Hopf algebroids and secondary characteristic classes

Abstract

We study a Hopf algebroid, $\mathcal{H}$, naturally associated to the groupoid $U_n^{\delta }⋉U_n$. 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 $\rho$-invariant of Atiyah–Patodi–Singer. Moreover, these cyclic classes are shown to extend to pair with the K-theory of the associated C${}^{\ast }$-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.