A priori bounds for geodesic diameter. Part I. Integral chains with coefficients in a complete normed commutative group

TL;DR

This paper develops a comprehensive framework for locally rectifiable and integral chains in Euclidean space using complete normed commutative groups, providing new algebraic insights and extending to a geometric approach for flat chains.

Contribution

It introduces a novel treatment of chains with coefficients in complete normed groups and explores algebraic and geometric extensions in the theory of rectifiable chains.

Findings

Develops a basic treatment of locally rectifiable chains in Euclidean space.

Introduces the concept of a complete normed commutative group bundle over the Grassmann manifold.

Provides new algebraic insights and extends the theory to a geometric approach for flat chains.

Abstract

As service to the community, we provide - for Euclidean space - a basic treatment of locally rectifiable chains and of the complex of locally integral chains. In this setting, we may beneficially develop the idea of a complete normed commutative group bundle over the Grassmann manifold whose fibre is the coefficient group of the chains. Our exposition also sheds new light on some algebraic aspects of the theory. Finally, we indicate an extension to a geometric approach to locally flat chains centring on locally rectifiable chains rather than completion procedures.