Characteristic classes for flat Diff(M)-foliations
In this work we relate the known results about the homotopy type of classifying spaces for smooth foliations, with the homology and cohomology of the discrete group of diffeomorphisms of a smooth compact connected oriented manifold. The Mather-Thurston Theorem forms a bridge between the results on the homotopy types of classifying spaces and the various homology and cohomology groups that are studied. We introduce the algebraic K-theory of a manifold M that is derived from the discrete group of diffeomorphisms of M, and observe that the calculations of homotopy groups in this work are about these K-theory groups. We include a variety of remarks and open problems related to the study of the diffeomorphism groups and their homological invariants using the Mather-Thurston Theorem.