hirsch-Wu invariants for embeddings and immersions
Recommended article: Tangential thickness of manifolds
Recommended article: Algebraic models of Poincaré embeddings
Recommended article: Isovariant mappings of degree 1 and the gap hypothesis
Recommended article: An Equivariant Sphere Theorem
Recommended article: Moduli of Riemann surfaces, Hurwitz-type spaces, and their superanalogues
description: scientific article; zbMATH DE number 1129397