Foliations with all leaves compact
Annales de l'Institut Fourier, Volume 26 (1976) no. 1, p. 265-282

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.

On définit la notion de volume dans un espace feuilleté. Si L est une feuille compacte, alors toute feuille rencontrant un petit voisinage de L a un volume très grand, ou sinon un volume qui est approximativement un multiple entier du volume de L. Si toutes les feuilles sont compactes il y a trois objets apparentés à étudier. Premièrement la topologie de l’espace des feuilles ; puis l’holonomie des feuilles ; enfin il s’agit de savoir si les feuilles ont un volume localement borné. Nous établissons diverses implications entre ces concepts et nous donnons des exemples. Nous démontrons un résultat énoncé par Ehresmann, et publié sans démonstration : dans une variété feuilletée, si une feuille compacte a des voisinages saturés arbitrairement petits, alors l’holonomie de cette feuille est finie.

     author = {Epstein, D. B. A.},
     title = {Foliations with all leaves compact},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {26},
     number = {1},
     year = {1976},
     pages = {265-282},
     doi = {10.5802/aif.607},
     zbl = {0313.57017},
     mrnumber = {54 \#8664},
     language = {en},
     url = {}
Epstein, D. B. A. Foliations with all leaves compact. Annales de l'Institut Fourier, Volume 26 (1976) no. 1, pp. 265-282. doi : 10.5802/aif.607.

[1] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie, Bruxelles (1950), 29-55. | MR 13,159e | Zbl 0054.07201

[2] D.B.A. Epstein, Periodic flows on 3-manifolds, Annals of Math., 95 (1972), 68-82. | MR 44 #5981 | Zbl 0231.58009

[3] A. Haefliger, Variétés feuilletées, Ann. Scuola Normale Sup. Pisa, 16 (1962), 367-397. | Numdam | MR 32 #6487 | Zbl 0122.40702

[4] A. Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comm. Math. Helv., 32 (1958), 248-329. | MR 20 #6702 | Zbl 0085.17303

[5] D. Montgomery and L. Zippin, Topological Transformation Groups, Inter-Science, New York (1955). | MR 17,383b | Zbl 0068.01904

[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. | MR 42 #3809 | Zbl 0203.26203

[7] A.W. Wadsley, Ph. D. Thesis, University of Warwick 1974.

[8] A.W. Wadsley, Geodesic foliations by circles, (available from University of Warwick). | Zbl 0336.57019

[9] A. Dress, Newman's theorems on transformation groups, Topology, 8 (1969) 203-207. | MR 38 #6629 | Zbl 0176.53201

[10] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Act. Sci. et Ind. N° 1183, Hermann, Paris (1952). | MR 14,1113a | Zbl 0049.12602

[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. | MR 14,192d | Zbl 0049.40401

[12] R.D. Edwards, K.C. Millet and D. Sullivan, Foliations with all leaves compact, (to appear).

[13] K.C. Millett, Compact Foliations, Springer-Verlag Lecture Notes 484, Differential Topology and Geometry Conference in Dijon 1974. | Zbl 0313.57018

[14] J. Dugundji, Topology, Allyn and Bacon (1970). | Zbl 0144.21501

[15] D. Sullivan, A counterexample to the periodic orbit conjecture, (I.H.E.S. preprint, 1975). | Numdam | Zbl 0372.58011