An analogy of the theorem of Hector and Duminy
type: Item
instance of: scholarly article
title: An analogy of the theorem of Hector and Duminy
zbMATH Open document ID: 0751.57014
published in: Hokkaido Mathematical Journal
publication date: 1992-09-26T00:00:00Z
review text: An unpublished theorem of G. Duminy asserts that, in a \(C^ 2\) foliation \(\mathcal F\) of codimension one on a closed manifold \(M\), a semiproper, exceptional leaf will have a holonomy contraction on its nonproper side. This leads to the striking result that a semiproper leaf in an exceptional minimal set has a Cantor set of ends. These results have been cited by various authors and were to have appeared in Duminy's thesis, under the direction of G. Hector, in the late 1970's. Unfortunately, this was never written. (The reviewer and J. Cantwell have written up a proof and circulated it informally.) In the paper under review, the author presents an analogous result in higher codimension. Not surprisingly, very restrictive hypotheses are required on the holonomy pseudogroup \(\Gamma\) and on the orbit \(\Gamma(x_ 0)\) which corresponds to the leaf in question. Roughly speaking, the pseudogroup is required to consist of similarity transformations and the ``exceptional and semiproper'' hypothesis on \(\Gamma(x_ 0)\), which makes no sense in higher codimension, is replaced by a hypothesis that ``\(\Gamma(x_ 0)\) is nonproper and with bubbles''. The conclusion is that there is a nontrivial element \(g\in \Gamma\) with \(g(x_ 0)=x_ 0\). This is a weaker analogy than would be hoped for, since it allows the possibility that \(g\) is a pure rotation. The author does not know if there is an example in which all holonomy elements fixing \(x_ 0\) are pure rotations.
reviewed by: Lawrence Conlon
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 55926
zbMATH Keywords: semiproper leaf in an exceptional minimal set
zbMATH Keywords: holonomy pseudogroup
zbMATH Keywords: similarity transformations
zbMATH Keywords: nonproper
zbMATH Keywords: bubbles
author: Kouichi Matsuda
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381413260
OpenAlex ID: W2083254919
DOI: 10.14492/HOKMJ/1381413260
description: scientific article; zbMATH DE number 55926