## 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.

### MSC:

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

### Keywords:

Morse form foliation; compact leaf; cohomology class
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### References:

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