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 (for some with ).[1] A set 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).[2]

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 , 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.

See also

References

  1. Werman, Michael; Wright, Matthew L. (2016-07-01). "Intrinsic Volumes of Random Cubical Complexes". Discrete & Computational Geometry. 56 (1): 93–113. arXiv:1402.5367. doi:10.1007/s00454-016-9789-z. ISSN 0179-5376.
  2. Kaczynski, Tomasz (2004). Computational homology. Mischaikow, Konstantin Michael,, Mrozek, Marian. New York: Springer. ISBN 9780387215976. OCLC 55897585.
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