Flat pairing and generalized Cheeger-Simons characters

Let h^{} be a multiplicative cohomology theory, h_{} its dual homology theory and \hat{h}^{} a differential refinement. We first construct the natural pairing between h_{} and the flat part of \hat{h}^{*}, generalizing the holonomy of a flat Deligne cohomology class. Then, in order to generalize the holonomy of any Deligne cohomology class, we define the generalized Cheeger-Simons characters. The latter are functions from suitably defined differential cycles to the cohomology ring of the point, such that the value on a trivial cycle only depends on the curvature.