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 [math]\displaystyle{ I\subsetneq\mathbf{R} }[/math] of the form
- [math]\displaystyle{ I = [l, l+1]\quad\text{or}\quad I=[l, l] }[/math]
for some [math]\displaystyle{ l\in\mathbf{Z} }[/math]. An elementary cube [math]\displaystyle{ Q }[/math] is the finite product of elementary intervals, i.e.
- [math]\displaystyle{ Q=I_1\times I_2\times \cdots\times I_d\subsetneq \mathbf{R}^d }[/math]
where [math]\displaystyle{ I_1,I_2,\ldots,I_d }[/math] are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube [math]\displaystyle{ [0,1]^n }[/math] embedded in Euclidean space [math]\displaystyle{ \mathbf{R}^d }[/math] (for some [math]\displaystyle{ n,d\in\mathbf{N}\cup\{0\} }[/math] with [math]\displaystyle{ n\leq d }[/math]).[2] A set [math]\displaystyle{ X\subseteq\mathbf{R}^d }[/math] 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 [math]\displaystyle{ Q }[/math], denoted [math]\displaystyle{ \dim Q }[/math]. The dimension of a cubical complex [math]\displaystyle{ X }[/math] is the largest dimension of any cube in [math]\displaystyle{ X }[/math].
If [math]\displaystyle{ Q }[/math] and [math]\displaystyle{ P }[/math] are elementary cubes and [math]\displaystyle{ Q\subseteq P }[/math], then [math]\displaystyle{ Q }[/math] is a face of [math]\displaystyle{ P }[/math]. If [math]\displaystyle{ Q }[/math] is a face of [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q\neq P }[/math], then [math]\displaystyle{ Q }[/math] is a proper face of [math]\displaystyle{ P }[/math]. If [math]\displaystyle{ Q }[/math] is a face of [math]\displaystyle{ P }[/math] and [math]\displaystyle{ \dim Q=\dim P-1 }[/math], then [math]\displaystyle{ Q }[/math] is a facet or primary face of [math]\displaystyle{ P }[/math].
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
References
- ↑ Kovalevsky, Vladimir. "Introduction to Digital Topology Lecture Notes". http://www.kovalevsky.de/Topology/Introduction_e.htm#a6.
- ↑ Werman, Michael; Wright, Matthew L. (2016-07-01). "Intrinsic Volumes of Random Cubical Complexes" (in en). Discrete & Computational Geometry 56 (1): 93–113. 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.
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