  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 CW-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.
