A topological index theorem for manifolds with corners

We define an analytic index and prove a topological index theorem for a non-compact manifold M ₀ with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K ₀(C ^*(M)), where C ^*(M) is a canonical C ^*-algebra associated to the canonical compactification M of M ₀. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah-Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K-theory groups K ₀(C ^*(M)) of the groupoid C ^*-algebra of the manifolds with corners M. We also prove that an elliptic operator P on M ₀ has an invertible perturbation P+R by a lower-order operator if and only if its analytic index vanishes.