Noncompact leaves of foliations of Morse forms
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.
scientific article; zbMATH DE number 1262483
scientific article