Summary
We study the influence of the π1 of a closed manifoldM n (n≧3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index ≠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 π1 M has no quotient isomorphic to ℤ*ℤ, then every nonexact form onM is weakly complete. We also say a form ω is complete if every path inM is homotopic to either a path transverse to ω or a path contained in a leaf of ω. Completeness of ω depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on π1 M and is related to finitely generated normal subgroups of π1 M with quotient ≃ℤ. If π1 M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.
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Levitt, G. 1-formes fermées singulières et groupe fondamental. Invent Math 88, 635–667 (1987). https://doi.org/10.1007/BF01391835
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DOI: https://doi.org/10.1007/BF01391835