Foliations with all leaves compact

D. B. A. Epstein

Annales de l'institut Fourier (1976)

  • Volume: 26, Issue: 1, page 265-282
  • ISSN: 0373-0956

Abstract

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The notion of the “volume" of a leaf in a foliated space is defined. If L is a compact leaf, then any leaf entering a small neighbourhood of L either has a very large volume, or a volume which is approximatively an integral multiple of the volume of L . If all leaves are compact there are three related objects to study. Firstly the topology of the quotient space obtained by identifying each leaf to a point ; secondly the holonomy of a leaf ; and thirdly whether the leaves have a locally bounded volume. We prove various implications relating these concepts and we also give some counterexamples. We give a proof of the result, published by Ehresmann without proof, that in a foliated manifold, if a compact leaf has arbitrarily small neighbourhoods, which are saturated by the leaves, then its holonomy is finite.

How to cite

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Epstein, D. B. A.. "Foliations with all leaves compact." Annales de l'institut Fourier 26.1 (1976): 265-282. <http://eudml.org/doc/74269>.

@article{Epstein1976,
abstract = {The notion of the “volume" of a leaf in a foliated space is defined. If $L$ is a compact leaf, then any leaf entering a small neighbourhood of $L$ either has a very large volume, or a volume which is approximatively an integral multiple of the volume of $L$. If all leaves are compact there are three related objects to study. Firstly the topology of the quotient space obtained by identifying each leaf to a point ; secondly the holonomy of a leaf ; and thirdly whether the leaves have a locally bounded volume. We prove various implications relating these concepts and we also give some counterexamples. We give a proof of the result, published by Ehresmann without proof, that in a foliated manifold, if a compact leaf has arbitrarily small neighbourhoods, which are saturated by the leaves, then its holonomy is finite.},
author = {Epstein, D. B. A.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {265-282},
publisher = {Association des Annales de l'Institut Fourier},
title = {Foliations with all leaves compact},
url = {http://eudml.org/doc/74269},
volume = {26},
year = {1976},
}

TY - JOUR
AU - Epstein, D. B. A.
TI - Foliations with all leaves compact
JO - Annales de l'institut Fourier
PY - 1976
PB - Association des Annales de l'Institut Fourier
VL - 26
IS - 1
SP - 265
EP - 282
AB - The notion of the “volume" of a leaf in a foliated space is defined. If $L$ is a compact leaf, then any leaf entering a small neighbourhood of $L$ either has a very large volume, or a volume which is approximatively an integral multiple of the volume of $L$. If all leaves are compact there are three related objects to study. Firstly the topology of the quotient space obtained by identifying each leaf to a point ; secondly the holonomy of a leaf ; and thirdly whether the leaves have a locally bounded volume. We prove various implications relating these concepts and we also give some counterexamples. We give a proof of the result, published by Ehresmann without proof, that in a foliated manifold, if a compact leaf has arbitrarily small neighbourhoods, which are saturated by the leaves, then its holonomy is finite.
LA - eng
UR - http://eudml.org/doc/74269
ER -

References

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  1. [1] C. EHRESMANN, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie, Bruxelles (1950), 29-55. Zbl0054.07201MR13,159e
  2. [2] D.B.A. EPSTEIN, Periodic flows on 3-manifolds, Annals of Math., 95 (1972), 68-82. Zbl0231.58009MR44 #5981
  3. [3] A. HAEFLIGER, Variétés feuilletées, Ann. Scuola Normale Sup. Pisa, 16 (1962), 367-397. Zbl0122.40702MR32 #6487
  4. [4] A. HAEFLIGER, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comm. Math. Helv., 32 (1958), 248-329. Zbl0085.17303MR20 #6702
  5. [5] D. MONTGOMERY and L. ZIPPIN, Topological Transformation Groups, Inter-Science, New York (1955). Zbl0068.01904MR17,383b
  6. [6] R.S. PALAIS, C1 -actions of compact Lie groups on compact manifolds are C1 -equivalent to C∞ -actions, Am. Jour. of Math., 92 (1970) 748-760. Zbl0203.26203MR42 #3809
  7. [7] A.W. WADSLEY, Ph. D. THESIS, University of Warwick1974. 
  8. [8] A.W. WADSLEY, Geodesic foliations by circles, (available from University of Warwick). Zbl0336.57019
  9. [9] A. DRESS, Newman's theorems on transformation groups, Topology, 8 (1969) 203-207. Zbl0176.53201MR38 #6629
  10. [10] G. REEB, Sur certaines propriétés topologiques des variétés feuilletées, Act. Sci. et Ind. N° 1183, Hermann, Paris (1952). Zbl0049.12602MR14,1113a
  11. [11] R.H. BING, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Annals of Math, 56 (1952), 354-362. Zbl0049.40401MR14,192d
  12. [12] R.D. EDWARDS, K.C. MILLET and D. SULLIVAN, Foliations with all leaves compact, (to appear). Zbl0356.57022
  13. [13] K.C. MILLETT, Compact Foliations, Springer-Verlag Lecture Notes 484, Differential Topology and Geometry Conference in Dijon 1974. Zbl0313.57018
  14. [14] J. DUGUNDJI, Topology, Allyn and Bacon (1970). Zbl0144.21501
  15. [15] D. SULLIVAN, A counterexample to the periodic orbit conjecture, (I.H.E.S. preprint, 1975). Zbl0372.58011

Citations in EuDML Documents

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  1. Hiromichi Nakayama, On strongly Hausdorff flows
  2. Elmar Vogt, A foliation of 𝐑 3 and other punctured 3-manifolds by circles
  3. Xavier Gomez-Mont, Integrals for holomorphic foliations with singularities having all leaves compact
  4. Thierry Barbot, Variétés affines radiales de dimension 3
  5. Robert A. Wolak, Foliated and associated geometric structures on foliated manifolds
  6. Jean-Marie Morvan, Georges Zafindratafa, Conformally flat submanifolds
  7. Kōjun Abe, Kazuhiko Fukui, On the first homology of automorphism groups of manifolds with geometric structures
  8. Paul Baird, Harmonic morphisms and circle actions on 3- and 4-manifolds
  9. André Haefliger, Feuilletages riemanniens
  10. Thierry Barbot, Feuilletages transversalement projectifs sur les variétés de Seifert

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