Reduction of longer chain complexes

Determine whether every length-K chain complex with K>D that is locally embedded in D-dimensional space is equivalent to a length-D chain complex.

Background

The classification established in the paper applies to codes obtained from length-D resolutions matched to a D-dimensional translation group. The authors leave unresolved whether chain complexes longer than the spatial dimension can, when locally embedded into that space, always be reduced up to the relevant equivalence to a length-D chain complex.