Gaël Meigniez

Submersions, fibrations and bundles

When does a submersion have the homotopy lifting property? When is it a locally trivial fibre bundle? We establish characterizations in terms of consistency in the topology of the neighbouring fibres.

Holonomy Groups of Solvable Lie Foliations

Following C. Ehresmann, to every foliated manifold one associates a pseudogroup of transformations that represents the transverse structure of the foliation. Conversely, every pseudogroup comes from a foliated manifold.

On completeness of transversely projective foliations

A CODIMENSION-I foliation on a differentiable n-manifold is called complete if the manifold is covered by the product of the real line and an (possibly noncompact) (n I)-manifold C, such that the leaves of the foliation are the images of the hypersurfaces constant x 1. Much of the study of a complete foliation reduces to the study of a group of diffeomorphisms of the real line. Is completeness stable under small perturbations of the foliation? The aim of this paper is to answer this question in the rcstrictcd frame of codimcnsion-I foliations which carry a transverse projective geometry. on 3-manifolds. In this paper, all manifolds, smooth maps and foliations arc of class C’. A trun.scerse projectiw utfus for a foliation 9, is a covering of the underlying manifold by a collection of 9-distinguished charts, for which the transverse coordinate transformations are projective, i.c. of the form .Y H ux + b/cx + d. A trunsuerse projeclioe structure is a maximal transverse projective atlas. A trunscersely projective fdiution is a foliation endowed with a transverse projective structure. We shall exclude from this work the following particular case. We call a transversely projective foliation reducible if it admits an atlas in which all the transverse coordinate transformations, considered as elements of PSL(2, [w), belong to some proper closed connected Lie subgroup. For example, let 9 be a transversely projective foliation which admits an atlas in which all the transverse coordinate transformations are of the form x H (IX + b (this is called a trunsversely uJfine foliurion). Then 9 is reducible. Here is our main result. We shall endow the set of all foliations on a tixcd manifold, with the C’fine topology in the sense of [3]. Roughly speaking: two foliations are close to each other when they admit C’-close atlases of distinguished charts.