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: Ch^k(X,dX) x Ch^{n-k-1}(X) --> S^1, given by (a,b) l-->(a*b)[X], induces isomorphisms: D : Ch^k(X,dX) --> Hom^{smooth}(Ch^{n-k-1}(X), S^1) D': Ch^{n-k-1}(X) --> Hom^{smooth}(Ch^k(X, dX), S^1) 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,dX). The relation of the sequence to the duality mappings is analyzed.