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 well-known that the coarse (naive) leaf space M=F, obtained from M by identifying each leaf to a point, contains very little information. In the literature, various models for a ner leaf space M=F are used for dening its cohomology. For example, one considers the cohomology of the classifying space of the foliation [2, 13, 17, 22], the sheaf cohomology of its holonomy groupoid [10, 18, 26], or the cyclic cohomology of its convolution algebra [7, 8]. Each of these methods has considerable drawbacks. E.g. they all involve non-Hausdor spaces 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 innite dimensional and non-Hausdor, it is not a CW-complex, 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 tobe replaced by or supplied with abstract non-trivial arguments. The same applies to the construction of \universal" characteristic classes in the cohomology of the classifying space of the Hae iger groupoid