1-formes ferm6es singuli6res et groupe fondamental by Levitt G.

By Levitt G.

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Levitt D~monstration Reprenons V et co' c o m m e ci-dessus. Puisque L f ( ~ o ' ) ~ ( ~ o ) , il nous suffit de m o n t r e r le r6sultat pour ~'. Les feuilles non compactes L de ~o' telles que L u Sing co' soit compact sont en n o m b r e fini. Leur union avec Sing~o' est un c o m p a c t K, et les composantes de V - K autres que N(~o') sont feuillet6es c o m m e un produit (feuille compacte • intervalle ouvert). Si L e s t une feuille contenue dans K, L s6pare V en deux composantes et ~o' est exacte dans l'une d'elles (car il n'y a pas de sous-vari6t6 [ d ] - e x a c t e non triviale).

Math. : VariSt6s feuillet6es. Ann. Sci. Norm. Super. Pisa, C1. , IV. Ser. : 3-manifolds. Ann. Math. Stud. 86 (1976) Princeton Univ. : Ergodicity of foliations with singularities. : On codimension one foliations defined by closed one forms with singularities. J. Math. Kyoto Univ. : Structure of codimension 1 foliations without holonomy on manifolds with abelian fundamental group. J. Math. Kyoto Univ. : Denjoy-Siegel theory of codimension one foliations. Sfigaku 32, 119-132 (1980) (en japonais). : Geometry and ergodicity of singular closed 1-forms.

1): tout chemin d'int6grale nuIle ~ extr6mit6s dans N(co) a ses extr6mit6s sur la m6me feuille; dans M, t o u s l e s niveaux f-l(c)c~/~-l(N(~o)) sent connexes; il existe une courbe ferm6e transverse C ~ N ( c o ) coupant toute feuille de N(co), telle que deux points quelconques de C dent la distance est une p6riode soient sur la m6me feuille. 1 m6ritent d'6tre appel6es faiblement compl6tes; en effet elles poss6dent les deux propri6t6s suivantes, que nous prenons c o m m e d~finition g~n~rale des formes faiblement completes (de rang >2): - si L est une feuille avec L u S i n g c o compact, alors /, s6pare M e n deux composantes, et co est exacte dans l'une d'elles; - N(co)/co~lR/P(co).

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