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Presence of minimal components in a Morse form foliation. (English) Zbl 1070.57016
Let $$M$$ be a connected closed oriented manifold, $$\omega$$ a Morse form on $$M$$ (a closed 1-form with Morse singularities) and $${\mathcal F}_\omega$$ the foliation defined by $$\omega$$. The form’s singularities give little information on the foliation topology. Consider the map $$[\omega]:H_1(M)\to\mathbb{R}$$, $$[\omega](z)=\int_z\omega$$ and $$rk\,\omega\overset{\text{def}}=rk(\text{im}[\omega])$$. Let $$\mathbb{H}_\omega$$ be the group generated by homology classes of all compact leaves and $$\overline H_\omega=\{z\in H_1(M)\mid z\circ \overline H_\omega=0\}$$. In this paper, the author obtains results for the presence of minimal components (non-compactifiable leaves) in a Morse form foliation. Theorem 1: “$${\mathcal F}_\omega$$ has a minimal component iff $$\overline H_\omega\not\subseteq \text{Ker}[\omega]$$.” In this paper, the author gives a sufficient condition simpler (practical) for the problem examined: Theorem 2: “Let $$h(M)$$ be the maximum rank of an isotropic subgroup in $$H^1(M,\mathbb{Z})$$. If $$rk\,\omega>h(M)$$ then $${\mathcal F}_\infty$$ has a minimal component.” If the form $$\omega$$ is in general position (i.e. with all periods being incommensurable) the following result follows from Theorem 2: Theorem 3: “Let $$\omega$$ be a Morse form in general position. If $$\cup:H^1(M,\mathbb{Z})\times H^1(M,\mathbb{Z})\to H^2(M,\mathbb{Z})$$ is non-trivial then $${\mathcal F}_\omega$$ has a minimal component.” Some important particular examples are studied, too.

##### MSC:
 57R30 Foliations in differential topology; geometric theory 58K65 Topological invariants on manifolds
##### Keywords:
Morse form foliation; minimal components; form rank; cup-product
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