Atiyah covering index theorem for Riemannian foliations

Abstract

We use the symbol calculus for foliations developed by the authors in 2017 to derive a cohomological formula for the Connes–Chern character of the Type II spectral triple given also by the authors in 2018. 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 L^2 -covering index theorem.