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 ${\displaystyle I\subsetneq \mathbf {R} }$ of the form

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

for some ${\displaystyle l\in \mathbf {Z} }$. 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 ${\displaystyle n\leq d}$).^[2] 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).^[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 ${\displaystyle Q}$, 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 facet or 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
• Abstract cell complex

References

This page was last edited on 8 December 2021, at 03:41