Reconstructing d-Manifold Subcomplexes of Cubes from Their $$(\\lfloor d/2 \\rfloor + 1)$$-Skeletons

Abstract

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 $d \\geq 3$.