Noncommutative topology and the world's simplest index theorem

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.