Cubical complex

Last updated December 01, 2021

In mathematics, a cubical complex (also called cubical set and Cartesian complex^[1]) 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 $I\subsetneq \mathbf {R}$ of the form

I=[l,l+1]\quad {\text{or}}\quad I=[l,l]

for some $l\in \mathbf {Z}$ . 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. Equivalently, an elementary cube is any translate of a unit cube $[0,1]^{n}$ embedded in Euclidean space $\mathbf {R} ^{d}$ (for some $n,d\in \mathbf {N} \cup \{0\}$ with $n\leq d$ ).^[2] A set $X\subseteq \mathbf {R} ^{d}$ 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).^[3]

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.

Related Research Articles

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Simplex Multi-dimensional generalization of triangle

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In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

A CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The C stands for "closure-finite", and the W for "weak" topology. A CW complex can be defined inductively.

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In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space $to itself by means of traces of the induced mappings on the homology groups of . It is named after Solomon Lefschetz, who first stated it in 1926.$

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In combinatorics, an abstract simplicial complex (ASC) is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles, their edges, and their vertices.

In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components.

In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).

In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito.

In mathematics the cotangent complex is roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M. André independently came up with a definition that works in all cases.

In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.

In mathematics, a Δ-setS, often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.

Clique complexes, flag complexes, and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques of an undirected graph.

In topology, a branch of mathematics, a collapse reduces a simplicial complex to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were invented by J. H. C. Whitehead. Collapses find applications in computational homology.

In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.

Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters.

This is a glossary of properties and concepts in algebraic topology in mathematics.

Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.

References

↑ Kovalevsky, Vladimir. "Introduction to Digital Topology Lecture Notes" . Retrieved November 30, 2021.
↑ 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.
↑ Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004). Computational Homology. New York: Springer. ISBN 9780387215976. OCLC 55897585.

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[1] Kovalevsky, Vladimir. "Introduction to Digital Topology Lecture Notes" . Retrieved November 30, 2021.

[2] 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.

[3] Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004). Computational Homology. New York: Springer. ISBN 9780387215976. OCLC 55897585.

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v t e Topology
Fields	General (point-set) Algebraic Combinatorial Continuum Differential Geometric low-dimensional Homology cohomology Set-theoretic
Key concepts	Open set / Closed set Continuity Space compact Connected Hausdorff metric uniform Homotopy homotopy group fundamental group Simplicial complex CW complex Manifold Second-countable space
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