Geometric cycles, index theory and twisted K-homology

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.