Cubical complex

In mathematics, a cubical complex or cubical set is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces.

All graphs are (homeomorphic to) 1-dimensional cubical complexes.

Definitions

An elementary interval is a subset ${\displaystyle I\subsetneq \mathbf {R} }$ of the form

${\displaystyle I=[\ell ,\ell +1]\quad {\text{or}}\quad I=[\ell ,\ell ]}$

for some ${\displaystyle \ell \in \mathbf {Z} }$. An elementary cube ${\displaystyle Q}$ is the finite product of elementary intervals, i.e.

${\displaystyle Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}}$

where ${\displaystyle I_{1},I_{2},\ldots ,I_{d}}$ are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube ${\displaystyle [0,1]^{n}}$ embedded in Euclidean space ${\displaystyle \mathbf {R} ^{d}}$ (for some ${\displaystyle n,d\in \mathbf {N} \cup \{0\}}$ with ${\displaystyle n\leq d}$).[1] A set ${\displaystyle X\subseteq \mathbf {R} ^{d}}$ is a cubical complex (or cubical set) if it can be written as a union of elementary cubes (or possibly, is homeomorphic to such a set).[2]

Related terminology

Elementary intervals of length 0 (containing a single point) are called degenerate, while those of length 1 are nondegenerate. The dimension of a cube is the number of nondegenerate intervals in ${\displaystyle Q}$, denoted ${\displaystyle \dim Q}$. The dimension of a cubical complex ${\displaystyle X}$ is the largest dimension of any cube in ${\displaystyle X}$.

If ${\displaystyle Q}$ and ${\displaystyle P}$ are elementary cubes and ${\displaystyle Q\subseteq P}$, then ${\displaystyle Q}$ is a face of ${\displaystyle P}$. If ${\displaystyle Q}$ is a face of ${\displaystyle P}$ and ${\displaystyle Q\neq P}$, then ${\displaystyle Q}$ is a proper face of ${\displaystyle P}$. If ${\displaystyle Q}$ is a face of ${\displaystyle P}$ and ${\displaystyle \dim Q=\dim P-1}$, then ${\displaystyle Q}$ is a primary face of ${\displaystyle P}$.

Algebraic topology

In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the singular homology, but is computable.

See also

• Simplicial complex
• Simplicial homology

References

1. Werman, Michael; Wright, Matthew L. (2016-07-01). "Intrinsic Volumes of Random Cubical Complexes". Discrete & Computational Geometry. 56 (1): 93–113. arXiv:1402.5367. doi:10.1007/s00454-016-9789-z. ISSN 0179-5376.
2. Kaczynski, Tomasz (2004). Computational homology. Mischaikow, Konstantin Michael,, Mrozek, Marian. New York: Springer. ISBN 9780387215976. OCLC 55897585.