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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.

57R30 Foliations in differential topology; geometric theory
58K65 Topological invariants on manifolds
Full Text: DOI
[1] Arnoux, P.; Levitt, G., Sur l’unique ergodicité des 1-formes fermées singulières, Invent. math., 84, 141-156, (1986) · Zbl 0577.58021
[2] De Leo, R., Existence and measure of ergodic foliations in the Novikov problem on semiclassical motion of electron, Russian math. surveys, 55, 1, 181-182, (2000) · Zbl 0984.37109
[3] Dynnikov, I., Geometry of stability zones in the Novikov problem on semiclassical motion of electron, Russian math. surveys, 54, 1, 21-60, (1999) · Zbl 0935.57040
[4] Henc”ˇ, D., Ergodicity of foliations with singularities, J. funct. anal., 75, 2, 349-361, (1987) · Zbl 0637.58021
[5] Latour, F., Existence de 1-formes fermées non singulierès dans une classe de cohomologie de de Rham, Inst. hautes études scient. publ. math., 80, 135-194, (1994) · Zbl 0837.58002
[6] Melnikova, I., Sufficient condition of non-compactifiability for a foliation on \(M_g^2\), Math. notes, 53, 3, 158-160, (1993)
[7] Melnikova, I., Maximal isotropic subspaces of a skew-symmetric bilinear map, Vestnik MGU, 4, 3-5, (1999) · Zbl 0957.57018
[8] I. Gelbukh, Structure of a Morse form foliation, submitted for publication · Zbl 1224.57010
[9] Novikov, S., The Hamiltonian formalism and a multivalued analog of Morse theory, Russian math. surveys, 37, 5, 1-56, (1982) · Zbl 0571.58011
[10] Novikov, S., The semiclassical electron in a magnetic field and lattice, Some problems of low dimensional “periodic” topology, geom. funct. anal., 5, 434-444, (1995) · Zbl 0853.57014
[11] Novikov, S.; Malcev, A., Topological phenomena in normal metals, Russian phys. surveys, 168, 3, 249-258, (1998)
[12] Pajitnov, A., The incidence coefficients in the Novikov complex are generically rational functions, Saint |St. Petersburg math. J., 9, 5, 969-1006, (1998)
[13] Tischler, D., On fibering certain foliated manifolds over \(S^1\), Topology, 9, 153-154, (1970) · Zbl 0177.52103
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