Differential K-theory as equivalence classes of maps to Grassmannians and unitary groups

TL;DR

This paper develops a differential K-theory model using geometric Chern forms, representing classes as maps into Grassmannians and unitary groups, and establishes circle-integration maps compatible with existing spectrum-based models.

Contribution

It introduces a new geometric model of differential K-theory based on equivalence classes of maps into Grassmannians and unitary groups, integrating classical homotopy theory.

Findings

Constructed a differential K-theory model with geometric Chern forms.

Produced circle-integration maps compatible with spectrum models.

Proved the model's equivalence to existing spectrum-based models.

Abstract

We construct a model of differential K-theory, using the geometrically defined Chern forms, whose cocycles are certain equivalence classes of maps into the Grassmannians and unitary groups. In particular, we produce the circle-integration maps for these models using classical homotopy-theoretic constructions, by incorporating certain differential forms which reconcile the incompatibility between these even and odd Chern forms. By the uniqueness theorem of Bunke and Schick, this model agrees with the spectrum-based models in the literature whose abstract Chern cocycles are compatible with the delooping maps on the nose.