The Number of Minimal Components and Homologically Independent Compact Leaves of a Weakly Generic Morse Form on a Closed Surface

Author: Irina Gelbukh

Rocky Mt. J. Math., to appear, 2013.

Abstract: On a closed orientable surface $M^2_g$ of genus $g$, we consider the foliation of a weakly generic Morse form $\omega$ on $M^2_g$ and show that for such forms $c(\omega) + m(\omega) = g - 1/2 k(\omega)$, where $c(\omega)$ is the number of homologically independent compact leaves of the foliation, $m(\omega)$ the number of its minimal components, and $k(\omega)$ the total number of singularities of $\omega$ surrounded by a minimal component. We also give lower bounds on $m(\omega)$ in terms of $k(\omega)$ and the form rank $rk \omega$ or the structure of $ker [\omega]$, where $[\omega]$ is the integration map.

Keywords: Surface; Morse form foliation; number of minimal components

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