The Atiyah-Singer index formula for subelliptic operators on contact manifolds, Part I

TL;DR

This paper extends the Atiyah-Singer index theorem to hypoelliptic operators on contact manifolds by constructing a noncommutative symbol class and proving the index formula using Connes' tangent groupoid approach.

Contribution

It introduces a K-theory class for hypoelliptic operators on contact manifolds and proves the Atiyah-Singer index formula in this noncommutative setting.

Findings

Constructed a symbol class in the K-theory of a noncommutative C*-algebra.

Established a canonical map to deRham cohomology for applying the index formula.

Proved the index formula holds for hypoelliptic operators on contact manifolds.

Abstract

The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and involves analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C*-algebra of symbols. There is a canonical map from this analytic K-theory group to the deRham cohomology of the manifold, which gives a class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes' tangent groupoid proof of the index theorem.