  In 1984, Dancis proved that any $d$-dimensional simplicial manifold is
determined by its $(\lfloor d/2 \rfloor + 1)$-skeleton. This paper adapts his
proof to the setting of cubical complexes that can be embedded into a cube of
arbitrary dimension. Under some additional conditions (for example, if the
cubical manifold is a sphere), the result can be tightened to the $\lceil d/2
\rceil$-skeleton when $d \geq 3$.
