Hopf cyclic cohomology and transverse characteristic classes

TL;DR

This paper refines the cyclic cohomological methods for computing Hopf cyclic cohomology related to infinite primitive Cartan-Lie pseudogroups and their characteristic classes, providing new models and explicit cocycles for foliations.

Contribution

It introduces a new model for Hopf cyclic cohomology of geometric Hopf algebras, enabling efficient transfer of classes and explicit cocycles in foliation groupoids.

Findings

Refined cyclic cohomological framework for Hopf algebras

Identification of a new Hopf cyclic complex model

Explicit cocycles for universal Hopf cyclic Chern classes

Abstract

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand-Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of foliation groupoids.