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.


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