We study twisted Spinc-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 Spinc-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.
Cite this article
Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology. J. Noncommut. Geom. 2 (2008), no. 4, pp. 497–552DOI 10.4171/JNCG/27