Geometric cycles, index theory and twisted K-homology

Abstract

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