cubical complex

In mathematics, a cubical complex (also called cubical set and Cartesian complex) 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 $I\subsetneq\mathbf$ of the form

: $I =$, l+1quad\text\quad I=, l/math>

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

: $Q=I_1\times I_2\times \cdots\times I_d\subsetneq \mathbf^d$

where $I_1,I_2,\ldots,I_d$ are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube ,1n embedded in Euclidean space $\mathbf^d$ (for some $n,d\in\mathbf\cup\$ with $n\leq d$). A set $X\subseteq\mathbf^d$ 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 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.

See also

* Simplicial complex
* Simplicial homology
* Abstract cell complex

References

{{Topology

Cubes
Topological spaces
Algebraic topology
Computational topology