zbMATH — the first resource for mathematics

Close cohomologous Morse forms with compact leaves. (English) Zbl 1289.57009
Let \(\omega \) be a Morse form, i.e., a closed \(1\)-form with non-degenerate singularities (zeros), on a smooth closed manifold \(M\). On the complement \(M\smallsetminus \text{Sing }\omega \) of the singular set of \(\omega \), consider the foliation \(\mathcal {F}_{\omega }=\ker \omega \). The author proves a number of results about the topology of this foliation and its dependence on \(\omega \) within a fixed cohomology class \([\omega ]=\Omega \in H^1(M;\mathbb R)\). Namely, the foliation \(\mathcal {F}_{\omega }\) is said to be compactifiable if all leaves are closed as subsets of \(M\smallsetminus \text{Sing }\omega \). It is shown that Morse forms with compactifiable foliations form an open subset of \(\Omega \). For such foliations consider the rank of the subgroup of \(H_{n-1}(M)\) generated by the fundamental classes of all compact leaves of \(\mathcal {F}_{\omega }\). It is then shown that this number, i.e., the number of homologically independent compact leaves, does not decrease under small perturbations of \(\omega \) within \(\Omega \). For generic forms (Morse forms with each singular leaf containing a unique singularity), it is even proved to be locally constant.

57R30 Foliations in differential topology; geometric theory
58K65 Topological invariants on manifolds
Full Text: DOI
[1] P. Arnoux, G. Levitt: Sur l’unique ergodicité des 1-formes fermées singulières (Unique ergodicity of closed singular 1-forms). Invent. Math. 84 (1986), 141–156. (In French.) · Zbl 0577.58021 · doi:10.1007/BF01388736
[2] M. Farber: Topology of Closed One-Forms. Mathematical Surveys and Monographs 108. AMS, Providence, RI, 2004.
[3] M. Farber, G. Katz, J. Levine: Morse theory of harmonic forms. Topology 37 (1998), 469–483. · Zbl 0911.58001 · doi:10.1016/S0040-9383(97)82730-9
[4] I. Gelbukh: Presence of minimal components in a Morse form foliation. Differ. Geom. Appl. 22 (2005), 189–198. · Zbl 1070.57016 · doi:10.1016/j.difgeo.2004.10.006
[5] I. Gelbukh: Ranks of collinear Morse forms. J. Geom. Phys. 61 (2011), 425–435. · Zbl 1210.57027 · doi:10.1016/j.geomphys.2010.10.010
[6] M. Golubitsky, V. Guillemin: Stable Mappings and Their Singularities. 2nd corr. printing. Graduate Texts in Mathematics, 14. Springer, New York, 1980. · Zbl 0434.58001
[7] M. W. Hirsch: Smooth regular neighborhoods. Ann. Math. (2) 76 (1962), 524–530. · Zbl 0151.32604 · doi:10.2307/1970372
[8] M. W. Hirsch: Differential Topology. Graduate Texts in Mathematics, 33. Springer, New York, 1976.
[9] G. Levitt: 1-formes fermées singuli‘eres et groupe fondamental. Invent. Math. 88 (1987), 635–667. (In French.) · Zbl 0594.57014 · doi:10.1007/BF01391835
[10] G. Levitt: Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie (Fundamental group of the leaf space of foliations without holonomy). J. Differ. Geom. 31 (1990), 711–761. (In French.) · Zbl 0714.57016
[11] E. K. Pedersen: Regular neighborhoods in topological manifolds. Mich. Math. J. 24 (1977), 177–183. · Zbl 0372.57010 · doi:10.1307/mmj/1029001881
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.