Noncommutative topology and the world's simplest index theorem

TL;DR

This paper explores how noncommutative topology methods can be used to prove an explicit index theorem for certain differential operators on 3-manifolds, extending to subelliptic operators on contact manifolds.

Contribution

It introduces a noncommutative approach to hypoelliptic index theory, providing explicit formulas and demonstrating the effectiveness of noncommutative topology in classical analysis.

Findings

Explicit index theorem for scalar second order differential operators on 3-manifolds.

Extension of the index theorem to subelliptic operators on contact manifolds.

Illustration of noncommutative topology's role in proving classical geometric results.

Abstract

This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how this theorem is a special case of a much more general index theorem for subelliptic operators on contact manifolds. Finally we discuss the noncommutative topology that is employed in the proof of this theorem. We present these results as an instance in which noncommutative topology is fruitful in proving a very explicit (analytic/geometric) classical result.