  We introduce an invariant linked to some foundational questions in geometric
measure theory and provide bounds on this invariant by decomposing an arbitrary
cycle into uniformly rectifiable pieces. Our invariant measures the difficulty
of cutting a nonorientable closed manifold or mod-2 cycle in $\mathbb{R}^n$
into orientable pieces, and we use it to answer some simple but long-open
questions on filling volumes and mod-$\nu$ currents.
