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

The Atiyah-Singer index theorem gives 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 naturally arising geometric operators are not elliptic, but subelliptic. A filtration on the algebra of differential operators that is adapted to these geometric structures, naturally leads to a symbolic calculus that is noncommutative, and a corresponding subelliptic theory can be developed.; For such subelliptic operators we construct a symbol class in the K-theory of a noncommutative C*-algebra naturally associated to the algebra of symbols. There is a canonical map from this noncommutative K-theory to the ordinary cohomology of the manifold, which gives a class to which the Atiyah-Singer formula can be applied. In this way we define the topological index of a subelliptic operator, and we prove that it is equal to its analytic index.