  I construct the real counterparts (which I call Borel-Bott classes) of the
R/Z classes constructed in "Characteristic classes in symplectic topology", to
appear, in the cohomology of volume-preserving and symplectomorhisms of a
compact (symplectic) manifold.I show that, for the symplectic action of the
mapping class group in the moduli space of stable vector bundles over a Riemann
surface, the restriction of the first constructed class from the
symplectomorphism group gives a generator for the second (bounded) cohomology
of the mapping class group.
