Smooth Banach structure on orbit spaces and leaf spaces

Abstract

We investigate the quotients of Banach manifolds with respect to free actions of pseudogroups of local diffeomorphisms. These quotient spaces are called H-manifolds since the corresponding simply transitive action of the pseudogroup on its orbits is regarded as a homogeneity condition. The importance of these structures stems from the fact that for every regular foliation without holonomy of a Banach manifold, the corresponding leaf space has the natural structure of an H-manifold. This is our main technical result, and one of its remarkable consequences is an infinite-dimensional version of Sophus Lie's third fundamental theorem, to the effect that every real Banach-Lie algebra can be integrated to an H-group, that is, a group object in the category of H-manifolds. In addition to these general results we discuss a wealth of examples of H-groups which are not Banach-Lie groups.