Classifying foliations of 3-manifolds via branched surfaces
type: Item
instance of: scholarly article
title: Classifying foliations of 3-manifolds via branched surfaces
zbMATH Open document ID: 1105.57026
published in: Topology and its Applications
publication date: 2007-01-09T00:00:00Z
review text: The author uses branched surfaces to define a pseudo-metric on the set of \(C^1\) codimension-one foliations on any closed manifold that are transverse to a nonsingular flow. She then defines two foliations to be \(b\)-equivalent if their distance measured with this pseudo-metric is zero. In this way, she obtains a discrete metric on the set of equivalence classes which has the property that foliations that are sufficiently close (up to \(b\)-equivalence) often share important topological properties, such as the existence of a compact leaf, tautness and so on. The techniques used in this paper are based on a construction that is described in an unpublished preprint by J. Christy and S. Goodman. This construction is reviewed in this paper. The construction gives a branched surface associated to a foliation \(F\), a nonsingular flow \(\phi\), and a set of compact surfaces embedded in leaves of \(F\) that satisfy certain general position requirements with respect to \(\phi\). The author in fact uses an amended version of Christy and Goodman's construction.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 5083217
zbMATH Keywords: branched surface
zbMATH Keywords: foliation
zbMATH Keywords: transverse flow
author: Sandra L. Shields
MaRDI profile type: Publication
full work available at URL: j.topol.2006.01.015
OpenAlex ID: W2141795894
Wikidata QID: Q124973099
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DOI: 10.1016/J.TOPOL.2006.01.015
description: scientific article; zbMATH DE number 5083217