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We construct a model of differential K-theory, using the geometricallydefined Chern forms, whose cocycles are certain equivalence classes of mapsinto the Grassmannians and unitary groups. In particular, we produce thecircle-integration maps for these models using classical homotopy-theoreticconstructions, by incorporating certain differential forms which reconcile theincompatibility between these even and odd Chern forms. By the uniquenesstheorem of Bunke and Schick, this model agrees with the spectrum-based modelsin the literature whose abstract Chern cocycles are compatible with thedelooping maps on the nose.

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