On codimension one foliations with Morse singularities on three-manifolds
A codimension one \(C^{\infty }\) foliation with isolated singularities on a manifold \(M\) is a pair \(\mathcal{ F} = (\mathcal{ F}_{0}, \sigma )\) where \(\sigma \subset M\) is a discrete subset of \(M\) and \(\mathcal{ F}_0\) is a codimension one \(C^{\infty}\) regular foliation of \(M- \sigma .\) \(\sigma\) is called the singular set of \(\mathcal{ F}.\) A point \(p \in \sigma\) is a Morse type singularity if there are a neighbourhood \(p \in U \subset M\) and a \(C^{\infty}\) function \(f:U \rightarrow \mathbb{R}\) such that \(\sigma \cap U = \{ p \}\) and \(\mathcal{ F}| U \) is defined by \(df =0\) and \(p\) is the non-degenerate critical point of \(f.\) Accordingly, the singular points of the foliation are called centers and saddles. A leaf of \(\mathcal{ F}\) is called a saddle connection if it contains separatrices of two distinct saddle singularities of \(\mathcal F .\) A codimension one \(C^{\infty}\) foliation with isolated singularities on \(M\) is called a Morse foliation if each singularity of \(\mathcal{ F}\) is of Morse type and \(\mathcal{ F}\) has no saddle connections. The main result of the paper is the following theorem: Theorem Let \(\mathcal F\) be an oriented Morse foliation on a closed oriented 3-manifold \(M.\) Suppose that the number \(k\) of centers and the number \(l\) of saddles of the foliation satisfy \(k > l.\) Then \(M\) is diffeomorphic to \(S^3.\) The proof is based on the Reeb Stability Theorem and an elimination procedure of certain pairs of center and saddle singularities. Finally, this procedure is used in place of the Poincaré-Bendixson theorem to obtain the following variant of the classical Haefliger theorem. Theorem Let \(\mathcal F\) be a Morse foliation on the 3-sphere with \(k\) centers and \(l\) sadddles and suppose that \(k > l-1.\) Then \(\mathcal F\) is an inverse modification of a Seifert fibration of \(S^3, \) i.e. a singular foliation of \(S^3\) by centers and spheres \(S^2,\) or one of the following possibilities happens: (1) There is a compact codimension one invariant subset whose holonomy is onesided. (2) There is a singular Reeb component. The results are illustrated by an example of a foliation with just one 2-2 saddle.
scientific article; zbMATH DE number 5132966