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Structure of a foliated neighbourhood

  • Jenny Harrison (a1)

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C. Ehresmann (2) has shown that if a leaf L of a smooth foliation has a foliated neighbourhood, then there exists a fibre bundle over L, normal to the leaves, with discrete structure group. Using the concept of a microbundle and the n-isotopy extension theorem, we find a similar result for both PL and TOP categories, and, in addition, show that the structure group can be chosen to be the holonomy group of L. As for applications we show that holonomy characterizes the foliated neighbourhood of a leaf (proved by Haefliger in the differentiable case (3)). In particular, if the holonomy group of a compact leaf L is trivial then the leaf has a trivial foliated neighbourhood, and if it is finite it has a neighbourhood of compact leaves which are covering spaces of L. Another corollary is the known result that a proper submersion with compact fibre is a fibration. Finally we use the fact that the constructed normal microbundle can be chosen to have its fibres contained in the leaves of a transverse foliation to demonstrate isotopy uniqueness of normal microbundles.

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(1)Davis, P. M. Sc. Dissertation, University of Warwick, 1973.
(2)Ehresmann, C. Les connexions infinitésimales dans un espace fibre différentiable. Colloque de Topologie de Bruxelles, 1950, CBRM, 2955.
(3)Haefliger, A.Structures feuilletées et cohomologie à valeur dans un faisceau de groupoldes. Comm. Math. Helv. 32 (1958), 248329.
(4)Haefliger, A.Variétés Feuilletées. Annali della Scuola Normale Superiore di Pisa, Serie III, Vol. XVI, Fasc. IV (1962), 367–96.
(5)Hudson, J. F. P.Extending PL isotopies. Proc. London Math. Soc. 16 (1966), 651–68.
(6)Rourke, C. P. and Sanderson, B. J.Introduction to PL topology (Springer-Verlag, 1972).
(7)Siebenmann, L. C.Deformation of homeomorphisms on stratified sets. Comm. Math. Helv. 47 (1972), 123–63.

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