Journal of Noncommutative Geometry

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Volume 3, Issue 1, 2009, pp. 1–25
DOI: 10.4171/JNCG/28

Published online: 2009-03-31

Hopf algebroids and secondary characteristic classes

Jerome Kaminker[1] and Xiang Tang[2]

(1) UC Davis, CA
(2) Washington University, St. Louis, MO

We study a Hopf algebroid, ℋ, naturally associated to the groupoid UδnUn. 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 [3] and show that the higher signatures associated to them are homotopy invariant.

Keywords: Riemannian foliation, secondary characteristic class, Hopf algebroid, cyclic cohomology, homotopy invariance

Kaminker Jerome, Tang Xiang: Hopf algebroids and secondary characteristic classes. J. Noncommut. Geom. 3 (2009), 1-25. doi: 10.4171/JNCG/28