  For a Lie group G and a smooth manifold W, we study the difference between
smooth actions of G on W and bundles over the classifying space of G with fiber
W and structure group Diff(W). In particular, we exhibit smooth manifold
bundles over BSU(2) that are not induced by an action. The main tool for
reaching this goal is a technical result that gives a constraint for the values
of tautological classes pulled back to the cohomology of BSU(2) along a map
induced by an action.
