Geometric construction of Hopf cyclic characteristic classes

TL;DR

This paper introduces a geometric method to explicitly construct Hopf cyclic characteristic classes for a Hopf algebra related to foliation theory, linking them to classical foliation characteristic classes.

Contribution

It provides an effective, Chern-Weil inspired construction of Hopf cyclic classes for H(n), clarifying their connection to foliation characteristic classes.

Findings

Explicit construction of Hopf cyclic classes for H(n)

Connection established between Hopf cyclic cohomology and Gelfand-Fuks cohomology

Method enhances understanding of index theory in foliation geometry

Abstract

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra H(n), playing the role of a `quantum structure group' for the `space of leaves' of a codimension n foliation, whose Hopf cyclic cohomology is canonically isomorphic to the Gelfand-Fuks cohomology of the Lie algebra of formal vector fields. However, with a few low-dimensional exceptions, no explicit construction was known for its Hopf cyclic classes. This paper provides an effective method for constructing the Hopf cyclic cohomology classes of H(n) and of H(n) relative to O(n), in the spirit of the Chern-Weil theory, which completely elucidates their relationship with the characteristic classes of foliations.