Characteristic classes of foliations via SAYD-twisted cocycles

TL;DR

This paper develops a method to compute characteristic classes of foliations as cyclic cocycles using SAYD-twisted cocycles, providing explicit calculations for codimension 2 cases and connecting to Hopf-cyclic cohomology.

Contribution

It introduces a new approach using SAYD-twisted cocycles and equivariant cup products to explicitly compute characteristic classes of foliations in cyclic cohomology.

Findings

Explicit computation of characteristic classes in codimension 2

Matching with known results in codimension 1

Construction of SAYD-twisted cyclic cocycles

Abstract

We have previously shown that the truncated Weil algebra of any Lie algebra is a Hopf-cyclic type complex with nontrivial coefficients. In this paper we apply this result to transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. Our result in codimension 1 matches with the only existing explicit computation done by Connes-Moscovici. In codimension 2 case, we carry out a constructive and explicit computation, by which we present the transverse fundamental class, the Godbillon-Vey class, and the other four residual classes as cyclic cocycles on the groupoid action algebra. The main object in charge in this new characteristic map is a SAYD-twisted cyclic cocycle of the same degree as the codimension. We construct such a cocycle by introducing an equivariant Hopf-cyclic cohomology and an equivariant cup product.