Hopf algebroids and secondary characteristic classes

Abstract

We study a Hopf algebroid, \mathcal H , naturally associated to the groupoid U^δ_n ⋉ U_n . We show that classes in the Hopf cyclic cohomology of \mathcal 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.