Noncompact leaves of foliations of Morse forms
type: Item
instance of: scholarly article
title: Noncompact leaves of foliations of Morse forms
zbMATH Open document ID: 0917.57022
DOI: 10.1007/BF02312769
author: I. A. Mel'nikova
published in: Mathematical Notes
publication date: 1999-08-05T00:00:00Z
review text: 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: \(r_k H_\omega+ s\leq {1\over 2} (|\Omega_1 |- |\Omega_0 |)+1\). \textit{P. Arnoux} and \textit{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.
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 1262483
zbMATH Keywords: singular points of a foliation
reviewed by: Costache Apreutesei
MaRDI profile type: Publication
cites work: Sur l'unique ergodicité des 1-formes fermées singulières. (Unique ergodicity of closed singular 1-forms)
full work available at URL: https://doi.org/10.1007/bf02312769
OpenAlex ID: W2024180947
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description: scientific article; zbMATH DE number 1262483