Codimension one foliations on solvable manifolds
type: Item
instance of: scholarly article
title: Codimension one foliations on solvable manifolds
zbMATH Open document ID: 0803.57006
DOI: 10.1007/BF02565839
author: Shigenori Matsumoto
published in: Commentarii Mathematici Helvetici
publication date: 1994-02-07T00:00:00Z
full work available at URL: https://eudml.org/doc/140320
review text: Throughout the paper \(M\) is a smooth connected manifold with solvable fundamental group \(\Pi\), \({\mathcal F}\) is a transversely oriented codimension one \(\text{C}^ r\) \((r\geq 0)\) foliation without closed leaves and such that every nontrivial leaf holonomy has an isolated fixed point, and \(\widetilde {\mathcal F}\) is the lift of \({\mathcal F}\) to the universal covering space \(\widetilde {M}\) of \(M\). The results read as follows: (1) There exists a \(\text{C}^ r\) submersion \(D\widetilde {M}\to \mathbb{R}\) constant along the leaves of \(\widetilde {F}\) and a homomorphism \(\varphi: \Pi\to \text{Diff}^ r (\mathbb{R})\) such that \(D(\gamma x)= \varphi (\gamma) D(x)\) for all \(x\in \widetilde {M}\) and \(\gamma\in \Pi\). (2) If all the leaves of \({\mathcal F}\) are dense in \(M\), then \({\mathcal F}\) is topologically conjugate to a transversely affine foliation. (3) If either \(\Pi\) is polycyclic or all the leaves of \({\mathcal F}\) have finitely generated fundamental groups, then \({\mathcal F}\) is complete, i.e. the leaf space \(\widetilde {M}/ \widetilde{\mathcal F}\) of \(\widetilde {\mathcal F}\) is homeomorphic to \(\mathbb{R}\).
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 495457
zbMATH Keywords: fundamental group
zbMATH Keywords: foliation
zbMATH Keywords: leaf holonomy
zbMATH Keywords: leaf space
reviewed by: Paweł Walczak
MaRDI profile type: Publication
OpenAlex ID: W2032024215
description: scientific article; zbMATH DE number 495457