Secondary characteristic classes and cyclic cohomology of Hopf algebras

Let X be a manifold on which a discrete (pseudo)group of diffeomorphisms Γ acts, and let E be a Γ-equivariant vector bundle on X. We give a construction of cyclic cocycles on the cross product algebra C0∞(X)⋊Γ representing the equivariant characteristic classes of E. Our formulas can be viewed as a higher-dimensional analogue of Connes’ Godbillon–Vey cyclic cocycle. An essential tool for our construction, which allows us to overcome difficulties arising in the higher-dimensional case, is Connes–Moscoviciʹs theory of cyclic cohomology of Hopf algebras.