A foliation of \({\mathbb{R}}^ 3\) and other punctured 3-manifolds by circles
type: Item
instance of: scholarly article
title: A foliation of \({\mathbb{R}}^ 3\) and other punctured 3-manifolds by circles
zbMATH Open document ID: 0688.57016
DOI: 10.1007/BF02698846
author: Elmar Vogt
published in: Publications Mathématiques
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104052
review text: About 14 years ago, D. B. A. Epstein posed the question: Can \({\mathbb{R}}^ 3\) be foliated by circles? The answer has been stubbornly elusive. In this paper, the author gives an elegant construction of such a foliation. This foliation is not of class \(C^ 1\), but it is differentiable. That is, the leaves are smooth, the tangent distribution is continuous, and the foliation charts can be chosen so that the coordinate changes are homeomorphisms having a derivative at each point. The author calls this a \(C^{1-}\)-foliation. Of course, the foliation has a ``bad set'' \(B_ 1\) where the length-of-leaf function V is not locally bounded. But \(V| B_ 1\) is locally bounded, so the Epstein hierarchy ends at \(B_ 1\). This construction is also used to prove that, if the 3-manifold M admits a \(C^{1-}\)-foliation by circles, so does \(M\setminus \{p\}\), where p is an interior point of M.
reviewed by: Lawrence Conlon
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4128089
zbMATH Keywords: compact foliation
zbMATH Keywords: foliation by circles
zbMATH Keywords: \({\mathbb{R}}^ 3\)
zbMATH Keywords: Epstein hierarchy
MaRDI profile type: Publication
cites work: Q5795840
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description: scientific article; zbMATH DE number 4128089