Topological Aspects of Differential Chains

TL;DR

This paper explores the topological properties of the space of differential chains on Riemannian manifolds, revealing differences from currents and establishing key structural features of these chains.

Contribution

It demonstrates that the space of differential chains is a separable ultrabornological (DF)-space, not reflexive, and provides new definitions and properties distinguishing it from currents.

Findings

Differential chains form a non-reflexive, separable ultrabornological (DF)-space.

Chains supported in finitely many points are dense in the space of differential chains.

The space exhibits properties not shared by current spaces, such as certain density and closure properties.

Abstract

In this paper we investigate the topological properties of the space of differential chains 'B(U) defined on an open subset U of a Riemannian manifold M. We show that '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 'B(U), and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of fundamental operators of the Cartan calculus on differential forms. The space has good properties some of which are not exhibited by currents B'(U) or D'(U). For example, chains supported in finitely many points are dense in 'B(U) for all open U in M, but not generally in the strong dual topology of B'(U).