Hopf algebroids and secondary characteristic classes
Jerome KaminkerUC Davis, CA
Xiang TangWashington University, St. Louis, MO
We study a Hopf algebroid, ℋ, naturally associated to the groupoid Uδn ⋉ Un. We show that classes in the Hopf cyclic cohomology of ℋ can be used to define secondary characteristic classes of trivialized flat Un-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  and show that the higher signatures associated to them are homotopy invariant.
Cite this article
Jerome Kaminker, Xiang Tang, Hopf algebroids and secondary characteristic classes. J. Noncommut. Geom. 3 (2009), no. 1, pp. 1–25DOI 10.4171/JNCG/28