  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. We show that
each rational stable characteristic class of oriented manifold bundles of even
dimension d is tautological, e.g., if d=2, then each rational stable
characteristic class is a polynomial in terms of Miller-Morita-Mumford classes.
