Explicit construction of Atiyah-Singer indices for maximally hypoelliptic operators on contact manifolds

TL;DR

This paper provides an explicit construction of the Atiyah-Singer index for maximally hypoelliptic operators on contact manifolds, extending index theory in non-elliptic settings using K-theory and groupoid methods.

Contribution

It introduces a new explicit construction of index maps for hypoelliptic operators on contact manifolds, building on Connes' tangent groupoid and Higson's symbol class methods.

Findings

Explicit index map construction for hypoelliptic operators

Extension of Atiyah-Singer index theorem to contact manifolds

Utilization of K-theory and groupoid techniques

Abstract

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. Enlightening from Alain Connes' tangent groupoid proof of the index theorem and van Erp's research for the Heisenberg index theory on contact manifolds, we give an explicit construction of a series of maps, whose induced map in K-theory is the Heisenberg Atiyah-Singer index map on contact manifolds. Our methods derive from Higson's construction for symbol class in K-theory.