Stability of compact leaves with trivial linear holonomy
type: Item
instance of: scholarly article
title: Stability of compact leaves with trivial linear holonomy
zbMATH Open document ID: 0683.57014
DOI: 10.1016/0040-9383(88)90006-7
published in: Topology
publication date: 1988-01-01T00:00:00Z
review text: The author deals with stability of a compact leaf of a foliation under \(C^ 1\)-perturbation. In the first place, he proves the following Theorem 1, which is a generalization of the results of \textit{W. Thurston} [Topology 13, 347-352 (1974; Zbl 0305.57025)] and of \textit{R. Langevin} and \textit{H. Rosenberg} [Topology 16, 107-111 (1977; Zbl 0346.57009)], which are also generalizations of the Reeb stability theorem. Let \(F_ 0\) denote a \(C^ 1\)-codimension q foliation of a compact smooth manifold M and \(L_ 0\) a compact leaf of \(F_ 0\). Theorem 1. Let \(g_ 1,...,g_ k\) be loops in \(L_ 0\) based at \(x_ 0\), whose real homology classes generate \(H_ 1(L_ 0;{\mathbb{R}})\). Suppose \(L_ 0\) has trivial linear holonomy. Then for any neighbourhood N of \(L_ 0\subset M\) there exist neighbourhoods V of \(F_ 0\) in \(Fol_ qM\) and U of \(x_ 0\) in the transverse disk \(D(x_ 0)\) such that for all F in V, (1) the perturbed holonomy \(H_ F(q_ i): U\to D(x_ 0)\) is defined \((i=1,...,k)\); (2) if for some \(x\in U\), \(H_ F(g_ i)(x)=x\) (for all i) then the leaf L of F passing through x is contained in N, diffeomorphic to \(L_ 0\) and hence compact. As applications of the above theorem, the author obtains several sufficient conditions under which compact leaves are stable. Let p: \(M\to B\) be a smooth fibration of M by the fibers of p. The fibers of p are said stable in the large if every foliation F of M sufficiently \(C^ 1\)- close to \(F_ 0\) \((=foliation\) by the fibers of p) has a compact leaf near to the one of the original fibers. A celebrated theorem of Seifert states that the fibers of the Hopf fibration \(S^ 3\to S^ 2\) are stable in the large. With the exception of {\S}7, most of the theorems in this paper concern this notion of stability in the large. We quote one of them only because it is easily understood. Theorem. Let p: \(M\to S^ 1\) be a fibration obtained as a suspension of a diffeomorphism f: \(L\to L\). If \(H_ 1(L;{\mathbb{R}})\cong {\mathbb{R…
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4119262
zbMATH Keywords: stability of a compact leaf of a foliation
zbMATH Keywords: linear holonomy
zbMATH Keywords: smooth fibration
zbMATH Keywords: suspension of a diffeomorphism
author: Paul A. Schweitzer
reviewed by: Tadayoshi Mizutani
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90006-7
OpenAlex ID: W2003541113
description: scientific article; zbMATH DE number 4119262