## 1-formes fermées singulières et groupe fondamental.(French)Zbl 0594.57014

We study the influence of the fundamental group of a closed manifold $$M^ n$$ (n$$\geq 3)$$ on the foliations of M defined by closed differential 1-forms with Morse singularities (of index $$\neq 0,n)$$. Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If $$\pi_ 1M$$ has no quotient isomorphic to $${\mathbb{Z}}*{\mathbb{Z}}$$, then every nonexact form on M is weakly complete. We also say a form $$\omega$$ is complete if every path in M is homotopic to either a path transverse to $$\omega$$ or a path contained in a leaf of $$\omega$$. Completeness of $$\omega$$ depends only on its de Rham cohomology class. The set of complete cohomology classes depends only on $$\pi_ 1M$$ and is related to finitely generated normal subgroups of $$\pi_ 1M$$ with quotient $$\simeq {\mathbb{Z}}$$. If $$\pi_ 1M$$ is nilpotent (or even polycyclic), every nonexact form on M is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.

### MSC:

 57R30 Foliations in differential topology; geometric theory 58A10 Differential forms in global analysis 58A12 de Rham theory in global analysis
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### References:

 [1] [AL] Arnoux, P., Levitt, G.: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math.84, 141-156 (1986) · Zbl 0577.58021 [2] [BL] Blank, S., Laudenbach, F.: Isotopie de formes fermées en dimension trois. Invent. Math.54, 103-177 (1979) · Zbl 0435.58002 [3] [BNS] Bieri, R., Neumann, W., Strebel, R.: A geometric invariant for discrete groups. Preprint [4] [FLP] Fathi, A., Laudenbach, F., Poenaru, V.: Travaux de Thurston sur les surfaces. Astérisque66-67 (1979), SMF Paris [5] [Fr] Fried, D.: Fibrations overS 1 with pseudo-Anosov monodromy. Exposé 14 de [FLP] [6] [FrL] Fried, D., Lee, R.: Realizing group automorphisms, in Group actions on manifolds. Contemp. Math.36, 427-432 (1985) [7] [Ha] Haefliger, A.: Variétes feuilletées. Ann. Sci. Norm. Super. Pisa, Cl. Sci., IV. Ser.16, 367-397 (1962) [8] [Hem] Hempel, J.: 3-manifolds. Ann. Math. Stud.86 (1976) Princeton Univ. press · Zbl 0345.57001 [9] [Hen] Hen?, D.: Ergodicity of foliations with singularities. Preprint IHES 1982 [10] [Im1] Imanishi, H.: On codimension one foliations defined by closed one forms with singularities. J. Math. Kyoto Univ.19, 285-291 (1979) · Zbl 0417.57010 [11] [Im2] Imanishi, H.: Structure of codimension 1 foliations without holomomy on manifolds with abelian fundamental group. J. Math. Kyoto Univ.19, 481-495 (1979) · Zbl 0452.57006 [12] [Im3] Imanishi, H.: Denjoy-Siegel theory of codimension one foliations. Sûgaku32, 119-132 (1980) (en japonais). MR 82k: 57017 [13] [Le] Levitt, G.: Geometry and ergodicity of singular closed 1-forms. Proc. V Escola Geom. Dif., São Paulo 1984, 109-118 [14] [Me] Meigniez, G.: Bouts d’un groupe dans une direction et feuilletages par 1-formes fermées (preprint) [15] [Mo] Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc.120, 286-294 (1965) · Zbl 0141.19407 [16] [Ne] Neumann, W.: Normal subgroups with infinite cyclic quotient. Math. Sci.4, 143-148 (1979) · Zbl 0414.20030 [17] [QR] Que, N., Roussarie, R.: Sur l’isotopie des formes fermées en dimension 3. Invent. Math.64, 69-87 (1981) · Zbl 0467.58004 [18] [Ro] Rosenberg, H.: Foliations by planes. Topology7, 131-138 (1968) · Zbl 0157.30504 [19] [Si] Sikorav, J.C.: Thèse d’État, Orsay 1987 [20] [St] Stallings, J.: On fibering certain 3-manifolds, Topology of 3-manifolds and related topics. Prentice Hall 1961, 95-100 [21] [Th1] Thurston, W.: A norm for the homology of 3-manifolds. Mem. Am. Math. Soc.339, 99-130 (1986) · Zbl 0585.57006 [22] [Th2] Thurston, W.: The geometry and topology of 3-manifolds. Princeton University notes [23] [Ti] Tischler, D.: On fibering certain foliated manifolds overS 1. Topology9, 153-154 (1970) · Zbl 0189.54502
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