A topological index theorem for manifolds with corners

E-ISSN： 1570-5846|148|2|640-668

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.148, Iss.2, 2012-03, pp. : 640-668

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Abstract

We define an analytic index and prove a topological index theorem for a non-compact manifold M₀ 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 K₀(C^*(M)), where C^*(M) is a canonical C^*-algebra associated to the canonical compactification M of M₀. 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 K₀(C^*(M)) of the groupoid C^*-algebra of the manifolds with corners M. We also prove that an elliptic operator P on M₀ has an invertible perturbation P+R by a lower-order operator if and only if its analytic index vanishes.