# Reconstructing $d$-manifold subcomplexes of cubes from their $(\lfloor d/2 \rfloor + 1)$-skeletons

## TL;DR

This paper extends Dancis's 1984 result from simplicial to cubical complexes, showing that certain cubical manifolds are uniquely determined by their lower-dimensional skeletons, with tighter bounds under specific conditions.

## Contribution

It adapts Dancis's proof to cubical complexes embedded in cubes, establishing new skeleton-determination results for cubical manifolds, especially spheres.

## Key findings

- Cubical manifolds are determined by their (loor{d/2}+1) -skeletons.
- Under certain conditions, the determination can be improved to the
ceil{d/2} -skeleton.
- The results generalize and tighten previous understanding of manifold reconstruction from 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 $d \geq 3$.