Czechoslovak Mathematical Journal, Vol. 63, No. 2, pp. 515-528, 2013

# Close cohomologous Morse forms with compact leaves

## Irina Gelbukh

Irina Gelbukh, Centro de Investigación en Computación (CIC), Instituto Politécnico Nacional (IPN), Av. Juan de Dios Bátiz, 07738, DF, México City, México, e-mail: gelbukh@member.ams.org

Abstract: We study the topology of foliations of close cohomologous Morse forms (smooth closed 1-forms with non-degenerate singularities) on a smooth closed oriented manifold. We show that if a closed form has a compact leave $\gamma$, then any close cohomologous form has a compact leave close to $\gamma$. Then we prove that the set of Morse forms with compactifiable foliations (foliations with no locally dense leaves) is open in a cohomology class, and the number of homologically independent compact leaves does not decrease under small perturbation of the form; moreover, for generic forms (Morse forms with each singular leaf containing a unique singularity; the set of generic forms is dense in the space of closed 1-forms) this number is locally constant.

Keywords: Morse form foliation, compact leaf, cohomology class

Classification (MSC 2010): 57R30, 58K65

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