Close cohomologous Morse forms with compact leaves. (English) Zbl 1289.57009

Let \(\omega \) be a Morse form, i.e., a closed \(1\)-form with non-degenerate singularities (zeros), on a smooth closed manifold \(M\). On the complement \(M\smallsetminus \text{Sing }\omega \) of the singular set of \(\omega \), consider the foliation \(\mathcal {F}_{\omega }=\ker \omega \). The author proves a number of results about the topology of this foliation and its dependence on \(\omega \) within a fixed cohomology class \([\omega ]=\Omega \in H^1(M;\mathbb R)\). Namely, the foliation \(\mathcal {F}_{\omega }\) is said to be compactifiable if all leaves are closed as subsets of \(M\smallsetminus \text{Sing }\omega \). It is shown that Morse forms with compactifiable foliations form an open subset of \(\Omega \). For such foliations consider the rank of the subgroup of \(H_{n-1}(M)\) generated by the fundamental classes of all compact leaves of \(\mathcal {F}_{\omega }\). It is then shown that this number, i.e., the number of homologically independent compact leaves, does not decrease under small perturbations of \(\omega \) within \(\Omega \). For generic forms (Morse forms with each singular leaf containing a unique singularity), it is even proved to be locally constant.


57R30 Foliations in differential topology; geometric theory
58K65 Topological invariants on manifolds
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