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Article

Keywords:
Morse form foliation; compact leaf; cohomology class
Summary:
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.
References:
[1] Arnoux, P., Levitt, G.: Sur l'unique ergodicité des 1-formes fermées singulières (Unique ergodicity of closed singular 1-forms). Invent. Math. 84 (1986), 141-156 French. DOI 10.1007/BF01388736 | MR 0830042 | Zbl 0577.58021
[2] Farber, M.: Topology of Closed One-Forms. Mathematical Surveys and Monographs 108. AMS, Providence, RI (2004). MR 2034601 | Zbl 1052.58016
[3] Farber, M., Katz, G., Levine, J.: Morse theory of harmonic forms. Topology 37 (1998), 469-483. DOI 10.1016/S0040-9383(97)82730-9 | MR 1604870 | Zbl 0911.58001
[4] Gelbukh, I.: Presence of minimal components in a Morse form foliation. Differ. Geom. Appl. 22 (2005), 189-198. DOI 10.1016/j.difgeo.2004.10.006 | MR 2122742 | Zbl 1070.57016
[5] Gelbukh, I.: Ranks of collinear Morse forms. J. Geom. Phys. 61 (2011), 425-435. DOI 10.1016/j.geomphys.2010.10.010 | MR 2746127 | Zbl 1210.57027
[6] Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. 2nd corr. printing. Graduate Texts in Mathematics, 14. Springer, New York (1980). MR 0341518 | Zbl 0434.58001
[7] Hirsch, M. W.: Smooth regular neighborhoods. Ann. Math. (2) 76 (1962), 524-530. DOI 10.2307/1970372 | MR 0149492 | Zbl 0151.32604
[8] Hirsch, M. W.: Differential Topology. Graduate Texts in Mathematics, 33. Springer, New York (1976). MR 1336822 | Zbl 0356.57001
[9] Levitt, G.: 1-formes fermées singulières et groupe fondamental. Invent. Math. 88 (1987), 635-667 French. DOI 10.1007/BF01391835 | MR 0884804 | Zbl 0594.57014
[10] Levitt, G.: Groupe fondamental de l'espace des feuilles dans les feuilletages sans holonomie (Fundamental group of the leaf space of foliations without holonomy). French J. Differ. Geom. 31 (1990), 711-761. MR 1053343 | Zbl 0714.57016
[11] Pedersen, E. K.: Regular neighborhoods in topological manifolds. Mich. Math. J. 24 (1977), 177-183. DOI 10.1307/mmj/1029001881 | MR 0482775 | Zbl 0372.57010
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