The de Rham-Federer theory of differential characters and character duality

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, many different spaces of currents are shown to yield the character groups. The fundamental exact sequences are derived, and a multiplication of de Rham-Federer characters is defined using new transversality/intersection results for flat and rectifiable currents. An equivalence of de Rham-Federer characters with the classical Cheeger-Simons characters is given, as in de Rham cohomology, via integration. This discussion rounds out the approach to characters introduced by Gillet-Soulé and Harris. The groups of differential characters have a natural topology and smooth Pontrjagin duals (introduced here) which fit into fundamental exact sequences. A principal result is the formulation and proof of duality for characters on oriented manifolds. It is shown that the pairing (a, b) → a * b([X]) defined 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 degreen - k - 1 where n = dim(X). Thorn homomorphisms, which refine the usual ones in de Rham theory and integral cohmology, are established for differential characters. Gysin maps for characters are similarly established. New examples, namely Morse sparks and Hodge sparks, are introduced and examined in detail.