2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164393We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $α: X \to K(\ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $α$-twisted $Spin^c$-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 $(X, \a)$ 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 $α: X \to K(\ZZ, 3)$, which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer families index theorem to twisted cases.Final version, 51 pages. To appear in Journal of Noncommutative GeometryK-Theory and HomologyAlgebraic Topology19K56, 55N22, 58J22text