cubicalkomplekseihin

Cubical complexes are a class of spaces studied in algebraic topology and computational topology. They consist of cubes of various dimensions that are glued together along their faces in a way that each face of a cube is either a subcell or identified with a face of another cube. Each cell is homeomorphic to a standard unit cube [0,1]^n, and the attaching maps identify faces via isometries. A cubical complex is finite if it has finitely many cubes; in general it may be countable or infinite. The dimension of a cubical complex is the largest dimension of its cubes.

Cubical complexes support a natural cellular structure and a corresponding chain complex. There are boundary operators that map n-cubes to (n−1)-cubes, yielding cubical homology. Subcomplexes can be formed by selecting cubes contained in a given subspace, and products of cubical complexes correspond to Cartesian products of the underlying spaces. Subdivision operations allow refining the complex by partitioning cubes into smaller cubes, preserving the cubical nature.

Typical examples include the standard n-cube, whose boundary is a cubical complex homeomorphic to the (n−1)-sphere, and more elaborate constructions such as the n-torus obtained by identifying opposite faces of an n-cube. Cubical complexes are also used to model digital images: grids of pixels (2D) or voxels (3D) form natural cubical complexes, enabling efficient computation of topological features.

Relation to other frameworks is a key feature. Compared with simplicial complexes, cubical complexes use axis-aligned cubes, which can simplify certain computations and make products and grid-based representations more natural. They are central in cubical homology and cubical persistent homology, and they interface with the broader theory of CW complexes and cubical sets.