A deformation theorem for tensor flat chains and applications (complement to ''Tensor rectifiable G-flat chains'')

TL;DR

This paper extends White's deformation theorem to G-flat tensor chains, establishing isometric isomorphisms between groups of normal tensor chains and subgroups of normal chains, with implications for mass equivalence and chain identification.

Contribution

It introduces an extension of White's deformation theorem to tensor chains and explores the isometric group isomorphisms related to coordinate slicing mass.

Findings

Normal tensor chains are identified with subgroups of normal chains.

The coordinate slicing mass is equivalent to the usual mass, with a proof using the deformation theorem.

Groups of tensor chains do not generally identify with subgroups of chains outside certain cases.

Abstract

In this note we extend White's deformation theorem for G-flat chains to the setting of G-flat tensor chains. As a corollary we obtain that the groups of normal tensor chains identify with some subgroups of normal chains. Moreover the corresponding natural group isomorphisms are isometric with respect to norms based on the coordinate slicing mass. The coordinate slicing mass of a k-chain is the integral of the mass of its 0-slices along all coordinate-planes of codimension k. The fact that this quantity is equivalent to the usual mass is not straightforward. To prove it, we use the deformation theorem and a partial extension of the restriction operator defined for all chains (not only of finite mass). On the contrary, except in some limit or degenerate cases, the whole groups of tensor chains and of finite mass tensor chains do not identify naturally with subgroups of chains.