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Abstract simplicial complex

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In <a href="Combinatorics" title="Combinatorics">combinatorics</a>, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a <a href="Family_of_sets" title="Family of sets">family of sets</a> that is closed under taking <a href="Subset" title="Subset">subsets</a>, 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 <a href="Simplicial_complex" title="Simplicial complex">simplicial complex</a>.<a href="#cite_note-Lee-1">[1]</a> For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).

In the context of <a href="Matroid" title="Matroid">matroids</a> and <a href="Greedoid" title="Greedoid">greedoids</a>, abstract simplicial complexes are also called <a href="Independence_system" title="Independence system">independence systems</a>.<a href="#cite_note-2">[2]</a>

An abstract simplex can be studied algebraically by forming its <a href="Stanley%E2%80%93Reisner_ring" title="Stanley–Reisner ring">Stanley–Reisner ring</a>; this sets up a powerful relation between <a href="Combinatorics" title="Combinatorics">combinatorics</a> and <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a>.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

A collection Δ of non-empty finite subsets of a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> S is called a set-family.

A set-family Δ is called an abstract simplicial complex if, for every set X in Δ, and every non-empty subset Y ⊆ X, the set Y also belongs to Δ.

The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊆ X, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex Δ is itself a face of Δ. The vertex set of Δ is defined as V(Δ) = ∪Δ, the union of all faces of Δ. The elements of the vertex set are called the vertices of the complex. For every vertex v of Δ, the set {v} is a face of the complex, and every face of the complex is a finite subset of the vertex set.

The maximal faces of Δ (i.e., faces that are not subsets of any other faces) are called facets of the complex. The dimension of a face X in Δ is defined as dim(X) = |X| − 1: faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The dimension of the complex dim(Δ) is defined as the largest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.

The complex Δ is said to be finite if it has finitely many faces, or equivalently if its vertex set is finite. Also, Δ is said to be pure if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In other words, Δ is pure if dim(Δ) is finite and every face is contained in a facet of dimension dim(Δ).

One-dimensional abstract simplicial complexes are mathematically equivalent to <a href="Simple_graph" class="mw-redirect" title="Simple graph">simple</a> <a href="Undirected_graph" class="mw-redirect" title="Undirected graph">undirected graphs</a>: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges.

A subcomplex of Δ is an abstract simplicial complex L such that every face of L belongs to Δ; that is, L ⊆ Δ and L is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of Δ is often called a simplex of Δ. (However, some authors use the term "simplex" for a face or, rather ambiguously, for both a face and the subcomplex associated with a face, by analogy with the non-abstract (geometric) <a href="Simplicial_complex" title="Simplicial complex">simplicial complex</a> terminology. To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes).

The <a href="D-skeleton" class="mw-redirect" title="D-skeleton">d-skeleton</a> of Δ is the subcomplex of Δ consisting of all of the faces of Δ that have dimension at most d. In particular, the <a href="Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">1-skeleton</a> is called the underlying graph of Δ. The 0-skeleton of Δ can be identified with its vertex set, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the 0-skeleton is a family of single-element sets).

The link of a face Y in Δ, often denoted Δ/Y or lkΔ(Y), is the subcomplex of Δ defined by

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<annotation encoding="application/x-tex">{\displaystyle \Delta /Y:=\{X\in \Delta \mid X\cap Y=\varnothing ,\,X\cup Y\in \Delta \}.}</annotation>

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</math><img src="./7f2cf80da0bfb1a5a453d05c74f7283da60ec8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.058ex; height:2.843ex;" alt="{\displaystyle \Delta /Y:=\{X\in \Delta \mid X\cap Y=\varnothing ,\,X\cup Y\in \Delta \}.}" loading="lazy"></dd></dl>

Note that the link of the empty set is Δ itself.

<div class="mw-heading mw-heading3"><h3 id="Simplicial_maps">Simplicial maps</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Simplicial_map" title="Simplicial map">Simplicial map</a></div>

Given two abstract simplicial complexes, Δ and Γ, a <a href="Simplicial_map" title="Simplicial map">simplicial map</a> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a>  f that maps the vertices of Δ to the vertices of Γ and that has the property that for any face X of Δ, the <a href="Image_(mathematics)" title="Image (mathematics)">image</a>  f (X) is a face of Γ. There is a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> SCpx with abstract simplicial complexes as objects and simplicial maps as <a href="Morphism" title="Morphism">morphisms</a>. This is equivalent to a suitable category defined using non-abstract <a href="Simplicial_complexes" class="mw-redirect" title="Simplicial complexes">simplicial complexes</a>.

Moreover, the categorical point of view allows us to tighten the relation between the underlying set S of an abstract simplicial complex Δ and the vertex set V(Δ) ⊆ S of Δ: for the purposes of defining a category of abstract simplicial complexes, the elements of S not lying in V(Δ) are irrelevant. More precisely, SCpx is equivalent to the category where:

<ul><li>an object is a set S equipped with a collection of non-empty finite subsets Δ that contains all singletons and such that if X is in Δ and Y ⊆ X is non-empty, then Y also belongs to Δ.</li>

<li>a morphism from (S, Δ) to (T, Γ) is a function f : S → T such that the image of any element of Δ is an element of Γ.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Geometric_realization">Geometric realization</h2></div>

We can associate to any abstract simplicial complex (ASC) K a <a href="Topological_space" title="Topological space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy">, called its geometric realization. There are several ways to define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Geometric_definition">Geometric definition</h3></div>

Every <a href="Geometric_simplicial_complex" class="mw-redirect" title="Geometric simplicial complex">geometric simplicial complex</a> (GSC) determines an ASC:<a href="#cite_note-:0-3">[3]</a>: 14 the vertices of the ASC are the vertices of the GSC, and the faces of the ASC are the vertex-sets of the faces of the GSC. For example, consider a GSC with 4 vertices {1,2,3,4}, where the maximal faces are the triangle between {1,2,3} and the lines between {2,4} and {3,4}. Then, the corresponding ASC contains the sets {1,2,3}, {2,4}, {3,4}, and all their subsets. We say that the GSC is the geometric realization of the ASC.

Every ASC has a geometric realization. This is easy to see for a finite ASC.<a href="#cite_note-:0-3">[3]</a>: 14 Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N:=|V(K)|}">

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<annotation encoding="application/x-tex">{\displaystyle N:=|V(K)|}</annotation>

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</math><img src="./1dab4f397e1ebd49c659cdb9da3652c3975bfe16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.765ex; height:2.843ex;" alt="{\displaystyle N:=|V(K)|}" loading="lazy">. Identify the vertices in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(K)}">

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<annotation encoding="application/x-tex">{\displaystyle V(K)}</annotation>

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</math><img src="./4681859a1f67ff73f7dd966ef4a05bc71d63e9f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.662ex; height:2.843ex;" alt="{\displaystyle V(K)}" loading="lazy"> with the vertices of an (N − 1)-dimensional simplex in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{N}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{N}}</annotation>

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</math><img src="./12d5be9beb2f7a56cdee3c6563c9453a913a0c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.37ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{N}}" loading="lazy">. Construct the GSC {<a href="Convex_hull" title="Convex hull">conv</a>(F): F is a face in K}. Clearly, the ASC associated with this GSC is identical to K, so we have indeed constructed a geometric realization of K. In fact, an ASC can be realized using much fewer dimensions. If an ASC is d-dimensional (that is, the maximum cardinality of a simplex in it is d+1), then it has a geometric realization in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2d+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2d+1}}</annotation>

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</math><img src="./7949bb1406374417dc6d47a49225626d50728772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.693ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2d+1}}" loading="lazy">, but might not have a geometric realization in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2d}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2d}}</annotation>

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</math><img src="./f603b63ebb84598a1484e9ae81c6d0994f7ba248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.592ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2d}}" loading="lazy"> <a href="#cite_note-:0-3">[3]</a>: 16 The special case d=1 corresponds to the well-known fact, that any <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be plotted in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> where the edges are straight lines that do not intersect each other except in common vertices, but not any <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be plotted in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

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</math><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"> in this way.

If K is the standard combinatorial n-simplex, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy"> can be naturally identified with Δn.

Every two geometric realizations of the same ASC, even in Euclidean spaces of different dimensions, are <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a>.<a href="#cite_note-:0-3">[3]</a>: 14 Therefore, given an ASC K, one can speak of the geometric realization of K.

<div class="mw-heading mw-heading3"><h3 id="Topological_definition">Topological definition</h3></div>

The construction goes as follows. First, define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy"> as a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{S}}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1]^{S}}</annotation>

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</math><img src="./f034b8ba7ce60e422fe66ec6ec411277793d55e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.945ex; height:3.176ex;" alt="{\displaystyle [0,1]^{S}}" loading="lazy"> consisting of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\colon S\to [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle t\colon S\to [0,1]}</annotation>

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</math><img src="./e8c72ad64f610a407bac29048577141ae0171bd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.639ex; height:2.843ex;" alt="{\displaystyle t\colon S\to [0,1]}" loading="lazy"> satisfying the two conditions:

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<annotation encoding="application/x-tex">{\displaystyle \{s\in S:t_{s}>0\}\in K}</annotation>

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</math><img src="./3b245048efe49aae95de282a346c8eaa315c0e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.703ex; height:2.843ex;" alt="{\displaystyle \{s\in S:t_{s}>0\}\in K}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{s\in S}t_{s}=1}</annotation>

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</math><img src="./58df96fc55fd1395fc74c02d34ea56293d6bf6d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.846ex; height:5.676ex;" alt="{\displaystyle \sum _{s\in S}t_{s}=1}" loading="lazy"></dd></dl>

Now think of the set of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{S}}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1]^{S}}</annotation>

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</math><img src="./f034b8ba7ce60e422fe66ec6ec411277793d55e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.945ex; height:3.176ex;" alt="{\displaystyle [0,1]^{S}}" loading="lazy"> with finite support as the <a href="Direct_limit" title="Direct limit">direct limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{A}}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1]^{A}}</annotation>

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</math><img src="./f56f4c21500c8a6384207a1e492aebc34d9ec6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.117ex; height:3.176ex;" alt="{\displaystyle [0,1]^{A}}" loading="lazy"> where A ranges over finite subsets of S, and give that direct limit the <a href="Final_topology" title="Final topology">induced topology</a>. Now give <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy"> the <a href="Subspace_topology" title="Subspace topology">subspace topology</a>.

<div class="mw-heading mw-heading3"><h3 id="Categorical_definition">Categorical definition</h3></div>

Alternatively, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation>

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</math><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"> denote the category whose objects are the faces of K and whose morphisms are inclusions. Next choose a <a href="Total_order" title="Total order">total order</a> on the vertex set of K and define a <a href="Functor" title="Functor">functor</a> F from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation>

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</math><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"> to the category of topological spaces as follows. For any face X in K of dimension n, let F(X) = Δn be the standard n-simplex. The order on the vertex set then specifies a unique <a href="Bijection" title="Bijection">bijection</a> between the elements of X and vertices of Δn, ordered in the usual way e0 < e1 < ... < en. If Y ⊆ X is a face of dimension m < n, then this bijection specifies a unique m-dimensional face of Δn. Define F(Y) → F(X) to be the unique <a href="Affine_transformation" title="Affine transformation">affine</a> linear <a href="Embedding" title="Embedding">embedding</a> of Δm as that distinguished face of Δn, such that the map on vertices is order-preserving.

We can then define the geometric realization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy"> as the <a href="Colimit" class="mw-redirect" title="Colimit">colimit</a> of the functor F. More specifically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}">

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<annotation encoding="application/x-tex">{\displaystyle |K|}</annotation>

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</math><img src="./029cb0f5f1e5c4b41c20a19ef1520d4542b0260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |K|}" loading="lazy"> is the <a href="Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> of the <a href="Disjoint_union" title="Disjoint union">disjoint union</a>

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<annotation encoding="application/x-tex">{\displaystyle \coprod _{X\in K}{F(X)}}</annotation>

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</math><img src="./30e3909ac24ffe210080e3162270fc9a02150d06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.874ex; height:5.676ex;" alt="{\displaystyle \coprod _{X\in K}{F(X)}}" loading="lazy"></dd></dl>

by the <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> that identifies a point y ∈ F(Y) with its image under the map F(Y) → F(X), for every inclusion Y ⊆ X.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

1. Let V be a finite set of <a href="Cardinality" title="Cardinality">cardinality</a> n + 1. The combinatorial n-simplex with vertex-set V is an ASC whose faces are all nonempty subsets of V (i.e., it is the <a href="Power_set" title="Power set">power set</a> of V). If V = S = {0, 1, ..., n}, then this ASC is called the standard combinatorial n-simplex.

2. Let G be an undirected graph. The <a href="Clique_complex" title="Clique complex">clique complex</a> of G is an ASC whose faces are all <a href="Clique_(graph_theory)" title="Clique (graph theory)">cliques</a> (complete subgraphs) of G. The independence complex of G is an ASC whose faces are all <a href="Independent_set_(graph_theory)" title="Independent set (graph theory)">independent sets</a> of G (it is the clique complex of the <a href="Complement_graph" title="Complement graph">complement graph</a> of G). Clique complexes are the prototypical example of <a href="Flag_complex" class="mw-redirect" title="Flag complex">flag complexes</a>. A flag complex is a complex K with the property that every set, all of whose 2-element subsets are faces of K, is itself a face of K.

3. Let H be a <a href="Hypergraph" title="Hypergraph">hypergraph</a>. A <a href="Matching_in_hypergraphs" title="Matching in hypergraphs">matching</a> in H is a set of edges of H, in which every two edges are <a href="Disjoint_sets" title="Disjoint sets">disjoint</a>. The matching complex of H is an ASC whose faces are all <a href="Matching_in_hypergraphs" title="Matching in hypergraphs">matchings</a> in H. It is the <a href="Independence_complex" title="Independence complex">independence complex</a> of the <a href="Line_graph_of_a_hypergraph" title="Line graph of a hypergraph">line graph</a> of H.

4. Let P be a <a href="Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset). The order complex of P is an ASC whose faces are all finite <a href="Total_order#Chains" title="Total order">chains</a> in P. Its <a href="Homology_(mathematics)" title="Homology (mathematics)">homology</a> groups and other <a href="Topological_property" title="Topological property">topological invariants</a> contain important information about the poset P.

5. Let M be a <a href="Metric_space" title="Metric space">metric space</a> and δ a real number. The <a href="Vietoris%E2%80%93Rips_complex" title="Vietoris–Rips complex">Vietoris–Rips complex</a> is an ASC whose faces are the finite subsets of M with diameter at most δ. It has applications in <a href="Homology_theory" class="mw-redirect" title="Homology theory">homology theory</a>, <a href="Hyperbolic_group" title="Hyperbolic group">hyperbolic groups</a>, <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a>, and <a href="Mobile_ad_hoc_network" class="mw-redirect" title="Mobile ad hoc network">mobile ad hoc networking</a>. It is another example of a flag complex.

6. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> be a square-free <a href="Monomial_ideal" title="Monomial ideal">monomial ideal</a> in a <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=K[x_{1},\dots ,x_{n}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=K[x_{1},\dots ,x_{n}]}</annotation>

</semantics>

</math><img src="./cafcb2abee81375c819a9b39da5621729975778f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.068ex; height:2.843ex;" alt="{\displaystyle S=K[x_{1},\dots ,x_{n}]}" loading="lazy"> (that is, an ideal generated by products of subsets of variables). Then the exponent vectors of those square-free monomials of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> that are not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> determine an abstract simplicial complex via the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \in \{0,1\}^{n}\mapsto \{i\in [n]:a_{i}=1\}}">

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \in \{0,1\}^{n}\mapsto \{i\in [n]:a_{i}=1\}}</annotation>

</semantics>

</math><img src="./26112483a09c7764f992cfd056afa64b1d771e82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.54ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \in \{0,1\}^{n}\mapsto \{i\in [n]:a_{i}=1\}}" loading="lazy">. In fact, there is a bijection between (non-empty) abstract simplicial complexes on n vertices and square-free monomial ideals in S. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\Delta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\Delta }}</annotation>

</semantics>

</math><img src="./ffc42fd837fec7b703ea5052b5b4af75b959fc27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.624ex; height:2.509ex;" alt="{\displaystyle I_{\Delta }}" loading="lazy"> is the square-free ideal corresponding to the simplicial complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> then the <a href="Quotient_ring" title="Quotient ring">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/I_{\Delta }}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S/I_{\Delta }}</annotation>

</semantics>

</math><img src="./f7dcc6addaccbd635b6d2c8b3af4950347c68864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.286ex; height:2.843ex;" alt="{\displaystyle S/I_{\Delta }}" loading="lazy"> is known as the <a href="Stanley%E2%80%93Reisner_ring" title="Stanley–Reisner ring">Stanley–Reisner ring</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Delta }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Delta }}</annotation>

</semantics>

</math><img src="./a650ae128f4d515b1448e592d71392d25d53ae7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle {\Delta }}" loading="lazy">.

7. For any <a href="Open_covering" class="mw-redirect" title="Open covering">open covering</a> C of a topological space, the <a href="Nerve_complex" title="Nerve complex">nerve complex</a> of C is an abstract simplicial complex containing the sub-families of C with a non-empty <a href="Intersection" title="Intersection">intersection</a>.

<div class="mw-heading mw-heading2"><h2 id="Enumeration">Enumeration</h2></div>

The number of abstract simplicial complexes on up to n labeled elements (that is on a set S of size n) is one less than the nth <a href="Dedekind_number" title="Dedekind number">Dedekind number</a>. These numbers grow very rapidly, and are known only for n ≤ 9; the Dedekind numbers are (starting with n = 0):

<dl><dd>1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365 (sequence <a href="https://oeis.org/A014466" class="extiw external" title="oeis:A014466">A014466</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). This corresponds to the number of non-empty <a href="Antichain" title="Antichain">antichains</a> of subsets of an n set.</dd></dl>

The number of abstract simplicial complexes whose vertices are exactly n labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993" (sequence <a href="https://oeis.org/A006126" class="extiw external" title="oeis:A006126">A006126</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), starting at n = 1. This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichain covers of an n-set and simplicial complexes on n elements described in terms of their maximal faces.

The number of abstract simplicial complexes on exactly n unlabeled elements is given by the sequence "1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210" (sequence <a href="https://oeis.org/A006602" class="extiw external" title="oeis:A006602">A006602</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), starting at n = 1.

<div class="mw-heading mw-heading2"><h2 id="Computational_problems">Computational problems</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Simplicial_complex_recognition_problem" title="Simplicial complex recognition problem">Simplicial complex recognition problem</a></div>

The <a href="Simplicial_complex_recognition_problem" title="Simplicial complex recognition problem">simplicial complex recognition problem</a> is: given a finite ASC, decide whether its geometric realization is homeomorphic to a given geometric object. This problem is <a href="Undecidable_problem" title="Undecidable problem">undecidable</a> for any d-dimensional manifolds for d ≥ 5.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_concepts">Relation to other concepts</h2></div>

An abstract simplicial complex with an additional property called the augmentation property or the exchange property yields a <a href="Matroid" title="Matroid">matroid</a>. The following expression shows the relations between the terms:

HYPERGRAPHS = SET-FAMILIES ⊃ INDEPENDENCE-SYSTEMS = ABSTRACT-SIMPLICIAL-COMPLEXES ⊃ MATROIDS.

<ul><li><a href="Kruskal%E2%80%93Katona_theorem" title="Kruskal–Katona theorem">Kruskal–Katona theorem</a></li>

<li><a href="Simplicial_set" title="Simplicial set">Simplicial set</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Lee-1"><a href="#cite_ref-Lee_1-0">^</a> <a href="John_M._Lee" title="John M. Lee">Lee, John M.</a>, Introduction to Topological Manifolds, Springer 2011, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4419-7939-5</bdi>, p153

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFKorte,_BernhardLovász,_LászlóSchrader,_Rainer1991" class="citation book cs1"><a href="Bernhard_Korte" title="Bernhard Korte">Korte, Bernhard</a>; <a href="L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">Lovász, László</a>; Schrader, Rainer (1991). Greedoids. Springer-Verlag. p. 9. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-18190-3</bdi>.</cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <a href="#cite_ref-:0_3-2">c</a> <a href="#cite_ref-:0_3-3">d</a> <cite id="CITEREFMatoušek2007" class="citation book cs1"><a href="Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Matoušek, Jiří</a> (2007). <a href="Using_the_Borsuk-Ulam_Theorem" class="mw-redirect" title="Using the Borsuk-Ulam Theorem">Using the Borsuk-Ulam Theorem</a>: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-00362-5</bdi>. <q>Written in cooperation with <a href="Anders_Bj%C3%B6rner" title="Anders Björner">Anders Björner</a> and <a href="G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Günter M. Ziegler</a></q></cite>

, Section 4.3

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFStillwell1993" class="citation cs2"><a href="John_Stillwell" title="John Stillwell">Stillwell, John</a> (1993), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=265lbM42REMC&pg=PA247">Classical Topology and Combinatorial Group Theory</a>, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780387979700</bdi></cite>.

</li>

</ol></div></div></div><div><div class="zim-footer">

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