Reconstructing d-Manifold Subcomplexes of Cubes from Their (⌊d/2⌋+1)-Skeletons

• Published in: Discrete & computational geometry Vol. 67; no. 2; pp. 492 - 502
• Main Author: Rowlands, Rowan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-03-2022; Springer Nature B.V
• Subjects: Combinatorics; Computational Mathematics and Numerical Analysis; Cubes; Geometry; Manifolds (mathematics); Mathematics; Mathematics and Statistics; Polytopes; Spheres; Manifolds; Poincaré duality; Cubical complexes; 52B05; 57P10; Simplicial complexes; 52B70; 05E45
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-021-00321-4

In 1984, Dancis proved that any
d
-dimensional simplicial manifold is determined by its
(
⌊
d
/
2
⌋
+
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
⌈
d
/
2
⌉
-skeleton when
d
≥
3
.