Presence of minimal components in a Morse form foliation
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.
scientific article; zbMATH DE number 2165117