Applying geometric K ‐cycles to fractional indices

A geometric model for twisted K ‐homology is introduced. It is modeled after the Mathai–Melrose–Singer fractional analytic index theorem in the same way as the Baum–Douglas model of K ‐homology was modeled after the Atiyah–Singer index theorem. A natural transformation from twisted geometric K ‐homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted K ‐homology in this model is an isomorphism for torsion twists on a finite C W ‐complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study T ‐duality for geometric cycles.