K-homology and Fredholm Operators II: Elliptic Operators
  This is an expository paper which gives a proof of the Atiyah-Singer index theorem for elliptic operators. Specifcally, we compute the geometric K-cycle that corresponds to the analytic K-cycle determined by the operator. This paper and its companion ("K-homology and index theory II: Dirac Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. Math. 2014. 
Paul Baum, Erik van Erp
math.DG