Cubical complex

In mathematics, a cubical complex is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analogously to simplicial complexes in the computation of the homology of topological spaces.

Definitions

Cubical sets

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 $l\in \mathbf {Z}$. Elementary intervals of length 0 (containing a single point) are called degenerate, while those of length 1 are nondegenerate.

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. The dimension of a cube is the number of nondegenerate intervals in $Q$, denoted $\operatorname {dim} Q$.

Algebraic topology

Main article: Cubical homology

In algebraic topology, cubical complexes are often useful for concrete calculations. For the definition of homology groups of a cubical complex, one can read the corresponding chain complex directly.

See also

• Simplicial complex
• Simplicial homology