*Preprint*

**Inserted:** 12 dec 2022

**Last Updated:** 12 dec 2022

**Year:** 2022

**Abstract:**

In this note we extend White's deformation theorem for $G$-flat chains to the setting of $G$-flat \emph{tensor} chains. As a corollary we obtain that the groups of \emph{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~\emph{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.

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