Elliptic operators on manifolds with singularities and K-homology
  It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds.   As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula for K-groups of algebras of pseudodifferential operators. 
A. Savin