Hopf algebroids and secondary characteristic classes

We study a Hopf algebroid, H, naturally associated to the groupoid U ^δ_n U_n. We show that classes in the Hopf cyclic cohomology of H can be used to define secondary characteristic classes of trivialized flat U_n-bundles. For example, there is a cyclic class which corresponds to the universal transgressed Chern character and which gives rise to the continuous part of the ρ-invariant of Atiyah-Patodi-Singer. Moreover, these cyclic classes are shown to extend to pair with the K-theory of the associated C^*-algebra. This point of view gives leads to homotopy invariance results for certain characteristic numbers. In particular, we define a subgroup of the cohomology of a group analogous to the Gelfand-Fuchs classes described by Connes [3] and show that the higher signatures associated to them are homotopy invariant.