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.

Definitions

An elementary interval is a subset of the form
for some. An elementary cube is the finite product of elementary intervals, i.e.
where are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube embedded in Euclidean space . A set is a cubical complex if it can be written as a union of elementary cubes.

Related terminology

Elementary intervals of length 0 are called degenerate, while those of length 1 are nondegenerate. The dimension of a cube is the number of nondegenerate intervals in, denoted. The dimension of a cubical complex is the largest dimension of any cube in.
If and are elementary cubes and, then is a face of. If is a face of and, then is a proper face of. If is a face of and, then is a primary face of.

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.