Remarks on stability for semiproper exceptional leaves
type: Item
instance of: scholarly article
title: Remarks on stability for semiproper exceptional leaves
zbMATH Open document ID: 0532.57014
author: Akira Seitoh
published in: Tokyo Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: There are two main results. Theorem A states that, for a closed, foliated 3-manifold M and a semiproper leaf L with finitely generated fundamental group, L is stable on a proper side if and only if the germinal holonomy group of L on that side is trivial. Theorem C gives a \(C^\infty\) example showing that the condition on the fundamental group is essential. For this, a method of G. Hector is used for constructing an exceptional minimal set. The author states no differentiability hypothesis in Theorem A, so one assumes that the weakest hypothesis used in this paper, \(C^ 0\) with \(C^\infty\) leaves, is intended. If the semiproper leaf L is proper, the theorem is due to T. Inaba, who assumed \(C^ 1\)-smoothness. If L is exceptional, the author shows that Inaba's proof extends without difficulty. Some remarks are in order. (1) Due to an unpublished result of G. Duminy, this extension is vacuous for \(C^ k\) foliations with \(k\geq 2\). By Duminy, in these cases a semiproper exceptional leaf L has a Cantor subset of ends. Since \(\dim(L)=2\), the fundamental group will not be finitely generated. (2) Inaba's proof used, in an essential way, Novikov's theorem that, in foliated 3-manifolds, a leaf with a vanishing cycle is compact. Novikov assumed \(C^ 2\)-smoothness, but the extension to \(C^ 1\) foliations was established by J. Franks in 1970. The \(C^ 0\) case has been proven recently by \textit{V. V. Solodov} [Mat. Sb., Nov. Ser. 119 (161), No.3 (11), 340-354 (1982; Zbl 0518.57014)]. Thus, Inaba's theorem and the author's extension are valid for \(C^ 0\)-foliated 3- manifolds.
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 3844656
zbMATH Keywords: exceptional leaf
zbMATH Keywords: stable leaf
zbMATH Keywords: semiproper leaf with finitely generated fundamental group
zbMATH Keywords: foliated 3-manifold
zbMATH Keywords: stable on a proper side
zbMATH Keywords: exceptional minimal set
reviewed by: Lawrence Conlon
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270214328
OpenAlex ID: W2046746527
DOI: 10.3836/TJM/1270214328
description: scientific article; zbMATH DE number 3844656