Characterization of differential K-theory by hexagon diagram
Jiahao Hu
TL;DR
The paper proves that differential K-theory is uniquely determined by its character diagram up to a unique natural equivalence, by endowing the diagram with canonical topologies and showing a canonical hexagon for the central group $\hat{K}$. It develops universal classes and a Chern–Simons integration framework, and uses approximation by manifolds to transfer cohomological data to a strictly exact, topologically rigid setting. A key outcome is a rigidity result identifying $K^{-1}_{\mathbf{R}/\mathbf{Z}}$ as the flat theory, complementing existing work and clarifying the role of torsion and identity components within differential K-theory. Overall, the methods adapt Bunke–Schick’s approach to a differential K-theory context, yielding a definitive axiomatisation and robust structural insights for differential refinements of generalized cohomology theories.
Abstract
Using a canonical topology on differential K-theory induced from the Frechét space topology on differential forms and the discrete topology on topological K-theory, we prove that differential K-theory is uniquely determined by the character diagram up to a unique natural equivalence, thus giving an affirmative answer to a question asked by Simons and Sullivan in \cite{SS10}. We further deduce rigidity results including that there is a unique way of realizing $\RR/\ZZ$-K-theory as the flat theory, strengthening the results of \cite{BS10}.