Axiomatic Characterization of Ordinary Differential Cohomology

Abstract

Cheeger-Simons differential characters, Deligne cohomology in the smooth category, the Hopkins-Singer construction of ordinary differential cohomology and the recent Harvey-Lawson constructions are each in two distinct ways Abelian group extensions of known functors. In one desciption these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short exact sequences mesh with two interlocking long exact sequences (the Bockstein sequence and a DeRham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Theorem 1.1 shows that on the category of smooth manifolds and smooth maps any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as above is unique up to unique natural equivalence. Theorem 1.2 shows the natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of Theorem 1.1 couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.