Cech-De Rham theory for leaf spaces of foliations

Abstract

This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations For a manifold M equipped with a foliation F it is wellknown that the coarse naive leaf space MF obtained from M by identifying each leaf to a point contains very little information In the literature various models for a ner leaf space MF are used for dening its cohomology For example one considers the cohomology of the classifying space of the foliation the sheaf cohomology of its holonomy groupoid or the cyclic cohomology of its convolution algebra Each of these methods has considerable drawbacks Eg they all involve nonHausdorspaces in an essential way More specically the classifying space which is probably the most common model for the ne leaf space is a space which in general is infinite dimensional and nonHausdor it is not a CWcomplex and it has lost all the smooth structure of the original foliation In particular it is not suitable for constructing cohomology theories with compact support For this reason the construction of characteristic classes in the cohomology of the classifying space of the foliation proceeds in a very indirect way and many of the standard geometrical constructions have to be replaced by or supplied with abstract nontrivial arguments The same applies to the construction of universal characteristic classes in the cohomology of the classifying space of the Haeiger groupoid q It is possible to construct interesting classes of foliated or transversal bundles over foliations by explicit geometrical methods but these classes are constructed in the cohomology of the manifold M rather than that of the leaf space MF