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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory