A transverse index theorem in the calculus of filtered manifolds

TL;DR

This paper develops a new index theorem for filtered manifolds with foliations, introducing transversally Rockland operators and establishing a Poincaré duality linking operators and their symbols.

Contribution

It defines transversally Rockland operators on filtered manifolds and constructs an equivariant KK-class, establishing a Poincaré duality in this setting.

Findings

Transversally Rockland operators yield a K-homology class.

Construction of an equivariant KK-class for transversally Rockland symbols.

Establishment of a Poincaré duality linking operators and their symbols.

Abstract

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols and show a Poincare duality type result linking the class of an operator and its symbol.