### 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) |
---|---|

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

## Fingerprint Dive into the research topics of 'A topological index theorem for manifolds with corners'. Together they form a unique fingerprint.

## Cite this

*Compositio Mathematica*,

*148*(2), 640-668. https://doi.org/10.1112/S0010437X11005458