Journal of the Mathematical Society of Japan

Compact foliations with finite transverse LS category

Steven HURDER and Paweł WALCZAK

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We prove that if $F$ is a foliation of a compact manifold $M$ with all leaves compact submanifolds, and the transverse saturated category of $F$ is finite, then the leaf space $M/F$ is compact Hausdorff. The proof is surprisingly delicate, and is based on some new observations about the geometry of compact foliations. The transverse saturated category of a compact Hausdorff foliation is always finite, so we obtain a new characterization of the compact Hausdorff foliations among the compact foliations as those with finite transverse saturated category.

Article information

J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1015-1046.

Received: 9 December 2016
First available in Project Euclid: 12 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R30: Foliations; geometric theory 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 57S15: Compact Lie groups of differentiable transformations

compact foliation transverse Lusternik–Schnirelmann category Epstein filtration


HURDER, Steven; WALCZAK, Paweł. Compact foliations with finite transverse LS category. J. Math. Soc. Japan 70 (2018), no. 3, 1015--1046. doi:10.2969/jmsj/76837683.

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  • R. Bishop and R. Crittenden, Geometry of manifolds, (reprint of the 1964 original), AMS Chelsea Publishing, Providence, RI, 2001.
  • C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Translated from the Portuguese by Sue E. Goodman, Progress in Mathematics, Birkhäuser Boston, MA, 1985.
  • A. Candel and L. Conlon, Foliations I, Amer. Math. Soc., Providence, RI, 2000.
  • H. Colman, Categoría LS en foliaciones, Publicaciones del Departamento de Topología y Geometría, 93, Universidade de Santiago de Compostele, 1998.
  • H. Colman, LS-categories for foliated manifolds, Foliations: Geometry and Dynamics (Warsaw, 2000), World Scientific Publishing Co. Inc., River Edge, N.J., 2002, 17–28.
  • H. Colman, Transverse Lusternik–Schnirelmann category of Riemannian foliations, Topology Appl., 141 (2004), 187–196.
  • H. Colman and S. Hurder, LS-category of compact Hausdorff foliations, Trans. Amer. Math. Soc., 356 (2004), 1463–1487.
  • H. Colman and E. Macias, Transverse Lusternik–Schnirelmann category of foliated manifolds, Topology, 40 (2000), 419–430.
  • M. do Carmo, Riemannian Geometry, 2nd edition, Birkhäuser, Boston, 1992.
  • R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 13–32.
  • D. B. A. Epstein, Periodic flows on 3-manifolds, Ann. of Math., 95 (1972), 66–82.
  • D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265–282.
  • D. B. A. Epstein and E. Vogt, A counterexample to the periodic orbit conjecture in codimension 3, Ann. of Math. (2), 108 (1978), 539–552.
  • C. Godbillon, Feuilletages: Etudes géométriques I, II, Publ. IRMA Strasbourg (1985–86), Progress in Math., 98, Birkhäuser, Boston, Mass., 1991.
  • A. Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geom., 15 (1981), 269–284.
  • G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A, B, Vieweg, Braunschweig, 1981.
  • H. Holmann, Seifertsche Faserräume, Math. Ann., 157 (1964), 138–166.
  • S. Hurder, Category and compact leaves, Topology Appl., 153 (2006), 2135–2154.
  • I. M. James, On category, in the sense of Lusternik–Schnirelmann, Topology, 17 (1978), 331–348.
  • I. M. James, Lusternik–Schnirelmann Category, Handbook of Algebraic Topology, Chapter 27, 1995, 1293–1310.
  • R. Langevin and P. Walczak, Transverse Lusternik–Schnirelmann category and non-proper leaves, Foliations: Geometry and Dynamics (Warsaw, 2000), World Scientific Publishing Co. Inc., River Edge, NJ, 2002, 351–354.
  • K. Millett, Compact foliations, Differential topology and geometry (Proc. Colloq., Dijon, 1974), Lect. Notes in Math., 484, Springer-Verlag, New York and Berlin, 1975, 277–287.
  • K. Millett, Generic properties of proper foliations, Fund. Math., 128 (1987), 131–138.
  • D. Montgomery, Pointwise periodic homeomorphisms, Amer. Journal Math., 59 (1937), 118–120.
  • M. H. A. Newman, A theorem on periodic transformations of spaces, Quart. Journal Math., 2 (1931), 1–9.
  • J. Plante, Foliations with measure preserving holonomy, Annals of Math., 102 (1975), 327–361.
  • G. Reeb, Sur certaines propiétés topologiques des variétés feuilletés, Act. Sci. Ind., 1183 (1952), 91–154.
  • H. Rummler, Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment. Math. Helv., 54 (1979), 224–239.
  • I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 359–363.
  • D. Sullivan, A counterexample to the periodic orbit conjecture, Publ. Math. Inst. Hautes Etudes Sci., 46, 5–14.
  • D. Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv., 54 (1979), 218–223.
  • I. Tamura, Topology of Foliations: An Introduction, 1976, Translations of Mathematical Monographs 97, Amer. Math. Soc., Providence, RI, 1992.
  • E. Vogt, Foliations of codimension 2 with all leaves compact, Manuscripta Math., 18 (1976), 187–212.
  • E. Vogt, Foliations of codimension 2 on closed 3, 4 and 5-manifolds, Math. Zeit., 157 (1977), 201–223.
  • E. Vogt, A periodic flow with infinite Epstein hierarchy, Manuscripta Math., 22 (1977), 403–412.
  • E. Vogt, A foliation of $\mathbf{R}^3$ and other punctured 3-manifolds by circles, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 215–232.