Quantitative nonorientability of embedded cycles

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.