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Noncompact leaves of foliations of Morse forms. (English. Russian original) Zbl 0917.57022
Math. Notes 63, No. 6, 760-763 (1998); translation from Mat. Zametki 63, No. 6, 862-865 (1998).
Let \(M\) be a compact connected oriented manifold of dimension \(n\) with a closed 1-form \(\omega\) having only Morse singularities (Morse form). Let \({\mathcal F}_\omega\) be a foliation with singularities on \(M\) and \([\gamma]\) the homology class of a nonsingular compact leaf \(\gamma\in{\mathcal F}_\omega\). The image of the set of nonsingular compact leaves generates a subgroup \(H_\omega\) in \(H_{n-1}(M)\). By \(\Omega_i\) denote the set of singular points of index \(i\). In this note an inequality involving the number \(s\) of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form \(\omega\) is obtained.
Theorem. The following inequality holds: \(r_k H_\omega+ s\leq {1\over 2} (|\Omega_1 |- |\Omega_0 |)+1\).
P. Arnoux and G. Levitt [Invent. Math. 84, 141-156 (1986; Zbl 0561.58024)] obtained an estimate of \(s\) in terms of characteristic of \(M:s\leq {1\over 2}\beta_1(M)\). These two estimates coincide for \(n=2\), and they are independent in the case \(n>2\). The method is based on some results of graph theory.

MSC:
57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
57R20 Characteristic classes and numbers in differential topology
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[1] I. A. Mel’nikova, ”Singular points of Morse forms and foliations,”Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. [Moscow Univ. Math. Bull.], No. 4, 37–40 (1996).
[2] P. Arnoux and G. Levitt, ”Sur l’unique ergodicité des 1-formes fermées singulières,”Invent. Math.,84, 141–156 (1986). · Zbl 0577.58021 · doi:10.1007/BF01388736
[3] I. A. Mel’nikova,Compact Foliations of Morse Forms [in Russian], Kandidat thesis in the physico-mathematical sciences, Moscow State University, Moscow (1996).
[4] F. Harary,Graph Theory, Addison-Wesley, Reading, Mass. (1969).
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