Close cohomologous Morse forms with compact leaves

Irina Gelbukh

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 2, page 515-528
  • ISSN: 0011-4642

Abstract

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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 γ , then any close cohomologous form has a compact leave close to γ . 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.

How to cite

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Gelbukh, Irina. "Close cohomologous Morse forms with compact leaves." Czechoslovak Mathematical Journal 63.2 (2013): 515-528. <http://eudml.org/doc/260718>.

@article{Gelbukh2013,
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.},
author = {Gelbukh, Irina},
journal = {Czechoslovak Mathematical Journal},
keywords = {Morse form foliation; compact leaf; cohomology class; Morse form foliation; compact leaf; cohomology class},
language = {eng},
number = {2},
pages = {515-528},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Close cohomologous Morse forms with compact leaves},
url = {http://eudml.org/doc/260718},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Gelbukh, Irina
TI - Close cohomologous Morse forms with compact leaves
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 515
EP - 528
AB - 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.
LA - eng
KW - Morse form foliation; compact leaf; cohomology class; Morse form foliation; compact leaf; cohomology class
UR - http://eudml.org/doc/260718
ER -

References

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  9. Levitt, G., 10.1007/BF01391835, Invent. Math. 88 (1987), 635-667 French. (1987) Zbl0594.57014MR0884804DOI10.1007/BF01391835
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