  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.
