Stable characteristic classes of smooth manifold bundles

Understand

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,...

• A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n.
• Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphism groups of oriented closed manifolds of dimension d.
• A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M.