Geometric Cycles, Index Theory and Twisted K-homology

We study twisted Spin-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-manifolds over (X,α), 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 analytical index. 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. CONTENTS