Atiyah covering index theorem for riemannian foliations

TL;DR

This paper proves an Atiyah $L^2$ covering index theorem for Riemannian foliations by deriving a cohomological formula for the Connes-Chern character using symbol calculus, linking K-theory and foliation geometry.

Contribution

It introduces a cohomological formula for the Connes-Chern character applicable to Riemannian foliations, extending the Atiyah covering index theorem in this context.

Findings

Derived a cohomological formula for the Connes-Chern character

Proved the Atiyah $L^2$ covering index theorem for Riemannian foliations

Established the equivalence of Connes-Chern characters on the image of the Baum-Connes map

Abstract

We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula for the Connes-Chern character of the semi-finite spectral triple. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the same map on the image of the maximal Baum-Connes map in K-theory, thereby proving an Atiyah $L^2$ covering index theorem.