Singular closed 1-forms and fundamental group. (1-formes fermées singulières et groupe fondamental.) (French) Zbl 0594.57014

Levitt, Gilbert

Invent. Math. 88, 635-667 (1987).

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.

Reviewer: Gilbert Levitt

MSC:

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

Keywords:

fundamental group; foliations defined by closed differential 1-forms with Morse singularities; nonexact form; leaf space; weakly complete forms; complete cohomology classes; nilpotent; polycyclic; irreducible 3- manifolds

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 · doi:10.1007/BF01388736
[2]	[BL] Blank, S., Laudenbach, F.: Isotopie de formes fermées en dimension trois. Invent. Math.54, 103-177 (1979) · Zbl 0435.58002 · doi:10.1007/BF01408934
[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 · doi:10.1090/S0002-9947-1965-0182927-5
[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 · doi:10.1007/BF01393934
[18]	[Ro] Rosenberg, H.: Foliations by planes. Topology7, 131-138 (1968) · Zbl 0157.30504 · doi:10.1016/0040-9383(68)90021-9
[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) · doi:10.1016/0040-9383(70)90037-6