Relative differential cohomology

Abstract

We briefly review the classical construction of the Cheeger-Simons characters, the Deligne cohomology groups and the differential K-theory groups, which are representatives of the absolute differential refinement of the corresponding cohomology theories. We present the axiomatic framework for the differential refinement of a generic cohomology theory in the absolute case, together with the important results of existence and uniqueness developed by Bunke and Schick. Motivated by the introduction of the relative Cheeger-Simons characters, we propose a suitable set of axioms for the relative differential extension of a cohomology theory, we construct a family of long exact sequences involving the differential groups and we extend to the relative case the results of existence and uniqueness. Furthermore, we generalize the notion of Cheeger-Simons character to any cohomology theory and we extend to the relative case the construction of the integration map.