Classe d'Euler et minimal exceptionnel
type: Item
instance of: scholarly article
title: Classe d'Euler et minimal exceptionnel
zbMATH Open document ID: 0607.57020
DOI: 10.1016/0040-9383(87)90024-3
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Sigma\) be a compact oriented surface different from \(S^ 2\). For a representation \(\phi\) : \(\pi\) \({}_ 1(\Sigma)\to Diff^+(S^ 1)\) let eu(\(\phi)\) denote the Euler class of the fibre bundle obtained by suspension of \(\phi\) evaluated on the fundamental class of \(\Sigma\). It is known that \(| eu(\phi)| \leq | \chi (\Sigma)|.\) The author shows that the inequality is sharp if \(\phi\) has a minimal exceptional set and takes on its values in the subgroup \(Diff^+_ 2(S^ 1)\) of \(C^ 2\)-diffeomorphisms. If, moreover, \(\phi\) is \(Diff^+_{\omega}(S^ 1)\)-valued (class \(C^{\omega})\), then \(eu(\phi)=0\). If \(eu(\phi)=\pm \chi (\Sigma)\) for a \(\phi\) : \(\pi\) \({}_ 1(\Sigma)\to Homeo^+(S^ 1)\) then every leaf of the suspended codimension 1 foliation is either a cylinder, or a plane, or a torus.
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 3982089
zbMATH Keywords: circle bundles
zbMATH Keywords: diffeomorphisms of the circle
zbMATH Keywords: Euler class
zbMATH Keywords: suspension
zbMATH Keywords: minimal exceptional set
zbMATH Keywords: suspended codimension 1 foliation
author: Étienne Ghys
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90024-3
OpenAlex ID: W2029301966
description: scientific article; zbMATH DE number 3982089