On codimension one foliations with Morse singularities on three-manifolds
type: Item
instance of: scholarly article
title: On codimension one foliations with Morse singularities on three-manifolds
zbMATH Open document ID: 1118.57024
published in: Topology and its Applications
publication date: 2007-03-12T00:00:00Z
review text: A codimension one \(C^{\infty }\) foliation with isolated singularities on a manifold \(M\) is a pair \(\mathcal{ F} = (\mathcal{ F}_{0}, \sigma )\) where \(\sigma \subset M\) is a discrete subset of \(M\) and \(\mathcal{ F}_0\) is a codimension one \(C^{\infty}\) regular foliation of \(M- \sigma .\) \(\sigma\) is called the singular set of \(\mathcal{ F}.\) A point \(p \in \sigma\) is a Morse type singularity if there are a neighbourhood \(p \in U \subset M\) and a \(C^{\infty}\) function \(f:U \rightarrow \mathbb{R}\) such that \(\sigma \cap U = \{ p \}\) and \(\mathcal{ F}| U \) is defined by \(df =0\) and \(p\) is the non-degenerate critical point of \(f.\) Accordingly, the singular points of the foliation are called centers and saddles. A leaf of \(\mathcal{ F}\) is called a saddle connection if it contains separatrices of two distinct saddle singularities of \(\mathcal F .\) A codimension one \(C^{\infty}\) foliation with isolated singularities on \(M\) is called a Morse foliation if each singularity of \(\mathcal{ F}\) is of Morse type and \(\mathcal{ F}\) has no saddle connections. The main result of the paper is the following theorem: Theorem Let \(\mathcal F\) be an oriented Morse foliation on a closed oriented 3-manifold \(M.\) Suppose that the number \(k\) of centers and the number \(l\) of saddles of the foliation satisfy \(k > l.\) Then \(M\) is diffeomorphic to \(S^3.\) The proof is based on the Reeb Stability Theorem and an elimination procedure of certain pairs of center and saddle singularities. Finally, this procedure is used in place of the Poincaré-Bendixson theorem to obtain the following variant of the classical Haefliger theorem. Theorem Let \(\mathcal F\) be a Morse foliation on the 3-sphere with \(k\) centers and \(l\) sadddles and suppose that \(k > l-1.\) Then \(\mathcal F\) is an inverse modification of a Seifert fibration of \(S^3, \) i.e. a singular foliation of \(S^3\) by centers and spheres \(S^2,\) or one of the following possibilit…
reviewed by: Robert A. Wolak
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 5132966
zbMATH Keywords: foliation
zbMATH Keywords: Morse singularity
zbMATH Keywords: Reeb stability theorem
zbMATH Keywords: codimension one foliation
author: César Camacho
author: Bruno Scárdua
MaRDI profile type: Publication
full work available at URL: j.topol.2006.10.005
OpenAlex ID: W2011431798
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DOI: 10.1016/J.TOPOL.2006.10.005
description: scientific article; zbMATH DE number 5132966