The analytic index of elliptic pseudodifferential operators on a singular foliation

TL;DR

This paper develops an analytic index theory for elliptic pseudodifferential operators on singular foliations, using KK-theory and introducing an adiabatic foliation to connect different constructions.

Contribution

It constructs the analytic index as a KK-theory element for elliptic operators on singular foliations and relates it to an adiabatic foliation on the product space.

Findings

Defined the analytic index as a KK-theory element.

Connected the index to an adiabatic foliation on M×R.

Extended previous pseudodifferential calculus to singular foliations.

Abstract

In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a KK-theory element, and prove that the same element can be obtained from an "adiabatic foliation" TF on $M \times \R$, which we introduce here.