Geometric cycles, index theory and twisted K-homology

Understand

We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $\alpha: X \to K(\ZZ, 3)$.

• We introduce the topological index and the analytical index on the bordism group of $\alpha$-twisted $Spin^c$-manifolds over $(X, \alpha)$, 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 $(X, \a)$ analogous to Baum-Douglas's geometric cycles.