View More View Less
  • 1 Moscow State University Department of Mathematics Moscow Russia
  • | 2 IPN CIC 07738 DF, Mexico Mexico
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

The numbers m ( ω ) of minimal components and c ( ω ) of homologically independent compact leaves of the foliation of a Morse form ω on a connected smooth closed oriented manifold M are studied in terms of the first non-commutative Betti number b1 ( M ). A sharp estimate 0 ≦ m ( ω ) + c ( ω ) ≦ b1 ( M ) is given. It is shown that all values of m ( ω ) + c ( ω ), and in some cases all combinations of m ( ω ) and c ( ω ) with this condition, are reached on a given M . The corresponding issues are also studied in the classes of generic forms and compactifiable foliations.

  • Arnoux, P. and Levitt, G. , Sur l’unique ergodicité des 1-formes fermées singulières, Invent. Math. , 84 (1986), no. 1, 141–156. MR87g :58004

    Levitt G. , 'Sur l’unique ergodicité des 1-formes fermées singulières ' (1986 ) 84 Invent. Math. : 141 -156.

    • Search Google Scholar
  • Farber, M. , Topology of closed one-forms , Math. Surv. and Monographs, AMS, v. 108, 2004. MR2005c :58023

  • Gelbukh, I. , Presence of minimal components in a Morse form foliation, Diff. Geom. Appl. , 22 (2005), no. 2, 189–198. MR2005m :57040

    Gelbukh I. , 'Presence of minimal components in a Morse form foliation ' (2005 ) 22 Diff. Geom. Appl. : 189 -198.

    • Search Google Scholar
  • Gelbukh, I. , On the structure of a Morse form foliation, Czechoslovak Mathematical Journal , 59 (2009), no. 1, 207–220.

    Gelbukh I. , 'On the structure of a Morse form foliation ' (2009 ) 59 Czechoslovak Mathematical Journal : 207 -220.

    • Search Google Scholar
  • Harary, F. , Graph theory , Addison-Wesley Publ. Comp., 1994. MR41 #1566

  • Imanishi, H. , On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ. , 19 (1979), no. 2, 285–291. MR80k :57050

    Imanishi H. , 'On codimension one foliations defined by closed one forms with singularities ' (1979 ) 19 J. Math. Kyoto Univ. : 285 -291.

    • Search Google Scholar
  • Katok, A. , Invariant measures for flows on oriented surfaces, Sov. Math., Dokl. , 14 (1973), no. 3, 1104–1108. MR48 #9771

    Katok A. , 'Invariant measures for flows on oriented surfaces ' (1973 ) 14 Sov. Math., Dokl. : 1104 -1108.

    • Search Google Scholar
  • Levitt, G. , 1-formes fermées singulières et groupe fondamental, Invent. Math. , 88 (1987), 635–667. MR88d :58004

    Levitt G. , '1-formes fermées singulières et groupe fondamental ' (1987 ) 88 Invent. Math. : 635 -667.

    • Search Google Scholar
  • Levitt, G. , Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie, J. Diff. Geom. , 31 (1990), 711–761. MR91d :57018

    Levitt G. , 'Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie ' (1990 ) 31 J. Diff. Geom. : 711 -761.

    • Search Google Scholar
  • Meľnikova, I. , An indicator of the noncompactness of a foliation on Mg 2 , Math. Notes , 53 : 3 (1993), 356–358. MR94h :57044

    Meľnikova I. , 'An indicator of the noncompactness of a foliation on Mg2 ' (1993 ) 53 Math. Notes : 356 -358.

    • Search Google Scholar
  • Meľnikova, I. , A test for non-compactness of the foliation of a Morse form, Russ. Math. Surveys , 50 : 2 (1995) 444–445. MR96f :57028

    Meľnikova I. , 'A test for non-compactness of the foliation of a Morse form ' (1995 ) 50 Russ. Math. Surveys : 444 -445.

    • Search Google Scholar
  • Meľnikova, I. , Non-compact leaves of a Morse form foliation, Math. Notes , 63 : 6 (1998), 760–763. MR2000e :57046

    Meľnikova I. , 'Non-compact leaves of a Morse form foliation ' (1998 ) 63 Math. Notes : 760 -763.

    • Search Google Scholar