Axioms for Differential Cohomology

Abstract

A differential cohomology theory is a type of extension of a cohomology theory E<super>*</super> restricted to smooth manifolds that encodes information that is not homotopy invariant. In particular, it takes values in graded abelian groups, and is equipped with natural transformations to both E<super>*</super> and closed differential forms with values in the graded vector space V = E<super>*</super>(point) &#215; <bold>R</bold>. Differential cohomology theories for certain choices of an underlying cohomolgy theory have been conjectured by Freed to be the proper home for certain types of quantized B-fields in superstring theory. In the case of ordinary integral cohomology, Simons and Sullivan showed they all were naturally isomorphic via a unique isomorphism. Bunke and Schick, under the assumptions that E<super>*</super> is countably generated in each degree and rationally even (i.e., E<super>2k+1</super>(point) &#215; <bold>Q</bold> = 0), arrive at the same result only when they also require the differential cohomology theories each have a degree -1 integration natural transformation that is compatible with the integration along the fiber map for forms and the suspension isomorphism for E<super>*</super>. We also construct such a natural isomorphism, and our only requirement of the cohomology theory is that it be finitely generated in each degree. However, we also require that the differential cohomology theory be defined on a particular type of category that is larger than the historical domain of the category of smooth manifolds with corners.