Applying geometric K-cycles to fractional indices

TL;DR

This paper introduces a new geometric model for twisted K-homology inspired by fractional index theorems, establishing isomorphisms and surjections in various cases and exploring T-duality for geometric cycles.

Contribution

It develops a novel geometric model for twisted K-homology based on K-cycles, extending the classical Baum-Douglas model to fractional indices and T-duality.

Findings

The model provides an isomorphism for torsion twists on finite CW-complexes.

The model yields a surjection for general twists on smooth manifolds.

T-duality for geometric cycles is analyzed within this framework.

Abstract

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.