# Foliations with all leaves compact

Annales de l'institut Fourier (1976)

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

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topEpstein, 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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- [7] A.W. WADSLEY, Ph. D. THESIS, University of Warwick1974.
- [8] A.W. WADSLEY, Geodesic foliations by circles, (available from University of Warwick). Zbl0336.57019
- [9] A. DRESS, Newman's theorems on transformation groups, Topology, 8 (1969) 203-207. Zbl0176.53201MR38 #6629
- [10] G. REEB, Sur certaines propriétés topologiques des variétés feuilletées, Act. Sci. et Ind. N° 1183, Hermann, Paris (1952). Zbl0049.12602MR14,1113a
- [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] R.D. EDWARDS, K.C. MILLET and D. SULLIVAN, Foliations with all leaves compact, (to appear). Zbl0356.57022
- [13] K.C. MILLETT, Compact Foliations, Springer-Verlag Lecture Notes 484, Differential Topology and Geometry Conference in Dijon 1974. Zbl0313.57018
- [14] J. DUGUNDJI, Topology, Allyn and Bacon (1970). Zbl0144.21501
- [15] D. SULLIVAN, A counterexample to the periodic orbit conjecture, (I.H.E.S. preprint, 1975). Zbl0372.58011

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