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

TL;DR

This paper refines the Chern-Dold character to a cocycle-level transformation, enabling additive structures in differential cohomology, bridging homotopy theoretic and singular cocycles.

Contribution

It introduces a new cocycle-level refinement of the Chern-Dold character, facilitating additive structures in differential cohomology.

Findings

Constructed a cocycle refinement of the Chern-Dold character.

Enabled addition of singular cocycles at the cocycle level.

Applied to develop additive structures in differential cohomology.

Abstract

The Chern-Dold character of a cohomology theory E is a canonical transformation $E\rightarrow HV$ 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.