Cech-De Rham theory for leaf spaces of foliations

We present a new ''Cech-De Rham'' model for the cohomology of the classifying space of a foliated manifold. This model enables us to lift the construction of known characteristic classes in the cohomology of the manifold to the cohomology of the classifying space, by standard geometric methods and without making any reference to (necessarily non-Hausdorff) groupoids and their classifying spaces. We also show how the Cech-De Rham model can be used to prove some other known formulas, as well as a version of Poincare duality for foliations.