Czechoslovak Mathematical Journal, Vol. 59, No. 1, pp. 207-220, 2009

On the structure of a Morse form foliation

I. Gelbukh

I. Gelbukh, CIC-IPN, 07738, DF, Mexico, e-mail:

Abstract: The foliation of a Morse form $\w$ 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 $\w$. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of $\rk\w$ and $\Sing\w$. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if $\w$ 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 folitation graph being a tree.

Keywords: number of minimal components, number of maximal components, compact leaves, foliation graph, rank of a form

Classification (MSC 2000): 57R30, 58K65

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