Hopf algebroids and secondary characteristic classes

TL;DR

This paper explores the use of Hopf cyclic cohomology of a specific Hopf algebroid to define secondary characteristic classes, linking algebraic structures to topological invariants and homotopy invariance.

Contribution

It introduces a novel connection between Hopf algebroid cyclic cohomology and secondary characteristic classes, extending to K-theory and proving homotopy invariance of certain higher signatures.

Findings

Cyclic classes correspond to universal transgressed Chern characters.

Extension of cyclic classes to K-theory of $C^{*}$-algebras.

Homotopy invariance of higher signatures associated with Gelfand-Fuchs classes.

Abstract

We study a Hopf algebroid, $\calh$, naturally associated to the groupoid $U_n^\delta\ltimes U_n$. We show that classes in the Hopf cyclic cohomology of $\calh$ 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 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, \cite{connes:transverse}, and show that the higher signatures associated to them are homotopy invariant.