An indicator of the noncompactness of a foliation on $M_g^2$

Author: Irina A. Mel'nikova

Mathematical Notes, vol. 53, no. 3, pp. 356-358, 1993.

Abstract: Let $\omega$ be a closed form on a manifold $M$ and possessing nondegenerate isolated singularities. A point $p \in M$ is called a regular singularity of $\omega$, if in some neighbourhood $O(p)\omega = df$, where $f$ is a Morse function having a singularity at $p$. The form $\omega$ determines a foliation $F_\omega$ on the set $M - Sing\omega$. Let $M = M^2_g$, the orientable closed surface of genus 2. The homology classes $[\gamma]$ of the nonsingular compact leaves of $F_\omega$ generate a subgroup of $H_1(M^2_g)$ denoted by $H_\omega$. If $[z_1], ..., [z_{2g}]$ is a basis of $H_1(M^2_g)$ we define $dirr\omega = rk_\mathbb{Q}{\int_{z_1}\omega, ..., \int_{z_{2g}}\omega} - 1$. By $M_\omega$ is denoted the set obtained by discarding all maximal neighbourhoods consisting of diffeomorphic compact leaves and all leaves which can be compactified by adding singular points. Theorem 1. $M_\omega = \emptyset \Leftrightarow rk H_\omega = g$. Theorem 2. If $\omega$ is a closed form with Morse singularities given on $M^2_g$ ($g \ne 0$) such that $dirr\omega \ge g$, then the foliation $F_\omega$ has a noncompact fiber. (The paper provides no abstract; this abstract is provided by a Zentralblatt review.)

Keywords: Morse form; foliations; compactness (The paper provides no keywords; these keywords are likely to match the topic.)

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