The index of operators on foliated bundles

TL;DR

This paper computes the equivariant cohomology Chern character of leafwise elliptic operator indices on foliated bundles using noncommutative geometry and pseudodifferential operator algebras.

Contribution

It introduces a novel approach to calculating the Chern character in the context of foliated bundles via noncommutative geometric techniques.

Findings

Explicit computation of the equivariant cohomology Chern character

Development of algebraic tools for analyzing noncommutative pseudodifferential algebras

Application of cyclic cohomology methods to foliation index theory

Abstract

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geometry.