K-homology and index theory on contact manifolds

Abstract

This paper applies K-homology to solve the index problem for a class of hypoelliptic (but not elliptic) operators on contact manifolds. K-homology is the dual theory to K-theory. We explicitly calculate the K-cycle (i.e., the element in geometric K-homology) determined by any hypoelliptic Fredholm operator in the Heisenberg calculus. The index theorem of this paper precisely indicates how the analytic versus geometric K-homology setting provides an effective framework for extending formulas of Atiyah–Singer type to non-elliptic Fredholm operators.