  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.
