Lefschetz-Pontrjagin duality for differential characters

A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing ℍ̂^k(X, ∂X) × ℍ̂^n-k-1(X) → S¹ given by (α, β) → (α * β) [X] induces isomorphisms D : ℍ̂^k(X, ∂X) → Hom_∞(ℍ̂^n-k-1(X), S¹) D′ : ℍ̂^n-k-1(X) → Hom_∞(ℍ̂^k(X, ∂X), S¹) onto the smooth Pontrjagin duals. In particular, D and D′ are injective with dense range in the group of all continuous homomorphisms into the circle. A coboundary map is introduced which yields a long sequence for the character groups associated to the pair (X, ∂X). The relation of the sequence to the duality mappings is analyzed.