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Properties of Morse forms that determine compact foliations on \(M_g^2\). (English. Russian original) Zbl 0898.57012
Math. Notes 60, No. 6, 714-716 (1996); translation from Mat. Zametki 60, No. 6, 942-945 (1996).
Summary: P. Arnoux and G. Levitt [Invent. Math. 84, 141-156 (1986; Zbl 0561.58024); ibid. 88, 635-667 (1987; Zbl 0594.57014)] showed that the topology of the foliation of a Morse form \(\omega\) on a compact manifold is closely related to the structure of the integration mapping \([\omega]: H_1(M)\to \mathbb{R}\). In this paper, the author considers the foliation of a Morse form on a two-dimensional manifold \(M_g^2\). He studies the relationship of the subgroup \(\text{Ker} [\omega] \subset H_1(M_g^2)\) with the topology of the foliation, considers the structure of the subgroup \(\text{Ker} [\omega]\) for a compact foliation and proves a criterion for the compactness of a foliation.
MSC:
57R30 Foliations in differential topology; geometric theory
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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[1] P. Arnoux and G. Levitt, Invent. Math.,84, 141–156 (1986). · Zbl 0577.58021 · doi:10.1007/BF01388736
[2] G. Levitt,Invent. Math.,88, 635–667 (1987). · Zbl 0594.57014 · doi:10.1007/BF01391835
[3] S. P. Novikov,Uspekhi Mat. Nauk [Russian Math. Surveys],37, No. 5, 3–49 (1982).
[4] I. A. Mel’nikova,Mat. Zametki [Math. Notes],53, No. 3, 158–160 (1983).
[5] I. A. Mel’nikova,Compact foliations of Morse forms, Kandidat thesis in the physico-mathematical sciences [in Russian], Moscow State Univ., Moscow (1996).
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