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Perturbations of Compact Foliations

Perturbations of compact foliations
Authors: Fukui, Kazuhiko;

Perturbations of Compact Foliations

Abstract

Consider a foliation of a compact manifold with all leaves compact. When must any \(C^ 0\)-perturbation of this foliation have a compact leaf? \textit{R. Langevin} and \textit{H. Rosenberg} [Lect. Notes Math. 652, 122-127 (1978; Zbl 0382.57011)] proved that any such perturbation of a fibration over a surface with nonzero Euler characteristic has a compact leaf if the fundamental group of the fiber is \({\mathbb{Z}}\) and the fundamental group of the base acts trivially on the fundamental group of the fiber. The present author generalizes this result to codimension-two foliations with the ''base'' being the leaf space of the foliation and an additional assumption of no ''isolated singular leaf''.

Keywords

perturbations of foliations, codimension-two foliations, generalized Seifert fibre space, 57R30, Foliations in differential topology; geometric theory, Stability theory for smooth dynamical systems, stability of compact leaves, foliation of a compact manifold with all leaves compact

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
Average
Average
Average
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gold