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On the structure of a Morse form foliation. (English) Zbl 1224.57010
Summary: The foliation of a Morse form \(\omega \) on a closed manifold \(M\) is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of \(M\) and \(\omega \). Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of \(\operatorname {rk} \omega \) and \(\operatorname {Sing} \omega \). The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that, if \(\omega \) has more centers than conic singularities, then \(b _1(M)=0\), and, thus, the foliation has no minimal components and homologically non-trivial compact leaves, its foliation graph being a tree.

MSC:
57R30 Foliations in differential topology; geometric theory
58K65 Topological invariants on manifolds
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