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

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

for some $\ell \in \mathbf {Z}$. 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 $n\leq d$). 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).

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 primary face of $P$.

Algebraic topology

Main article: Cubical homology

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