Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



LS-category of compact Hausdorff foliations

Authors: Hellen Colman and Steven Hurder
Journal: Trans. Amer. Math. Soc. 356 (2004), 1463-1487
MSC (2000): Primary 55M30, 57R30; Secondary 57S05, 57N80
Published electronically: November 4, 2003
MathSciNet review: 2034314
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in $\{1,2, \ldots, \infty\}$. A foliation with all leaves compact and Hausdorff leaf space $M/\mathcal{F}$ is called compact Hausdorff. The transverse saturated category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}M$ of a compact Hausdorff foliation is always finite.

In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(M)$ in terms of the geometry of $\mathcal{F}$ and the Epstein filtration of the exceptional set $\mathcal{E}$. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that

\begin{displaymath}\max \{\operatorname{cat}(M/{\mathcal{F}}), \operatorname{ca...{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(\mathcal{E}) + q.\end{displaymath}

We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

References [Enhancements On Off] (What's this?)

  • 1. F. Bonahon and L. Siebenmann.
    The classification of Seifert fibred $3$-orbifolds.
    In Low-dimensional topology (Chelwood Gate, 1982), Lond. Math. Soc. Lect. Note Ser., vol. 95, Cambridge Univ. Press, Cambridge (1985), pp. 19-85. MR 87k:57012
  • 2. G.E. Bredon.
    Introduction to compact transformation groups.
    Academic Press, New York, 1972. MR 54:1265
  • 3. C. Camacho and A. Neto.
    Geometric Theory of Foliations.
    Progress in Math. Birkhausser, Boston, Basel and Stuttgart, 1985. MR 87a:57029
  • 4. A. Candel and L. Conlon.
    Foliations 1.
    Amer. Math. Soc., Providence, RI, 2000. MR 2002f:57058
  • 5. H. Colman.
    Categoría LS en foliaciones.
    Publicaciones del Departamento de Topología y Geometría, N 93, 1998, Universidade de Santiago de Compostele.
  • 6. H. Colman.
    LS-categories for foliated manifolds.
    Foliations: Geometry and Dynamics (Warsaw, 2000)
    World Scientific Publishing, River Edge, NJ. 2002:17-28. MR 2002m:55007
  • 7. H. Colman.
    Equivariant LS-category for finite group actions.
    In Proceedings of the AMS Conference: Lusternik-Schnirelmann Category in the New Millennium, July 2001, ed. O. Cornea et. al, Contemp. Math., vol. 316, Amer. Math. Soc., Providence, R.I., 2002, 35-40.
  • 8. H. Colman.
    Transverse category of Riemannian foliations.
    submitted, 2003.
  • 9. H. Colman and E. Macias.
    Transverse LS-category of foliated manifolds.
    Topology, Vol.40, 419-430, 2000. MR 2002c:55010
  • 10. R. Edwards, K. Millett, and D. Sullivan.
    Foliations with all leaves compact.
    Topology, 16:13-32, 1977. MR 55:11268
  • 11. D. B. A. Epstein.
    Periodic flows on 3-manifolds.
    Annals of Math., 95:68-82, 1972. MR 44:5981
  • 12. D. B. A. Epstein.
    Foliations with all leaves compact.
    Ann. Inst. Fourier (Grenoble)., 26:265-282, 1976. MR 54:8664
  • 13. D. B. A. Epstein and E. Vogt
    A counter-example to the Periodic Orbit Conjecture in codimension $3$.
    Annals of Math., 108:539-552, 1978. MR 80c:57014
  • 14. E. Fadell.
    The equivariant Lusternik-Schnirelmann method for invariant functionals and relative cohomological index theories.
    In Méthodes Topologiques en Analyse Non-Lineaire, ed. A. Granas, Montreal, 1985, 41-70. MR 86f:58005
  • 15. E. Fadell and S. Husseini.
    Relative cohomological index theories.
    Adv. Math, 64:1-31, 1987. MR 88f:55006
  • 16. J. Feldman and C.C Moore.
    Ergodic equivalence relations, cohomology and von Neumann algebras, I.
    Trans. Amer. Math. Soc., 234:289-324, 1977. MR 58:28261a
  • 17. A. Haefliger.
    Groupoïdes d'holonomie et classifiants.
    Astérisque, 116:70-97, Société Mathématique de France 1984. MR 86c:57026a
  • 18. A. Haefliger.
    Complexes of groups and orbihedra.
    Group theory from a geometrical viewpoint (Trieste, 1990), World Sci. Publishing, River Edge, NJ, 1991, 504-540. MR 93m:20048
  • 19. H. Holmann.
    Seifertsche Faserräume.
    Math. Ann., 157:138-166, 1964. MR 30:587
  • 20. S. Hurder.
    Category and compact leaves.
    submitted, 2003.
  • 21. S. Hurder and A. Katok.
    Ergodic theory and Weil measures for foliations.
    Annals of Math., pages 221-275, 1987. MR 89d:57042
  • 22. S. Hurder and P. Walczak.
    Compact foliations with finite transverse category.
    preprint, 2002.
  • 23. I.M. James.
    On category, in the sense of Lusternik-Schnirelmann.
    Topology 17:331-348, 1978. MR 80i:55001
  • 24. I.M. James.
    Lusternik-Schnirelmann Category.
    Handbook of Algebraic Topology, (ed. I. M. James), Elsevier Science, Amsterdam, 1995, 1293-1310. MR 97a:55003
  • 25. I.M. James and J. Morris.
    Fibrewise category.
    Proceedings of the Royal Society of Edinburgh, 119A:177-190, 1991. MR 92g:55005
  • 26. M. Jankins and W.D. Neumann.
    Lectures on Seifert Manifolds.
    Brandeis Lecture Notes No. 2, March 1983. MR 85j:57015
  • 27. K.B. Lee and F. Raymond.
    The role of Seifert fiber spaces in transformation groups.
    In Group actions on manifolds (Boulder, Colo., 1983), Contemp. Math Vol. 36, pages 367-425, 1985. Amer. Math. Soc., Providence, RI. MR 86j:57015
  • 28. L. Lusternik and L. Schnirelmann.
    Méthodes topologiques dans les Problèmes Variationnels.
    Hermann, Paris, 1934.
  • 29. W. Marzantowicz.
    A G-Lusternik-Schnirelmann category of space with an action of a compact Lie group.
    Topology, 28:403-412, 1989. MR 91c:55002
  • 30. K. Millett.
    Compact foliations.
    In Foliations: Dijon 1974, Lect. Notes in Math. volume 484, pages 277-287,1975. Springer-Verlag, New York and Berlin. MR 52:11944
  • 31. C. C. Moore and C. Schochet.
    Analysis on Foliated Spaces.
    Math. Sci. Res. Inst. Publ., 9, Springer-Verlag, 1988. MR 89h:58184
  • 32. D. Montgomery.
    Compact groups of transformations.
    In Differential Analysis, Bombay Colloq., 1964, pages 43-56, 1964. Oxford Univ. Press, London. MR 31:6238
  • 33. R. Palais.
    The Classification of G-spaces.
    Memoirs of the Amer. Math Soc., no. 36. Amer. Math. Soc., Providence, RI 1960. MR 31:1664
  • 34. G. Reeb.
    Sur certaines propiétés topologiques des variétés feuilletés.
    Act. Sci. et Ind. 1183:91-154, 1952, Hermann, Paris. MR 14:1113a
  • 35. C. Robinson.
    Dynamical Systems. Stability, Symbolic Dynamics, and Chaos.
    CRC Press, 1995. MR 97e:58064
  • 36. D.Ruelle and D.Sullivan Currents, flows, and diffeomorphisms. Topology 14, 319-327, 1975. MR 54:3759
  • 37. I. Satake.
    On a generalization of the notion of manifold.
    Proc. Nat. Acad. Sci. U.S.A., 42:359-363, 1956. MR 18:144a
  • 38. R. Schultz.
    Homotopy invariants and ${G}$-manifolds: a look at the past fifteen years.
    In Group actions on manifolds (Boulder, Colo., 1983), Contemp. Math Vol. 36, pages 17-81, 1985. Amer. Math. Soc., Providence, RI. MR 86e:57037
  • 39. D. Sullivan.
    A counterexample to the periodic orbit conjecture.
    Publ. Math. Inst. Hautes Etudes Sci., 46:5-14, 1976. MR 58:18492
  • 40. W. P. Thurston.
    The Geometry and Topology of Three-Manifolds.
    Lecture notes, Princeton University, Princeton, N.J. 1979.
  • 41. T. tom Dieck.
    Transformation groups.
    Walter de Gruyter & Co., Berlin, 1987. MR 89c:57048
  • 42. K. Varadarajan.
    On fibrations and category.
    Math. Zeitschrift, 88:267-273, 1965. MR 31:5199
  • 43. E. Vogt.
    Foliations of codimension $2$ with all leaves compact.
    Manuscripta Math., 18:187-212, 1976. MR 53:4091
  • 44. E. Vogt.
    Foliations of codimension 2 on closed 3, 4 and 5-manifolds.
    Math. Zeit., 157:201-223, 1977. MR 57:17665
  • 45. E. Vogt.
    A periodic flow with infinite Epstein hierarchy.
    Manuscripta Math., 22:403-412, 1977. MR 57:17664

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55M30, 57R30, 57S05, 57N80

Retrieve articles in all journals with MSC (2000): 55M30, 57R30, 57S05, 57N80

Additional Information

Hellen Colman
Affiliation: Department of Mathematics, University of Illinois at Chicago, 322 SEO (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045

Steven Hurder
Affiliation: Department of Mathematics, University of Illinois at Chicago, 322 SEO (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045

Received by editor(s): August 1, 2002
Published electronically: November 4, 2003
Additional Notes: The first author was partially supported by grants from the Xunta Galicia, Spain, and University of Sheffield, EU RTN1-1999-00176, Modern Homotopy Theory
The second author was partially supported by NSF Grant DMS-9704768
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society