  The simplest case of a manifold with singularities is a manifold M with
boundary, together with an identification of the boundary with a product M1 x
P, where P is a fixed manifold. The associated singular space is obtained by
collapsing P to a point. When P = Z/k or S^1, we show how to attach to such a
space a noncommutative C*-algebra that captures the extra structure. We then
use this C*-algebra to give a new proof of the Freed-Melrose Z/k-index theorem
and a proof of an index theorem for manifolds with S^1 singularities. Our
proofs apply to the real as well as to the complex case. Applications are given
to the study of metrics of positive scalar curvature.
