On the structure of a Morse form foliation. (English) Zbl 1224.57010

Gelbukh, I.

Czech. Math. J. 59, No. 1, 207-220 (2009).

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 \(\operatorname {rk} \omega \) and \(\operatorname {Sing} \omega \). 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 foliation graph being a tree.

MSC:

57R30	Foliations in differential topology; geometric theory
58K65	Topological invariants on manifolds

Keywords:

number of minimal components; number of maximal components; compact leaf; foliation graph; rank of a form

References:

[1]	P. Arnoux and G. Levitt: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math. 84 (1986), 141–156. · Zbl 0577.58021 · doi:10.1007/BF01388736
[2]	M. Farber, G. Katz and J. Levine: Morse theory of harmonic forms. Topology 37 (1998), 469–483. · Zbl 0911.58001 · doi:10.1016/S0040-9383(97)82730-9
[3]	I. Gelbukh: Presence of minimal components in a Morse form foliation. Diff. Geom. Appl. 22 (2005), 189–198. · Zbl 1070.57016 · doi:10.1016/j.difgeo.2004.10.006
[4]	I. Gelbukh: Ranks of collinear Morse forms. Submitted. · Zbl 1210.57027
[5]	F. Harary: Graph theory. Addison-Wesley Publ. Comp., Massachusetts, 1994. · Zbl 0795.05118
[6]	K. Honda: A note on Morse theory of harmonic 1-forms. Topology 38 (1999), 223–233. · Zbl 0959.58014 · doi:10.1016/S0040-9383(98)00018-4
[7]	H. Imanishi: On codimension one foliations defined by closed one forms with singularities. J. Math. Kyoto Univ. 19 (1979), 285–291. · Zbl 0417.57010
[8]	A. Katok: Invariant measures of flows on oriented surfaces. Sov. Math. Dokl. d14 (1973), 1104–1108. · Zbl 0298.28013
[9]	G. Levitt: 1-formes fermées singulières et groupe fondamental. Invent. Math. 88 (1987), 635–667. · Zbl 0594.57014 · doi:10.1007/BF01391835
[10]	G. Levitt: Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie. J. Diff. Geom. 31 (1990), 711–761. · Zbl 0714.57016
[11]	I. Mel’nikova: A test for non-compactness of the foliation of a Morse form. Russ. Math. Surveys 50 (1995), 444–445. · Zbl 0859.58005 · doi:10.1070/RM1995v050n02ABEH002092
[12]	I. Mel’nikova: Maximal isotropic subspaces of skew-symmetric bilinear map. Vestnik MGU 4 (1999), 3–5.
[13]	S. Novikov: The Hamiltonian formalism and a multivalued analog of Morse theory. Russian Math. Surveys 37 (1982), 1–56. · Zbl 0571.58011 · doi:10.1070/RM1982v037n05ABEH004020
[14]	A. Pazhitnov: The incidence coefficients in the Novikov complex are generically rational functions. Sankt-Petersbourg Math. J. 9 (1998), 969–1006.
[15]	D. Tischler: On fibering certain foliated manifolds over S 1. Topology 9 (1970), 153–154. · doi:10.1016/0040-9383(70)90037-6