Stability and instability of certain foliations of 4-manifolds by closed orientable surfaces
type: Item
instance of: scholarly article
title: Stability and instability of certain foliations of 4-manifolds by closed orientable surfaces
zbMATH Open document ID: 0623.57017
author: Kazuhiko Fukui
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: The author studies the stability of foliations of 4-manifolds \(M\) by closed orientable surfaces. The main results are: Theorem A. Let \(F\) be a foliation with all leaves tori and only reflection leaves as singular leaves. Then we can regard a union of reflection leaves as \(T^2\times [0,1]/h\), where h is a diffeomorphism of \(T^2\). If the induced automorphism \(h_*: H_1(T^2; {\mathbb{Z}})\to H_1(T^2; {\mathbb{Z}})\) is equal to \(\begin{pmatrix} -1&0\\ 0&-1\end{pmatrix}\), then \(F\) is \(C^1\)-stable. Theorem B. Let \(F\) be a foliation of \(M\) without singular leaves. Then \(F\) is \(C^r\)-unstable (r\(\geq 0)\) if one of the following is satisfied: (1) \(M/F\) is homeomorphic to the 2-sphere and the genus of a generic leaf \(\geq 2\), (2) \(M/F\) is homeomorphic to the projective plane and the genus of a generic leaf \(\geq 4\), (3) \(M/F\) is neither homeomorphic to the 2-sphere nor the projective plane and the genus of a generic leaf \(\geq 6\). Theorem C. Let \(F\) be a foliation of \(M\) with generic leaf of genus g and \(B=M/F\) be the leaf space. Suppose \(F\) has \(m\) rotation leaves with holonomy groups \(Z_{k_ i}\) \((i=1,2,...,m)\) and \(m_ j\) dihedral leaves with holonomy groups \(D_{\ell_{j,K}}\) \((K=1,2,...,m_ j)\) which corresond to points of \(\partial_ jB\) for each \(j\) \((1\leq j\leq n')\). If \(g \geq \max(3\max (K_i;\ 1\leq i\leq m)+1\), \(8\max (\ell_{j,k};\ 1\leq j\leq n',\ 1\leq K\leq m_ j)+1,7\varepsilon),\) then \(F\) is \(C^r\)-unstable \((r\geq 0)\), where \(\varepsilon=0\) or \(1\) and \(F\) has no reflection leaves if and only if \(\varepsilon =0\).
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4011447
zbMATH Keywords: existence of compact leaves
zbMATH Keywords: foliations of 4-manifolds M by closed orientable surfaces
zbMATH Keywords: foliation with all leaves tori
zbMATH Keywords: singular leaves
reviewed by: Fabiano G. B. Brito
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195177067
OpenAlex ID: W2012364432
DOI: 10.2977/PRIMS/1195177067
description: scientific article; zbMATH DE number 4011447