Geometric cycles, index theory and twisted K-homology

Abstract

We study twisted ${\mathrm{Spin}}^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $\alpha :X\to K(Z,3)$. We introduce the topological index and the analytical index on the bordism group of $\alpha$-twisted ${\mathrm{Spin}}^c$-manifolds over $(X,\alpha )$, 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 $(X,\alpha )$ 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 $\alpha :X\to K(Z,3)$, which generalizes the Connes–Skandalis index theorem for foliations and the Atiyah–Singer families index theorem to twisted cases.