Hopf algebras and universal Chern classes

Abstract

We construct a variant ${\mathcal{K}}_n$ of the Hopf algebra ${\mathcal{H}}_n$, which acts directly on the noncommutative model for the space of leaves of a general foliation rather than on its frame bundle. We prove that the Hopf cyclic cohomology of ${\mathcal{K}}_n$ is isomorphic to that of the pair $({\mathcal{H}}_n,{\mathfrak{gl}}_n)$ and thus consists of the universal Hopf cyclic Chern classes. We also realize these classes in terms of geometric cocycles.