## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 640-668 |

Number of pages | 29 |

Journal | Compositio Mathematica |

Volume | 148 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2012 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory