Journal of the Mathematical Society of Japan

Compact foliations with finite transverse LS category

Steven HURDER and Paweł WALCZAK

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1528790547

Digital Object Identifier
doi:10.2969/jmsj/76837683

Mathematical Reviews number (MathSciNet)
MR3830797

Zentralblatt MATH identifier
06966972

Subjects
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

Keywords
compact foliation transverse Lusternik–Schnirelmann category Epstein filtration

Citation

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. https://projecteuclid.org/euclid.jmsj/1528790547


Export citation

References

  • 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.