We construct a model of differential K-theory 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 by using certain differential forms that witness the incompatibility between the even and odd universal Chern forms. By the uniqueness theorem of Bunke and Schick, this model agrees with the spectrum based models in the literature whose nongeometrically defined Chern cocycles are compatible with the delooping maps of the spectrum. These constructions favor geometry over homotopy theory.

The first and second authors were supported in part by grants from The City University of New York PSC-CUNY Research Award Program. The third author was partially supported by the NSF grant DMS-1309099 and would like to thank the Max Planck Institute and the Hausdorff Institute for Mathematics for their support and hospitality during his visits. All three authors gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University, at which some of the research for this paper was performed.