A test for non-compactness of the foliation of a Morse form

Abstract

In this paper we study foliations determined by a closed 1-form with Morse singularities on smooth compact manifolds. The problem of the topological structure of the level surfaces of such forms was posed by Novikov in [1], and has been studied in [2]-[5]. In the present article we investigate the problem of the existence of a non-compact leaf, verify a test for non-compactness of a foliation in terms of the degree of irrationality of the form ω, and show that the non-compactness of a foliation is a case of general position. We consider a compact manifold Μ and a closed 1-form ω with Morse singularities defined on it. The closed form ω determines a foliation of codimension 1 on the set Μ — Sing ω. Correspondingly, a foliation ίΓω with singularities is obtained on Μ by adjoining the singular points to Μ — Sing ω. We say that a leaf 7 6 i w is compact if it is a non-singular compact leaf or can be compactified by adding singular points. The foliation 3~ω is said to be compact if all its leaves are compact. Definition 1. Let 7 be a non-singular compact leaf of 3ω, and consider the map 7 —ψ [η e Ηη-ι(Μ). Then the image of the set of compact leaves under this map generates a subgroup of Hn^i(M). Denote it by Ηω. The foliation &ω is characterized by the condition that 7 0 7 ' = 0 for 7,7' £ J w . We consider the group Hn-i(M) and the intersection operation for homology classes,