Properties of Morse forms that determine compact foliations on $M_g^2$

Author: Irina Mel'nikova

Math. Notes, vol. 60, no. 6, pp. 714-716, 1996.

Abstract: In [1, 2] P. Arnoux and G. Levitt showed that the topology of the foliation of a Morse form $\omega$ on a compact manifold is closely related to the structure of the integration mapping $[\omega]: H_1(M) \to R$. In this paper we consider the foliation of a Morse form on a two-dimensional manifold $M_g^2$. We study the relationship of the subgroup $Ker [\omega] \subset H_1(M_g^2)$ with the topology of the foliation. We consider the structure of the subgroup $Ker [\omega]$ for a compact foliation and prove a criterion for the compactness of a foliation.

Keywords: Two-dimensional manifold, foliation, Morse form, integration over cycles

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