Cubical complex

In mathematics, a cubical complex (also called cubical set and Cartesian complex^[1]) 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.

Definitions

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

I=[l,l+1]\quad {\text{or}}\quad I=[l,l]

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

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

where $I_{1},I_{2},\ldots ,I_{d}$ are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube $[0,1]^{n}$ embedded in Euclidean space $\mathbf {R} ^{d}$ (for some $n,d\in \mathbf {N} \cup \{0\}$ with $n\leq d$ ).^[2] A set $X\subseteq \mathbf {R} ^{d}$ 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).^[3]

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 $Q$ , denoted $\dim Q$ . The dimension of a cubical complex $X$ is the largest dimension of any cube in $X$ .

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

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.

References

^ Kovalevsky, Vladimir. "Introduction to Digital Topology Lecture Notes". Archived from the original on 2020-02-23. Retrieved November 30, 2021.
^ 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.
^ Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004). Computational Homology. New York: Springer. ISBN 9780387215976. OCLC 55897585.