Properties of Morse forms that determine compact foliations on \(M_g^2\). (English. Russian original) Zbl 0898.57012

Mel’nikova, I. A.

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

Keywords:

integration over cycles; Morse form; two-dimensional manifold; compact foliation

Citations:

Zbl 0561.58024; Zbl 0607.57021; Zbl 0577.58021; Zbl 0594.57014

References:

[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).