On the codimension-one foliation theorem of W. Thurston FRANC¸OIS LAUDENBACH 7 0 Abstract. This article has been withdrawn due to a mistake which is explained in version2. 0 2 n a J 8 We consider a 3-simplex σ in an affine space E. Let x1,x2,x3,x4 be its vertices; the edges are oriented by the ordering of the vertices. Let F be the 2-face opposite to x . We are looking at ] i i T germs of codimension-one foliation along σ (or along a subcomplex of σ) which are transversal G to σ and to all its faces of positive dimension. . h at If such a foliation H is given along the three 2-faces F2,F3,F4 through x1 and if H does not m trace spiralling leaves on F2 ∪F3 ∪F4, then H extends to σ transversally to F1. If H is only [ given along F2 ∪ F4 (resp. F3 ∪ F4), then H extends to F3 (resp. F2) with no spiralling on 2 F2 ∪F3 ∪F4, and hence to σ. v 7 But, on contrary of what is claimed on version 1 of this paper, it is in general not true when 9 4 H is given along F2 ∪F3. It is only true when an extra condition is fulfilled: The separatrices 9 of x2 in F3 and of x3 in F2 cross F2∩F3 = [x1,x4] respectively at points y2 and y3 which lie in 0 the order y < y . 6 2 3 0 / h The first place where this extension argument is misused is corollary 4.5. Moreover the state- t a ment of this corollary is wrong. Let us explain why. m : Let σpl ⊂ E be a so-called pleated 3-simplex associated to σ and H be a germ of codimension- v i one foliation transversal to its simplices. We recall that σpl and σ have the same boundary X and we assume that H traces spiralling leaves on ∂σ, making the pleating necessary according r a to the Reeb stability theorem. Let x ∗ σpl be the (abstract) cone on σpl. If dimE is large enough, it embeds into E. Certainly H does not extend to x∗σpl, contradicting the statement of corollary 4.5. Indeed, if it does, then we get a foliation of x∗∂σpl = x∗∂σ transversal to all faces. 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 impossible due to the spiralling leaves on ∂σ. Laboratoire de math´ematiques Jean Leray, UMR 6629 du CNRS, Facult´e des Sciences et Techniques Universit´e de Nantes, 2, rue de la Houssini`ere, F-44322 Nantes cedex 3, France E-mail address: [email protected] 1991 Mathematics Subject Classification. 57R30. Key words and phrases. Foliations, Γ-structures, transverse geometry. 1