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:

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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