Graph manifolds and taut foliations
type: Item
instance of: scholarly article
title: Graph manifolds and taut foliations
zbMATH Open document ID: 0896.57017
author: Mark Brittenham
author: Ramin Naimi
author: Rachel Roberts
published in: Journal of Differential Geometry
publication date: 1998-09-22T00:00:00Z
review text: Recall that a manifold \(M\) is called a graph manifold if there is a collection \(\mathcal T\) of disjoint tori embedded in \(M\) such that the manifold obtained by splitting \(M\) open along \(\mathcal T\) is a (non-necessarily connected) Seifert-fibered space. Recall also that a foliation \({\mathcal F}\) is taut if every leaf has a closed loop passing through it which is everywhere transverse to the leaves of \(\mathcal F\). In this paper, the authors prove the following results: (1) There exist infinitely many graph manifolds which admit foliations without Reeb components but no taut foliations. (2) There exist three Seifert-fibered spaces \(M\) for which every taut foliation must have a (non-separating) torus leaf, and each space admits taut foliations. (3) There exist infinitely many graph manifolds \(M\) which admit \(C^0\) taut foliations with no compact leaf, but every \(C^2\) foliation must have a (separating) torus leaf. In particular, each manifold admits no \(C^2\) taut foliation. (4) There exists infinitely many graph manifolds \(M\) which admit \(C^0\) taut foliations with no compact leaf, but no Anosov flows. In the last part of the paper, the authors propose several interesting open questions.
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1145700
zbMATH Keywords: taut foliation
zbMATH Keywords: Reeb component
zbMATH Keywords: graph manifold
zbMATH Keywords: Seifert-fibered space
zbMATH Keywords: essential lamination
zbMATH Keywords: Anosov flow
zbMATH Keywords: pretzel knot
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214459838
OpenAlex ID: W1608605840
Wikidata QID: Q115175782
DOI: 10.4310/JDG/1214459838
description: scientific article; zbMATH DE number 1145700