Lefschetz-Pontrjagin duality for differential characters

Abstract

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 ℍ̂k(X, ∂X) × ℍ̂n-k-1(X) → S1 given by (α, β) → (α * β) [X] induces isomorphisms D : ℍ̂k(X, ∂X) → Hom∞(ℍ̂n-k-1(X), S1) D′ : ℍ̂n-k-1(X) → Hom∞(ℍ̂k(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.