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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
##### Keywords:
singular points of a foliation
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##### References:
 [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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