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An indicator of the noncompactness of a foliation on $$M_ g^ 2$$. (English. Russian original) Zbl 0809.57018
Math. Notes 53, No. 3, 356-358 (1993); translation from Mat. Zametki 53, No. 3, 158-160 (1993).
Let $$\omega$$ be a closed form on a manifold $$M$$ and possessing nondegenerate isolated singularities. A point $$p \in M$$ is called a regular singularity of $$\omega$$, if in some neighbourhood $$O(p) \omega = df$$, where $$f$$ is a Morse function having a singularity at $$p$$. The form $$\omega$$ determines a foliation $$F_ \omega$$ on the set $$M-\text{Sing} \omega$$.
Let $$M = M^ 2_ g$$, the orientable closed surface of genus 2. The homology classes $$[\gamma]$$ of the nonsingular compact leaves of $$F_ \omega$$ generate a subgroup of $$H_ 1 (M^ 2_ g)$$ denoted by $$H_ \omega$$. If $$[z_ 1], \dots, [z_{2g}]$$ is a basis of $$H_ 1 (M^ 2_ g)$$ we define $$\text{dirr} \omega = rk_ \mathbb{Q} \{\int_{z_ 1} \omega, \dots, \int_{z_{2g}} \omega\} - 1$$.
By $$M_ \omega$$ is denoted the set obtained by discarding all maximal neighbourhoods consisting of diffeomorphic compact leaves and all leaves which can be compactified by adding singular points.
Theorem 1. $$M_ \omega = \emptyset \Leftrightarrow \text{rk} H_ \omega = g$$.
Theorem 2. If $$\omega$$ is a closed form with Morse singularities given on $$M^ 2_ g$$ $$(g \neq 0)$$ such that $$\text{dirr} \omega \geq g$$, then the foliation $$F_ \omega$$ has a noncompact fiber.

##### MSC:
 57R30 Foliations in differential topology; geometric theory 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010)
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##### References:
 [1] H. Damir, J. Funct. Anal.,75, No. 2, 349-361 (1987). · Zbl 0637.58021 · doi:10.1016/0022-1236(87)90100-5 [2] S. P. Novikov, Usp. Mat. Nauk,37, No. 5, 3-49 (1982).
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