# Article

 Title: On the structure of a Morse form foliation (English) Author: Gelbukh, I. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 59 Issue: 1 Year: 2009 Pages: 207-220 Summary lang: English . Category: math . Summary: The foliation of a Morse form $\omega$ on a closed manifold $M$ is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of $M$ and $\omega$. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of $\mathop{\rm rk}\omega$ and ${\rm Sing} \omega$. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if $\omega$ has more centers than conic singularities then $b_1(M)=0$ and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree. (English) Keyword: number of minimal components Keyword: number of maximal components Keyword: compact leaves Keyword: foliation graph Keyword: rank of a form MSC: 57R30 MSC: 58K65 idZBL: Zbl 1224.57010 idMR: MR2486626 . Date available: 2010-07-20T15:01:51Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/140474 . Reference: [1] Arnoux, P., Levitt, G.: Sur l'unique ergodicité des 1-formes fermées singulières.Invent. Math. 84 (1986), 141-156. Zbl 0577.58021, MR 0830042, 10.1007/BF01388736 Reference: [2] Farber, M., Katz, G., Levine, J.: Morse theory of harmonic forms.Topology 37 (1998), 469-483. Zbl 0911.58001, MR 1604870, 10.1016/S0040-9383(97)82730-9 Reference: [3] Gelbukh, I.: Presence of minimal components in a Morse form foliation.Diff. Geom. Appl. 22 (2005), 189-198. Zbl 1070.57016, MR 2122742, 10.1016/j.difgeo.2004.10.006 Reference: [4] Gelbukh, I.: Ranks of collinear Morse forms.Submitted. Reference: [5] Harary, F.: Graph theory.Addison-Wesley Publ. Comp., Massachusetts (1994). MR 0256911 Reference: [6] Honda, K.: A note on Morse theory of harmonic 1-forms.Topology 38 (1999), 223-233. Zbl 0959.58014, MR 1644028, 10.1016/S0040-9383(98)00018-4 Reference: [7] Imanishi, H.: On codimension one foliations defined by closed one forms with singularities.J. Math. Kyoto Univ. 19 (1979), 285-291. Zbl 0417.57010, MR 0545709 Reference: [8] Katok, A.: Invariant measures of flows on oriented surfaces.Sov. Math. Dokl. 14 (1973), 1104-1108. Zbl 0298.28013 Reference: [9] Levitt, G.: 1-formes fermées singulières et groupe fondamental.Invent. Math. 88 (1987), 635-667. Zbl 0594.57014, MR 0884804, 10.1007/BF01391835 Reference: [10] Levitt, G.: Groupe fondamental de l'espace des feuilles dans les feuilletages sans holonomie.J. Diff. Geom. 31 (1990), 711-761. Zbl 0714.57016, MR 1053343 Reference: [11] Mel'nikova, I.: A test for non-compactness of the foliation of a Morse form.Russ. Math. Surveys 50 (1995), 444-445. Zbl 0859.58005, 10.1070/RM1995v050n02ABEH002092 Reference: [12] Mel'nikova, I.: Maximal isotropic subspaces of skew-symmetric bilinear map.Vestnik MGU 4 (1999), 3-5. MR 1716286 Reference: [13] Novikov, S.: The Hamiltonian formalism and a multivalued analog of Morse theory.Russian Math. Surveys 37 (1982), 1-56. MR 0676612, 10.1070/RM1982v037n05ABEH004020 Reference: [14] Pazhitnov, A.: The incidence coefficients in the Novikov complex are generically rational functions.Sankt-Petersbourg Math. J. 9 (1998), 969-1006. MR 1604381 Reference: [15] Tischler, D.: On fibering certain foliated manifolds over $S^1$.Topology 9 (1970), 153-154. Zbl 0177.52103, MR 0256413, 10.1016/0040-9383(70)90037-6 .

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