Orbit spaces and leaf spaces of foliations as generalized manifolds

Abstract: The paper gives a categorical approach to generalized manifolds such as orbit spaces and leaf spaces of foliations. It is suggested to consider these spaces as sets equipped with some additional structure which generalizes the notion of atlas. The approach is compared with the known ones that use the Grothendieck topos, the Haefliger classifying space and the Connes non-commutative geometry. The main aim of this paper is to indicate that the suggested approach permits to simplify essentially the modern theory of the leaf space of foliations and the orbit space of diffeomorphism groups, and to obtain some new results on the characteristic classes of foliations. As an application, it is shown that the first Chern class is non-trivial for the leaf space of the Reeb foliation and its geometrical meaning is indicated.