Topological Aspects of Differential Chains

Abstract

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U)$, and prove that it is a separable ultrabornological (DF)-space.

Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus—the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or $\mathcal{D}'(U)$. For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U⊂M, but not generally in the strong dual topology of $\mathcal{B}'(U)$.