Refinements of the Chern-Dold Character: Cocycle Additions in Differential Cohomology

Abstract

The Chern-Dold character of a cohomology theory E is a canonical transformation $E\to HV$ to ordinary cohomology. A spectrum representing E gives homotopy theoretic cocycles for E, while HV can be represented by singular cocycles. We construct a refinement of the Chern-Dold character to a transformation of the cocycle categories that takes the homotopical composition to the addition of singular cocycles. This is applied to construct additive structures at the level of differential cocycles for generalized differential cohomology.