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A topological index theorem for manifolds with corners

Part of:Miscellaneous topics - Partial differential equationsInfinite-dimensional manifolds

Published online by Cambridge University Press:  09 November 2011

Bertrand Monthubert and
Victor Nistor
Bertrand Monthubert
Affiliation:
Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse CEDEX 4, France (email: bertrand@monthubert.net)
Victor Nistor
Affiliation:
Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA (email: nistor@math.psu.edu)Inst. of Math. of the Romanian Academy, PO BOX 1-764, 014700 Bucharest, Romania
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Abstract

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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.

Keywords

index theorymanifold with poly-cylindrical endsmanifold with cornersLie groupoid

MSC classification

Secondary:58B34: Noncommutative geometry (à la Connes)35S05: Pseudodifferential operators

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012, pp. 640 - 668
Copyright
Copyright © Foundation Compositio Mathematica 2011

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