## Abstract

We study twisted Spin^{c}-manifolds over a paracompact Hausdorff space X with a twisting α :X → K(ℤ; 3). We introduce the topological index and the analytical index on the bordism group of α-twisted Spin^{c}-manifolds over .(X; α), taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this article is to establish the equality between the topological index and the analytical index for closed smooth manifolds. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of .(X;α) analogous to Baum-Douglas's geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold (.X,F ) with a twisting α X → K(.ℤ 3), which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer families index theorem to twisted cases.

Original language | English |
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Pages (from-to) | 497-552 |

Number of pages | 56 |

Journal | Journal of Noncommutative Geometry |

Volume | 2 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2008 |