Differential characters for K-theory

We describe a sequence of results that begins with the introduction of differential characters on singular cycles in the seventies motivated by the search for invariants of geometry or more generally bundles with connections. The sequence passes through an Eilenberg-Steenrod type uniqueness result for ordinary differential cohomology using these characters and a construction of a differential K-theory using Grothendieck’s construction on classes of complex bundles with connection. The last element of the sequence returns full circle with a differential character definition of differential K-theory. The cycles in this definition of characters for differential K-theory are closed smooth manifolds provided with complex structures and hermitian connections on their stable tangent bundles.