This article has been withdrawn due to a mistake which is explained in
version 2.Comment: 1 pag

Laudenbach, Francois

English

arXiv

1 pageThis article has been withdrawn due to a mistake which is explained in version 2

HAL-Univ-Nantes

On the codimension-one foliation theorem of W.ThurstonFrancois LaudenbachTo cite this version:Francois Laudenbach. On the codimension-one foliation theorem of W. Thurston. 1 page.2007. <hal-00095915v2>HAL Id: hal-00095915https://hal.archives-ouvertes.fr/hal-00095915v2Submitted on 8 Jan 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00095915, version 2 - 8 Jan 2007On the codimension-one foliation theorem of W. ThurstonFRANC¸OIS LAUDENBACHAbstract. This article has been withdrawn due to a mistake which is explained in version 2.We consider a 3-simplex σ in an affine space E. Let x1, x2, x3, x4 be its vertices; the edges areoriented by the ordering of the vertices. Let Fi be the 2-face opposite to xi. We are looking atgerms of codimension-one foliation along σ (or along a subcomplex of σ) which are transversalto σ and to all its faces of positive dimension.If such a foliation H is given along the three 2-faces F2, F3, F4 through x1 and if H does nottrace spiralling leaves on F2 ∪ F3 ∪ F4, then H extends to σ transversally to F1. If H is onlygiven along F2 ∪ F4 (resp. F3 ∪ F4), then H extends to F3 (resp. F2) with no spiralling onF2 ∪ F3 ∪ F4, and hence to σ.But, on contrary of what is claimed on version 1 of this paper, it is in general not true whenH is given along F2 ∪ F3. It is only true when an extra condition is fulfilled: The separatricesof x2 in F3 and of x3 in F2 cross F2 ∩ F3 = [x1, x4] respectively at points y2 and y3 which lie inthe order y2 < y3.The first place where this extension argument is misused is corollary 4.5. Moreover the state-ment of this corollary is wrong. Let us explain why.Let σpl ⊂ E be a so-called pleated 3-simplex associated to σ and H be a germ of codimension-one foliation transversal to its simplices. We recall that σpl and σ have the same boundaryand we assume that H traces spiralling leaves on ∂σ, making the pleating necessary accordingto the Reeb stability theorem. Let x ∗ σpl be the (abstract) cone on σpl. If dimE is largeenough, it embeds into E. Certainly H does not extend to x ∗ σpl, contradicting the statementof corollary 4.5. Indeed, if it does, then we get a foliation of x ∗ ∂σpl = x ∗ ∂σ transversal to allfaces. Proposition 4.4 states that, if all 3-faces through x in the 4-simplex x ∗ σ are foliated,then the foliation extends to the face opposite to x, which is σ itself. But this is impossibledue to the spiralling leaves on ∂σ.Laboratoire de mathe´matiques Jean Leray, UMR 6629 du CNRS, Faculte´ des Sciences etTechniques Universite´ de Nantes, 2, rue de la Houssinie`re, F-44322 Nantes cedex 3, FranceE-mail address : Francois.Laudenbach@univ-nantes.fr1991 Mathematics Subject Classification. 57R30.Key words and phrases. Foliations, Γ-structures, transverse geometry.1

On the codimension-one foliation theorem of W. Thurston

Abstract

Similar works

Full text

Available Versions

HAL-Univ-Nantes