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decoding="async" width="200" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/300px-Simplicial_complex_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/400px-Simplicial_complex_example.svg.png 2x" data-file-width="532" data-file-height="517" /></a><figcaption>Geometric realization of a 3-dimensional abstract simplicial complex</figcaption></figure> <p>In <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, an <b>abstract simplicial complex</b> (ASC), often called an <b>abstract complex</b> or just a <b>complex</b>, is a <a href="/wiki/Family_of_sets" title="Family of sets">family of sets</a> that is closed under taking <a href="/wiki/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="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a>.<sup id="cite_ref-Lee_1-0" class="reference"><a href="#cite_note-Lee-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 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). </p><p>In the context of <a href="/wiki/Matroid" title="Matroid">matroids</a> and <a href="/wiki/Greedoid" title="Greedoid">greedoids</a>, abstract simplicial complexes are also called <b><a href="/wiki/Independence_system" title="Independence system">independence systems</a></b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>An abstract simplex can be studied algebraically by forming its <a href="/wiki/Stanley%E2%80%93Reisner_ring" title="Stanley–Reisner ring">Stanley–Reisner ring</a>; this sets up a powerful relation between <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A collection <span class="texhtml">Δ</span> of non-empty finite subsets of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>S</i> is called a set-family. </p><p>A set-family <span class="texhtml">Δ</span> is called an <b>abstract simplicial complex</b> if, for every set <span class="texhtml mvar" style="font-style:italic;">X</span> in <span class="texhtml">Δ</span>, and every non-empty subset <span class="texhtml"><i>Y</i> ⊆ <i>X</i></span>, the set <span class="texhtml mvar" style="font-style:italic;">Y</span> also belongs to <span class="texhtml">Δ</span>. </p><p>The finite sets that belong to <span class="texhtml">Δ</span> are called <b>faces</b> of the complex, and a face <span class="texhtml mvar" style="font-style:italic;">Y</span> is said to belong to another face <span class="texhtml mvar" style="font-style:italic;">X</span> if <span class="texhtml"><i>Y</i> ⊆ <i>X</i></span>, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex <span class="texhtml">Δ</span> is itself a face of <span class="texhtml">Δ</span>. The <b>vertex set</b> of <span class="texhtml">Δ</span> is defined as <span class="texhtml"><i>V</i>(Δ) = ∪Δ</span>, the union of all faces of <span class="texhtml">Δ</span>. The elements of the vertex set are called the <b>vertices</b> of the complex. For every vertex <i>v</i> of <span class="texhtml">Δ</span>, the set {<i>v</i>} is a face of the complex, and every face of the complex is a finite subset of the vertex set. </p><p>The maximal faces of <span class="texhtml">Δ</span> (i.e., faces that are not subsets of any other faces) are called <b>facets</b> of the complex. The <b>dimension of a face</b> <span class="texhtml mvar" style="font-style:italic;">X</span> in <span class="texhtml">Δ</span> is defined as <span class="texhtml">dim(<i>X</i>) = |<i>X</i>| − 1</span>: faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The <b>dimension of the complex</b> <span class="texhtml">dim(Δ)</span> 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. </p><p>The complex <span class="texhtml">Δ</span> is said to be <b>finite</b> if it has finitely many faces, or equivalently if its vertex set is finite. Also, <span class="texhtml">Δ</span> is said to be <b>pure</b> if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In other words, <span class="texhtml">Δ</span> is pure if <span class="texhtml">dim(Δ)</span> is finite and every face is contained in a facet of dimension <span class="texhtml">dim(Δ)</span>. </p><p>One-dimensional abstract simplicial complexes are mathematically equivalent to <a href="/wiki/Simple_graph" class="mw-redirect" title="Simple graph">simple</a> <a href="/wiki/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. </p><p>A <b>subcomplex</b> of <span class="texhtml">Δ</span> is an abstract simplicial complex <i>L</i> such that every face of <i>L</i> belongs to <span class="texhtml">Δ</span>; that is, <span class="texhtml"><i>L</i> ⊆ Δ</span> and <i>L</i> is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of <span class="texhtml">Δ</span> is often called a <b>simplex</b> of <span class="texhtml">Δ</span>. (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="/wiki/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). </p><p>The <b><a href="/wiki/D-skeleton" class="mw-redirect" title="D-skeleton"><i>d</i>-skeleton</a></b> of <span class="texhtml">Δ</span> is the subcomplex of <span class="texhtml">Δ</span> consisting of all of the faces of <span class="texhtml">Δ</span> that have dimension at most <i>d</i>. In particular, the <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">1-skeleton</a> is called the <b>underlying graph</b> of <span class="texhtml">Δ</span>. The 0-skeleton of <span class="texhtml">Δ</span> 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). </p><p>The <b>link</b> of a face <span class="texhtml mvar" style="font-style:italic;">Y</span> in <span class="texhtml">Δ</span>, often denoted <span class="texhtml">Δ/<i>Y</i></span> or <span class="texhtml">lk<sub>Δ</sub>(<i>Y</i>)</span>, is the subcomplex of <span class="texhtml">Δ</span> defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta /Y:=\{X\in \Delta \mid X\cap Y=\varnothing ,\,X\cup Y\in \Delta \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Y</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>∈</mo> <mi mathvariant="normal">Δ</mi> <mo>∣</mo> <mi>X</mi> <mo>∩</mo> <mi>Y</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>X</mi> <mo>∪</mo> <mi>Y</mi> <mo>∈</mo> <mi mathvariant="normal">Δ</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta /Y:=\{X\in \Delta \mid X\cap Y=\varnothing ,\,X\cup Y\in \Delta \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2cf80da0bfb1a5a453d05c74f7283da60ec8f2" 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 \}.}" /></span></dd></dl> <p>Note that the link of the empty set is <span class="texhtml">Δ</span> itself. </p> <div class="mw-heading mw-heading3"><h3 id="Simplicial_maps">Simplicial maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=2" title="Edit section: Simplicial maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Simplicial_map" title="Simplicial map">Simplicial map</a></div> <p>Given two abstract simplicial complexes, <span class="texhtml">Δ</span> and <span class="texhtml">Γ</span>, a <b><a href="/wiki/Simplicial_map" title="Simplicial map">simplicial map</a></b> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"> <i>f</i> </span> that maps the vertices of <span class="texhtml">Δ</span> to the vertices of <span class="texhtml">Γ</span> and that has the property that for any face <span class="texhtml mvar" style="font-style:italic;">X</span> of <span class="texhtml">Δ</span>, the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> <span class="texhtml"> <i>f</i> (<i>X</i>)</span> is a face of <span class="texhtml">Γ</span>. There is a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> <b>SCpx</b> with abstract simplicial complexes as objects and simplicial maps as <a href="/wiki/Morphism" title="Morphism">morphisms</a>. This is equivalent to a suitable category defined using non-abstract <a href="/wiki/Simplicial_complexes" class="mw-redirect" title="Simplicial complexes">simplicial complexes</a>. </p><p>Moreover, the categorical point of view allows us to tighten the relation between the underlying set <i>S</i> of an abstract simplicial complex <span class="texhtml">Δ</span> and the vertex set <span class="texhtml"><i>V</i>(Δ) ⊆ <i>S</i></span> of <span class="texhtml">Δ</span>: for the purposes of defining a category of abstract simplicial complexes, the elements of <i>S</i> not lying in <span class="texhtml"><i>V</i>(Δ)</span> are irrelevant. More precisely, <b>SCpx</b> is equivalent to the category where: </p> <ul><li>an object is a set <i>S</i> equipped with a collection of non-empty finite subsets <span class="texhtml">Δ</span> that contains all singletons and such that if <span class="texhtml mvar" style="font-style:italic;">X</span> is in <span class="texhtml">Δ</span> and <span class="texhtml"><i>Y</i> ⊆ <i>X</i></span> is non-empty, then <span class="texhtml mvar" style="font-style:italic;">Y</span> also belongs to <span class="texhtml">Δ</span>.</li> <li>a morphism from <span class="texhtml">(<i>S</i>, Δ)</span> to <span class="texhtml">(<i>T</i>, Γ)</span> is a function <span class="texhtml"><i>f</i> : <i>S</i> → <i>T</i></span> such that the image of any element of <span class="texhtml">Δ</span> is an element of <span class="texhtml">Γ</span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Geometric_realization">Geometric realization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=3" title="Edit section: Geometric realization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can associate to any abstract simplicial complex (ASC) <i>K</i> a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span>, called its <b>geometric realization</b>. There are several ways to define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_definition">Geometric definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=4" title="Edit section: Geometric definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every <a href="/wiki/Geometric_simplicial_complex" class="mw-redirect" title="Geometric simplicial complex">geometric simplicial complex</a> (GSC) determines an ASC:<i><sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></i><sup class="reference nowrap"><span title="Page: 14">: 14 </span></sup> 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 <b>geometric realization</b> of the ASC. </p><p>Every ASC has a geometric realization. This is easy to see for a finite ASC.<i><sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></i><sup class="reference nowrap"><span title="Page: 14">: 14 </span></sup> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N:=|V(K)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N:=|V(K)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dab4f397e1ebd49c659cdb9da3652c3975bfe16" 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)|}" /></span>. Identify the vertices in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4681859a1f67ff73f7dd966ef4a05bc71d63e9f1" 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)}" /></span> with the vertices of an (<i>N-1</i>)-dimensional simplex in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d5be9beb2f7a56cdee3c6563c9453a913a0c92" 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}}" /></span>. Construct the GSC {<a href="/wiki/Convex_hull" title="Convex hull">conv</a>(F): F is a face in K}. Clearly, the ASC associated with this GSC is identical to <i>K</i>, so we have indeed constructed a geometric realization of <i>K.</i> In fact, an ASC can be realized using much fewer dimensions. If an ASC is <i>d</i>-dimensional (that is, the maximum cardinality of a simplex in it is <i>d</i>+1), then it has a geometric realization in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2d+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2d+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7949bb1406374417dc6d47a49225626d50728772" 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}}" /></span>, but might not have a geometric realization in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f603b63ebb84598a1484e9ae81c6d0994f7ba248" 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}}" /></span> <i><sup id="cite_ref-:0_3-2" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 16">: 16 </span></sup></i> The special case <i>d</i>=1 corresponds to the well-known fact, that any <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be plotted in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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}}" /></span> where the edges are straight lines that do not intersect each other except in common vertices, but not any <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be plotted in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" 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}}" /></span> in this way. </p><p>If <i>K</i> is the standard combinatorial <i>n</i>-simplex, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span> can be naturally identified with <span class="texhtml">Δ<sup><i>n</i></sup></span>. </p><p>Every two geometric realizations of the same ASC, even in Euclidean spaces of different dimensions, are <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a>.<i><sup id="cite_ref-:0_3-3" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></i><sup class="reference nowrap"><span title="Page: 14">: 14 </span></sup> Therefore, given an ASC <i>K,</i> one can speak of <i>the</i> geometric realization of <i>K</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Topological_definition">Topological definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=5" title="Edit section: Topological definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The construction goes as follows. First, define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span> as a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]^{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f034b8ba7ce60e422fe66ec6ec411277793d55e4" 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}}" /></span> consisting of functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\colon S\to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\colon S\to [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c72ad64f610a407bac29048577141ae0171bd5" 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]}" /></span> satisfying the two conditions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s\in S:t_{s}>0\}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{s\in S:t_{s}>0\}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b245048efe49aae95de282a346c8eaa315c0e46" 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}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{s\in S}t_{s}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{s\in S}t_{s}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58df96fc55fd1395fc74c02d34ea56293d6bf6d3" 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}" /></span></dd></dl> <p>Now think of the set of elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]^{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f034b8ba7ce60e422fe66ec6ec411277793d55e4" 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}}" /></span> with finite support as the <a href="/wiki/Direct_limit" title="Direct limit">direct limit</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56f4c21500c8a6384207a1e492aebc34d9ec6d1" 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}}" /></span> where <i>A</i> ranges over finite subsets of <i>S</i>, and give that direct limit the <a href="/wiki/Final_topology" title="Final topology">induced topology</a>. Now give <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span> the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Categorical_definition">Categorical definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=6" title="Edit section: Categorical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternatively, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">K</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2" 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}}}" /></span> denote the category whose objects are the faces of <span class="texhtml mvar" style="font-style:italic;">K</span> and whose morphisms are inclusions. Next choose a <a href="/wiki/Total_order" title="Total order">total order</a> on the vertex set of <span class="texhtml mvar" style="font-style:italic;">K</span> and define a <a href="/wiki/Functor" title="Functor">functor</a> <i>F</i> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">K</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2" 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}}}" /></span> to the category of topological spaces as follows. For any face <i>X</i> in <i>K</i> of dimension <i>n</i>, let <span class="texhtml"><i>F</i>(<i>X</i>) = Δ<sup><i>n</i></sup></span> be the standard <i>n</i>-simplex. The order on the vertex set then specifies a unique <a href="/wiki/Bijection" title="Bijection">bijection</a> between the elements of <span class="texhtml mvar" style="font-style:italic;">X</span> and vertices of <span class="texhtml">Δ<sup><i>n</i></sup></span>, ordered in the usual way <span class="texhtml"><i>e</i><sub>0</sub> < <i>e</i><sub>1</sub> < ... < <i>e<sub>n</sub></i></span>. If <span class="texhtml"><i>Y</i> ⊆ <i>X</i></span> is a face of dimension <span class="texhtml"><i>m</i> < <i>n</i></span>, then this bijection specifies a unique <i>m</i>-dimensional face of <span class="texhtml">Δ<sup><i>n</i></sup></span>. Define <span class="texhtml"><i>F</i>(<i>Y</i>) → <i>F</i>(<i>X</i>)</span> to be the unique <a href="/wiki/Affine_transformation" title="Affine transformation">affine</a> linear <a href="/wiki/Embedding" title="Embedding">embedding</a> of <span class="texhtml">Δ<sup><i>m</i></sup></span> as that distinguished face of <span class="texhtml">Δ<sup><i>n</i></sup></span>, such that the map on vertices is order-preserving. </p><p>We can then define the geometric realization <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span> as the <a href="/wiki/Colimit" class="mw-redirect" title="Colimit">colimit</a> of the functor <i>F</i>. More specifically <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |K|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |K|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cb0f5f1e5c4b41c20a19ef1520d4542b0260b" 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|}" /></span> is the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> of the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \coprod _{X\in K}{F(X)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∐</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \coprod _{X\in K}{F(X)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e3909ac24ffe210080e3162270fc9a02150d06" 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)}}" /></span></dd></dl> <p>by the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> that identifies a point <span class="texhtml"><i>y</i> ∈ <i>F</i>(<i>Y</i>)</span> with its image under the map <span class="texhtml"><i>F</i>(<i>Y</i>) → <i>F</i>(<i>X</i>)</span>, for every inclusion <span class="texhtml"><i>Y</i> ⊆ <i>X</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=7" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>1. Let <i>V</i> be a finite set of <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> <span class="texhtml"><i>n</i> + 1</span>. The <b>combinatorial <i>n</i>-simplex</b> with vertex-set <i>V</i> is an ASC whose faces are all nonempty subsets of <i>V</i> (i.e., it is the <a href="/wiki/Power_set" title="Power set">power set</a> of <i>V</i>). If <span class="texhtml"><i>V</i> = <i>S</i> = {0, 1, ..., <i>n</i>},</span> then this ASC is called the <b>standard combinatorial <i>n</i>-simplex</b>. </p><p>2. Let <i>G</i> be an undirected graph. The <b><a href="/wiki/Clique_complex" title="Clique complex">clique complex</a></b> <b>of <i>G</i></b> is an ASC whose faces are all <a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">cliques</a> (complete subgraphs) of <i>G</i>. The <b>independence complex of <i>G</i></b> is an ASC whose faces are all <a href="/wiki/Independent_set_(graph_theory)" title="Independent set (graph theory)">independent sets</a> of <i>G</i> (it is the clique complex of the <a href="/wiki/Complement_graph" title="Complement graph">complement graph</a> of G). Clique complexes are the prototypical example of <a href="/wiki/Flag_complex" class="mw-redirect" title="Flag complex">flag complexes</a>. A <b>flag complex</b> is a complex <i>K</i> with the property that every set, all of whose 2-element subsets are faces of <i>K</i>, is itself a face of <i>K</i>. </p><p>3. Let <i>H</i> be a <a href="/wiki/Hypergraph" title="Hypergraph">hypergraph</a>. A <a href="/wiki/Matching_in_hypergraphs" title="Matching in hypergraphs">matching</a> in <i>H</i> is a set of edges of <i>H</i>, in which every two edges are <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a>. The <b>matching complex of <i>H</i></b> is an ASC whose faces are all <a href="/wiki/Matching_in_hypergraphs" title="Matching in hypergraphs">matchings</a> in <i>H</i>. It is the <a href="/wiki/Independence_complex" title="Independence complex">independence complex</a> of the <a href="/wiki/Line_graph_of_a_hypergraph" title="Line graph of a hypergraph">line graph</a> of <i>H</i>. </p><p>4. Let <i>P</i> be a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset). The <b>order complex</b> of <i>P</i> is an ASC whose faces are all finite <a href="/wiki/Total_order#Chains" title="Total order">chains</a> in <i>P</i>. Its <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> groups and other <a href="/wiki/Topological_property" title="Topological property">topological invariants</a> contain important information about the poset <i>P</i>. </p><p>5. Let <i>M</i> be a <a href="/wiki/Metric_space" title="Metric space">metric space</a> and <i>δ</i> a real number. The <b><a href="/wiki/Vietoris%E2%80%93Rips_complex" title="Vietoris–Rips complex">Vietoris–Rips complex</a></b> is an ASC whose faces are the finite subsets of <i>M</i> with diameter at most <i>δ</i>. It has applications in <a href="/wiki/Homology_theory" class="mw-redirect" title="Homology theory">homology theory</a>, <a href="/wiki/Hyperbolic_group" title="Hyperbolic group">hyperbolic groups</a>, <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, and <a href="/wiki/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. </p><p>6. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" 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}" /></span> be a square-free <a href="/wiki/Monomial_ideal" title="Monomial ideal">monomial ideal</a> in a <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=K[x_{1},\dots ,x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=K[x_{1},\dots ,x_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafcb2abee81375c819a9b39da5621729975778f" 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}]}" /></span> (that is, an ideal generated by products of subsets of variables). Then the exponent vectors of those square-free monomials of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" 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}" /></span> that are not in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" 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}" /></span> determine an abstract simplicial complex via the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \in \{0,1\}^{n}\mapsto \{i\in [n]:a_{i}=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \in \{0,1\}^{n}\mapsto \{i\in [n]:a_{i}=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26112483a09c7764f992cfd056afa64b1d771e82" 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\}}" /></span>. In fact, there is a bijection between (non-empty) abstract simplicial complexes on <span class="texhtml"> <i>n</i></span> vertices and square-free monomial ideals in <span class="texhtml"> <i>S</i></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\Delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\Delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc42fd837fec7b703ea5052b5b4af75b959fc27" 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 }}" /></span> is the square-free ideal corresponding to the simplicial complex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" 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 }" /></span> then the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/I_{\Delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S/I_{\Delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7dcc6addaccbd635b6d2c8b3af4950347c68864" 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 }}" /></span> is known as the <a href="/wiki/Stanley%E2%80%93Reisner_ring" title="Stanley–Reisner ring">Stanley–Reisner ring</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a650ae128f4d515b1448e592d71392d25d53ae7d" 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 }}" /></span>. </p><p>7. For any <a href="/wiki/Open_covering" class="mw-redirect" title="Open covering">open covering</a> <i>C</i> of a topological space, the <b><a href="/wiki/Nerve_complex" title="Nerve complex">nerve complex</a></b> of <i>C</i> is an abstract simplicial complex containing the sub-families of <i>C</i> with a non-empty <a href="/wiki/Intersection" title="Intersection">intersection</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Enumeration">Enumeration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=8" title="Edit section: Enumeration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The number of abstract simplicial complexes on up to <i>n</i> labeled elements (that is on a set <i>S</i> of size <i>n</i>) is one less than the <i>n</i>th <a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind number</a>. These numbers grow very rapidly, and are known only for <span class="texhtml"><i>n</i> ≤ 9</span>; the Dedekind numbers are (starting with <i>n</i> = 0): </p> <dl><dd>1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365 (sequence <span class="nowrap external"><a href="//oeis.org/A014466" class="extiw" title="oeis:A014466">A014466</a></span> in the <a href="/wiki/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="/wiki/Antichain" title="Antichain">antichains</a> of subsets of an <span class="texhtml"> <i>n</i></span> set.</dd></dl> <p>The number of abstract simplicial complexes whose vertices are exactly <i>n</i> labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993" (sequence <span class="nowrap external"><a href="//oeis.org/A006126" class="extiw" title="oeis:A006126">A006126</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), starting at <i>n</i> = 1. This corresponds to the number of antichain covers of a labeled <i>n</i>-set; there is a clear bijection between antichain covers of an <i>n</i>-set and simplicial complexes on <i>n</i> elements described in terms of their maximal faces. </p><p>The number of abstract simplicial complexes on exactly <i>n</i> unlabeled elements is given by the sequence "1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210" (sequence <span class="nowrap external"><a href="//oeis.org/A006602" class="extiw" title="oeis:A006602">A006602</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), starting at <i>n</i> = 1. </p> <div class="mw-heading mw-heading2"><h2 id="Computational_problems">Computational problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=9" title="Edit section: Computational problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Simplicial_complex_recognition_problem" title="Simplicial complex recognition problem">Simplicial complex recognition problem</a></div> <p>The <a href="/wiki/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="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a> for any <i>d</i>-dimensional manifolds for <i>d</i> ≥ 5.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_concepts">Relation to other concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=10" title="Edit section: Relation to other concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An abstract simplicial complex with an additional property called the <b>augmentation property</b> or the <b>exchange property</b> yields a <b><a href="/wiki/Matroid" title="Matroid">matroid</a></b>. The following expression shows the relations between the terms: </p><p>HYPERGRAPHS = SET-FAMILIES ⊃ INDEPENDENCE-SYSTEMS = ABSTRACT-SIMPLICIAL-COMPLEXES ⊃ MATROIDS. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kruskal%E2%80%93Katona_theorem" title="Kruskal–Katona theorem">Kruskal–Katona theorem</a></li> <li><a href="/wiki/Simplicial_set" title="Simplicial set">Simplicial set</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abstract_simplicial_complex&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Lee-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lee_1-0">^</a></b></span> <span class="reference-text"><a href="/wiki/John_M._Lee" title="John M. Lee">Lee, John M.</a>, Introduction to Topological Manifolds, Springer 2011, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4419-7939-5" title="Special:BookSources/1-4419-7939-5">1-4419-7939-5</a>, p153</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKorte,_BernhardLovász,_LászlóSchrader,_Rainer1991" class="citation book cs1"><a href="/wiki/Bernhard_Korte" title="Bernhard Korte">Korte, Bernhard</a>; <a href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">Lovász, László</a>; Schrader, Rainer (1991). <i>Greedoids</i>. Springer-Verlag. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-18190-3" title="Special:BookSources/3-540-18190-3"><bdi>3-540-18190-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Greedoids&rft.pages=9&rft.pub=Springer-Verlag&rft.date=1991&rft.isbn=3-540-18190-3&rft.au=Korte%2C+Bernhard&rft.au=Lov%C3%A1sz%2C+L%C3%A1szl%C3%B3&rft.au=Schrader%2C+Rainer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbstract+simplicial+complex" class="Z3988"></span></span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_3-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMatoušek2007" class="citation book cs1"><a href="/wiki/Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Matoušek, Jiří</a> (2007). <i><a href="/wiki/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</i> (2nd ed.). Berlin-Heidelberg: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-00362-5" title="Special:BookSources/978-3-540-00362-5"><bdi>978-3-540-00362-5</bdi></a>. <q>Written in cooperation with <a href="/wiki/Anders_Bj%C3%B6rner" title="Anders Björner">Anders Björner</a> and <a href="/wiki/G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Günter M. Ziegler</a></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Using+the+Borsuk-Ulam+Theorem%3A+Lectures+on+Topological+Methods+in+Combinatorics+and+Geometry&rft.place=Berlin-Heidelberg&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=2007&rft.isbn=978-3-540-00362-5&rft.aulast=Matou%C5%A1ek&rft.aufirst=Ji%C5%99%C3%AD&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbstract+simplicial+complex" class="Z3988"></span> , Section 4.3</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStillwell1993" class="citation cs2"><a href="/wiki/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"><i>Classical Topology and Combinatorial Group Theory</i></a>, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387979700" title="Special:BookSources/9780387979700"><bdi>9780387979700</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Topology+and+Combinatorial+Group+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.pages=247&rft.pub=Springer&rft.date=1993&rft.isbn=9780387979700&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D265lbM42REMC%26pg%3DPA247&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbstract+simplicial+complex" class="Z3988"></span>.</span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Abstract_simplicial_complex&oldid=1270517241">https://en.wikipedia.org/w/index.php?title=Abstract_simplicial_complex&oldid=1270517241</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Algebraic_topology" title="Category:Algebraic topology">Algebraic topology</a></li><li><a href="/wiki/Category:Families_of_sets" title="Category:Families of sets">Families of sets</a></li><li><a href="/wiki/Category:Simplicial_sets" title="Category:Simplicial sets">Simplicial sets</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 19 January 2025, at 22:41<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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