The de Rham-Federer Theory of Differential Characters and Character Duality

In the first part of this paper the theory of differential characters is developed completely from a de Rham Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular connections. As in de Rham Federer cohomology, there are many different spaces of currents which yield the character groups. The fundamental exact sequences in the theory are easily derived from methods of geometric measure theory. A multiplication of de Rham-Federer characters is defined using transversality results for flat and rectifiable currents established in the appendix. It is shown that there is a natural equivalence of ring functors from de Rham Federer characters to the classical Cheeger-Simons characters given, as in de Rham cohomology, via integration. This discussion rounds out the approach to differential character theory introduced by Gillet-Soulé and Harris. The groups of differential characters have an obvious topology and natural smooth Pontrjagin duals (introduced here). It is shown that the dual groups sit in two exact sequences which resemble the fundamental exact sequences for the character groups themselves. They are essentially the smooth duals of the fundamental sequences with roles interchanged. A principal result here is the formulation and proof of duality for characters on oriented manifolds. It is shown that the pairing (a, b) 7→ a ∗ b([X ]) given by multiplication and evaluation on the fundamental cycle, gives an isomorphism of the group of differential characters of degree k with the dual to characters in degree n− k− 1 where n = dim(X). A number of examples are examined in detail. It is also shown that there are natural Thom homomorphisms for differential characters. In fact, given an orthogonal connection on an oriented vector bundle π : E → X , there is a canonical Thom spark γ on E which generates the Thom image as a free module over the characters on X . One has dγ = τ − [X ] where τ is a canonical Thom form on E. Under the two basic differentials in the theory, the Thom map becomes the usual ones in de Rham theory and in integral cohomology. Gysin maps are also defined for differential characters. Many examples including Morse sparks, Hodge sparks and characteristic sparks are examined in detail. Research of all authors was partially supported by the NSF