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<title>Simplicial complex</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Simplicial complex</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>simplicial complex</b> is a structured <a href="Set_(mathematics)" title="Set (mathematics)">set</a> composed of <a href="Point_(geometry)" title="Point (geometry)">points</a>, <a href="Line_segment" title="Line segment">line segments</a>, <a href="Triangle" title="Triangle">triangles</a>, and their <i>n</i>-dimensional counterparts, called <a href="Simplex" title="Simplex">simplices</a>, such that all the faces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of a <a href="Simplicial_set" title="Simplicial set">simplicial set</a> appearing in modern simplicial <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a>. The purely <a href="Combinatorics" title="Combinatorics">combinatorial</a> counterpart to a simplicial complex is an <a href="Abstract_simplicial_complex" title="Abstract simplicial complex">abstract simplicial complex</a>. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a <b>geometric simplicial complex</b>.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 7">: 7 </span></sup>

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

<p>A <b>simplicial complex</b> <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> is a set of <a href="Simplex" title="Simplex">simplices</a> that satisfies the following conditions:

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<ol><li>Every <a href="Simplex#Elements" title="Simplex">face</a> of a simplex from <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> is also in <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span>.</li>

<li>The non-empty <a href="Set_intersection" class="mw-redirect" title="Set intersection">intersection</a> of any two simplices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}}</annotation>

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</math></span><img src="./4c67d9da8521c1515f476ab0fa8544d50adf2518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.409ex; height:2.509ex;" alt="{\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}}" loading="lazy"></span> is a face of both <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}}</annotation>

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</math></span><img src="./7fa0e56273a1cb32709b442e2421e9f947522b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{2}}</annotation>

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</math></span><img src="./8d4b9cd9efc54bcfd04e0a2231913c13f10798d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{2}}" loading="lazy"></span>.</li></ol>

<p>See also the definition of an <a href="Abstract_simplicial_complex" title="Abstract simplicial complex">abstract simplicial complex</a>, which loosely speaking is a simplicial complex without an associated geometry.

</p><p>A <b>simplicial <i>k</i>-complex</b> <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> is a simplicial complex where the largest dimension of any simplex in <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> equals <i>k</i>. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any <a href="Tetrahedra" class="mw-redirect" title="Tetrahedra">tetrahedra</a> or higher-dimensional simplices.

</p><p>A <b>pure</b> or <b>homogeneous</b> simplicial <i>k</i>-complex <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> is a simplicial complex where every simplex of dimension less than <i>k</i> is a face of some simplex <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \in {\mathcal {K}}}">

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

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</math></span><img src="./99e2bc7bd864fd07cb7b9d0b07871041888c9cca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle \sigma \in {\mathcal {K}}}" loading="lazy"></span> of dimension exactly <i>k</i>. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a <i>non</i>-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as <a href="Triangulation_(topology)" title="Triangulation (topology)">triangulations</a> and provide a definition of <a href="Polytope" title="Polytope">polytopes</a>.

</p><p>A <b>facet</b> is a maximal simplex, i.e., any simplex in a complex that is <i>not</i> a face of any larger simplex.<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> (Note the difference from a <a href="Face_of_a_simplex" class="mw-redirect" title="Face of a simplex">"face" of a simplex</a>). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of) <a href="Simplicial_polytope" title="Simplicial polytope">simplicial polytopes</a> this coincides with the meaning from polyhedral combinatorics.

</p><p>Sometimes the term <i>face</i> is used to refer to a simplex of a complex, not to be confused with a face of a simplex.

</p><p>For a simplicial complex <a href="Embedding" title="Embedding">embedded</a> in a <i>k</i>-dimensional space, the <i>k</i>-faces are sometimes referred to as its <b>cells</b>. The term <i>cell</i> is sometimes used in a broader sense to denote a set <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a> to a simplex, leading to the definition of <a href="Cell_complex" class="mw-redirect" title="Cell complex">cell complex</a>.

</p><p>The <b>underlying space</b>, sometimes called the <b>carrier</b> of a simplicial complex, is the <a href="Union_(set_theory)" title="Union (set theory)">union</a> of its simplices. It is usually denoted by <span class="mwe-math-element mwe-math-element-inline"><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}}|}">

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

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</math></span><img src="./11c3d14dc6d6aef85b7627280fb27dcb8134fa83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |{\mathcal {K}}|}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><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}}\|}">

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

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</math></span><img src="./6732a46eadde5743f53a43a4906c557a1e846590.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.096ex; height:2.843ex;" alt="{\displaystyle \|{\mathcal {K}}\|}" loading="lazy"></span>.

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

<p>The <a href="Relative_interior" title="Relative interior">relative interiors</a> of all simplices in <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> form a partition of its underlying space <span class="mwe-math-element mwe-math-element-inline"><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}}|}">

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

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</math></span><img src="./11c3d14dc6d6aef85b7627280fb27dcb8134fa83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |{\mathcal {K}}|}" loading="lazy"></span>: for each point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in |{\mathcal {K}}|}">

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

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</math></span><img src="./47627095ff358a8c57a793d3365ac215eb7e2a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.235ex; height:2.843ex;" alt="{\displaystyle x\in |{\mathcal {K}}|}" loading="lazy"></span>, there is exactly one simplex in <span class="mwe-math-element mwe-math-element-inline"><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}}}">

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

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</math></span><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"></span> containing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in its relative interior. This simplex is called the <b>support</b> of <i>x</i> and denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {supp} (x)}">

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</math></span><img src="./f51f2ca254170ce4199b3dcbdfcbde2b9680fe92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle \operatorname {supp} (x)}" loading="lazy"></span>.<sup id="cite_ref-:02_3-0" class="reference"><a href="#cite_note-:02-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 9">: 9 </span></sup>

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<div class="mw-heading mw-heading2"><h2 id="Closure,_star,_and_link">Closure, star, and link</h2></div>

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</style><span class="tmp-color" style="color:#fc3">simplices</span> and their <span class="tmp-color" style="color:#093"><b>closure</b></span>.</div>

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<div class="gallerytext">A <span class="tmp-color" style="color:#fc3">vertex</span> and its <span class="tmp-color" style="color:#093"><b>star</b></span>.</div>

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<div class="gallerytext">A <span class="tmp-color" style="color:#fc3">vertex</span> and its <span class="tmp-color" style="color:#093"><b>link</b></span>.</div>

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<p>Let <i>K</i> be a simplicial complex and let <i>S</i> be a collection of simplices in <i>K</i>.

</p><p>The <b>closure</b> of <i>S</i> (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} \ S}">

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

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</math></span><img src="./89546936aab3c956ab8d638a1130f1cc7490db3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.405ex; height:2.176ex;" alt="{\displaystyle \mathrm {Cl} \ S}" loading="lazy"></span>) is the smallest simplicial subcomplex of <i>K</i> that contains each simplex in <i>S</i>. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} \ S}">

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</math></span><img src="./89546936aab3c956ab8d638a1130f1cc7490db3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.405ex; height:2.176ex;" alt="{\displaystyle \mathrm {Cl} \ S}" loading="lazy"></span> is obtained by repeatedly adding to <i>S</i> each face of every simplex in <i>S</i>.

</p><p>The <b><a href="Star_(simplicial_complex)" class="mw-redirect" title="Star (simplicial complex)">star</a></b> of <i>S</i> (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {st} \ S}">

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

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</math></span><img src="./30f17e9cae51e6a6ae79bd3896449ee422532868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.901ex; height:2.176ex;" alt="{\displaystyle \mathrm {st} \ S}" loading="lazy"></span>) is the union of the stars of each simplex in <i>S</i>. For a single simplex <i>s</i>, the star of <i>s</i> is the set of simplices in <i>K</i> that have <i>s</i> as a face. The star of <i>S</i> is generally not a simplicial complex itself, so some authors define the <b>closed star</b> of S (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {St} \ S}">

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

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</math></span><img src="./53ee1334c1e07780cbd222b7b1cd12565da0b10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.277ex; height:2.176ex;" alt="{\displaystyle \mathrm {St} \ S}" loading="lazy"></span>) as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} \ \mathrm {st} \ S}">

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

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</math></span><img src="./e3430a787437f9088b1692331e964c1b53eceb6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.806ex; height:2.176ex;" alt="{\displaystyle \mathrm {Cl} \ \mathrm {st} \ S}" loading="lazy"></span> the closure of the star of S.

</p><p>The <b><a href="Link_(geometry)" class="mw-redirect" title="Link (geometry)">link</a></b> of <i>S</i> (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Lk} \ S}">

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

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</math></span><img src="./b4c6ae7464b1dcd1ec83a20d3639cef57589b457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.76ex; height:2.176ex;" alt="{\displaystyle \mathrm {Lk} \ S}" loading="lazy"></span>) equals <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} {\big (}\mathrm {st} (S){\big )}\setminus \mathrm {st} {\big (}\mathrm {Cl} (S){\big )}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Cl} {\big (}\mathrm {st} (S){\big )}\setminus \mathrm {st} {\big (}\mathrm {Cl} (S){\big )}}</annotation>

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</math></span><img src="./ca65d8ee8825a04793a8be00f27a423520759c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.363ex; height:3.176ex;" alt="{\displaystyle \mathrm {Cl} {\big (}\mathrm {st} (S){\big )}\setminus \mathrm {st} {\big (}\mathrm {Cl} (S){\big )}}" loading="lazy"></span>. It is the closed star of <i>S</i> minus the stars of all faces of <i>S</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Algebraic_topology">Algebraic topology</h2></div>

/* start https://en.wikipedia.org/ */

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

<p>In <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>, simplicial complexes are often useful for concrete calculations. For the definition of <a href="Homology_group" class="mw-redirect" title="Homology group">homology groups</a> of a simplicial complex, one can read the corresponding <a href="Chain_complex" title="Chain complex">chain complex</a> directly, provided that consistent orientations are made of all simplices. The requirements of <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a> lead to the use of more general spaces, the <a href="CW_complex" title="CW complex">CW complexes</a>. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at <a href="Polytope" title="Polytope">Polytope</a> of simplicial complexes as subspaces of <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> made up of subsets, each of which is a <a href="Simplex" title="Simplex">simplex</a>. That somewhat more concrete concept is there attributed to <a href="Pavel_Sergeevich_Alexandrov" class="mw-redirect" title="Pavel Sergeevich Alexandrov">Alexandrov</a>. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a <a href="Compact_space" title="Compact space">compact</a> <a href="Topological_space" title="Topological space">topological space</a> which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a <a href="Polyhedron" title="Polyhedron">polyhedron</a> (see <a href="#CITEREFSpanier1966">Spanier 1966</a>, <a href="#CITEREFMaunder1996">Maunder 1996</a>, <a href="#CITEREFHiltonWylie1967">Hilton & Wylie 1967</a>).

</p>

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

<p><a href="Combinatorics" title="Combinatorics">Combinatorialists</a> often study the <b><i>f</i>-vector</b> of a simplicial d-complex Δ, which is the <a href="Integer" title="Integer">integer</a> sequence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{0},f_{1},f_{2},\ldots ,f_{d+1})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (f_{0},f_{1},f_{2},\ldots ,f_{d+1})}</annotation>

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</math></span><img src="./ce4bb31b83fdd2b658e0b3995ae60672f0b45639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.968ex; height:2.843ex;" alt="{\displaystyle (f_{0},f_{1},f_{2},\ldots ,f_{d+1})}" loading="lazy"></span>, where <i>f</i><sub><i>i</i></sub> is the number of (<i>i</i>−1)-dimensional faces of Δ (by convention, <i>f</i><sub>0</sub> = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the <a href="Octahedron" title="Octahedron">octahedron</a>, then its <i>f</i>-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its <i>f</i>-vector is (1, 18, 23, 8, 1). A complete characterization of the possible <i>f</i>-vectors of simplicial complexes is given by the <a href="Kruskal%E2%80%93Katona_theorem" title="Kruskal–Katona theorem">Kruskal–Katona theorem</a>.

</p><p>By using the <i>f</i>-vector of a simplicial <i>d</i>-complex Δ as coefficients of a <a href="Polynomial" title="Polynomial">polynomial</a> (written in decreasing order of exponents), we obtain the <b>f-polynomial</b> of Δ. In our two examples above, the <i>f</i>-polynomials would be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}+6x^{2}+12x+8}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{3}+6x^{2}+12x+8}</annotation>

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</math></span><img src="./2e1aeb971646db4123ef50a6da9adc110bb2615b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.269ex; height:2.843ex;" alt="{\displaystyle x^{3}+6x^{2}+12x+8}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+18x^{3}+23x^{2}+8x+1}">

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{4}+18x^{3}+23x^{2}+8x+1}</annotation>

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</math></span><img src="./48992f938a1cb7b7c9e868bbf48ac5e43adf496e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.818ex; height:2.843ex;" alt="{\displaystyle x^{4}+18x^{3}+23x^{2}+8x+1}" loading="lazy"></span>, respectively.

</p><p>Combinatorists are often quite interested in the <b>h-vector</b> of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging <i>x</i> − 1 into the <i>f</i>-polynomial of Δ. Formally, if we write <i>F</i><sub>Δ</sub>(<i>x</i>) to mean the <i>f</i>-polynomial of Δ, then the <b>h-polynomial</b> of Δ is

</p>

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<annotation encoding="application/x-tex">{\displaystyle F_{\Delta }(x-1)=h_{0}x^{d+1}+h_{1}x^{d}+h_{2}x^{d-1}+\cdots +h_{d}x+h_{d+1}}</annotation>

</semantics>

</math></span><img src="./cf22ade34975cbcf828541fde6bb74fb36d89d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.199ex; height:3.176ex;" alt="{\displaystyle F_{\Delta }(x-1)=h_{0}x^{d+1}+h_{1}x^{d}+h_{2}x^{d-1}+\cdots +h_{d}x+h_{d+1}}" loading="lazy"></span></dd></dl>

<p>and the <i>h</i>-vector of Δ is

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<annotation encoding="application/x-tex">{\displaystyle (h_{0},h_{1},h_{2},\cdots ,h_{d+1}).}</annotation>

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</math></span><img src="./40222c64fecc91eb71bef92d2c24335688966e7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.413ex; height:2.843ex;" alt="{\displaystyle (h_{0},h_{1},h_{2},\cdots ,h_{d+1}).}" loading="lazy"></span></dd></dl>

<p>We calculate the h-vector of the octahedron boundary (our first example) as follows:

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<annotation encoding="application/x-tex">{\displaystyle F(x-1)=(x-1)^{3}+6(x-1)^{2}+12(x-1)+8=x^{3}+3x^{2}+3x+1.}</annotation>

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</math></span><img src="./9cacb790cd6c6442aea3c71320b89b74345748bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.537ex; height:3.176ex;" alt="{\displaystyle F(x-1)=(x-1)^{3}+6(x-1)^{2}+12(x-1)+8=x^{3}+3x^{2}+3x+1.}" loading="lazy"></span></dd></dl>

<p>So the <i>h</i>-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this <i>h</i>-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial <a href="Polytope" title="Polytope">polytope</a> (these are the <a href="Dehn%E2%80%93Sommerville_equations" title="Dehn–Sommerville equations">Dehn–Sommerville equations</a>). In general, however, the <i>h</i>-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting <i>h</i>-vector is (1, 3, −2).

</p><p>A complete characterization of all simplicial polytope <i>h</i>-vectors is given by the celebrated <a href="G-theorem" class="mw-redirect" title="G-theorem">g-theorem</a> of <a href="Richard_P._Stanley" title="Richard P. Stanley">Stanley</a>, Billera, and Lee.

</p><p>Simplicial complexes can be seen to have the same geometric structure as the <a href="Contact_graph" title="Contact graph">contact graph</a> of a <a href="Sphere_packing" title="Sphere packing">sphere packing</a> (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Triangulation_(topology)" title="Triangulation (topology)">Triangulation (topology)</a></div>

<p>A triangulation of a <a href="Topological_space" title="Topological space">topological space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is a <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a> <span class="mwe-math-element mwe-math-element-inline"><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:|{\mathcal {T}}|\rightarrow X}">

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<annotation encoding="application/x-tex">{\displaystyle t:|{\mathcal {T}}|\rightarrow X}</annotation>

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</math></span><img src="./324e70f1dc924e77dbff7a74a42dfe1a555f4d04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.6ex; height:2.843ex;" alt="{\displaystyle t:|{\mathcal {T}}|\rightarrow X}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><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 {T}}}">

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

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</math></span><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy"></span> is a simplicial complex.

</p><p>Topological spaces do not necessarily admit a triangulation and if they do, it is never unique. <a href="Topological_manifold" title="Topological manifold">Topological manifolds</a> of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\leq 3}">

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<annotation encoding="application/x-tex">{\displaystyle d\leq 3}</annotation>

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</math></span><img src="./f2a21883c7b86c050d8230f596f8ff519c3295d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\leq 3}" loading="lazy"></span> are always triangulable,<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><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> but not necessarily for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d>3}">

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</math></span><img src="./58db34887c6f7dafa930e0a4f17834a8e14469df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d>3}" loading="lazy"></span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable manifolds</a> of any dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle d\geq 1}</annotation>

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</math></span><img src="./33476a779aa3e8bc64413c16f777ef15e9d78df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 1}" loading="lazy"></span> admit triangulations.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<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>

<p>The <a href="Simplicial_complex_recognition_problem" title="Simplicial complex recognition problem">simplicial complex recognition problem</a> is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem is <a href="Undecidable_problem" title="Undecidable problem">undecidable</a> for any <i>d</i>-dimensional manifolds for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 5}">

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<annotation encoding="application/x-tex">{\displaystyle d\geq 5}</annotation>

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</math></span><img src="./5bb1d60405583dd796868783ba7cdc8a9591ba06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 5}" loading="lazy"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 9–11">: 9–11 </span></sup>

</p>

<ul><li><a href="Abstract_simplicial_complex" title="Abstract simplicial complex">Abstract simplicial complex</a></li>

<li><a href="Barycentric_subdivision" title="Barycentric subdivision">Barycentric subdivision</a></li>

<li><a href="Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li>

<li><a href="Delta_set" title="Delta set">Delta set</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Polygonal_chain" title="Polygonal chain">Polygonal chain</a> – 1 dimensional simplicial complex</li>

<li><a href="Tucker's_lemma" title="Tucker's lemma">Tucker's lemma</a></li>

<li><a href="Simplex_tree" title="Simplex tree">Simplex tree</a></li></ul>

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

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</style><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). <i><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</i> (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</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"><cite id="CITEREFDe_LoeraRambauSantos2010" class="citation cs2"><a href="Jes%C3%BAs_A._De_Loera" title="Jesús A. De Loera">De Loera, Jesús A.</a>; Rambau, Jörg; <a href="Francisco_Santos_Leal" title="Francisco Santos Leal">Santos, Francisco</a> (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SxY1Xrr12DwC&pg=PA493"><i>Triangulations: Structures for Algorithms and Applications</i></a>, Algorithms and Computation in Mathematics, vol. 25, Springer, p. 493, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783642129711</bdi></cite></span>

</li>

<li id="cite_note-:02-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-:02_3-0">^</a></b></span> <span class="reference-text"><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). <i><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</i> (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</span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFEdwin_Moise1977" class="citation cs2">Edwin Moise (1977), <i>Geometric Topology in Dimensions 2 and 3</i>, New York: Springer Verlag</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFRado" class="citation web cs1 cs1-prop-foreign-lang-source">Rado, Tibor. <a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~v1ranick/papers/rado.pdf">"Über den Begriff der Riemannschen Fläche"</a> <span class="cs1-format">(PDF)</span> (in German).</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohn_M._Lee2000" class="citation cs2">John M. Lee (2000), <i>Introduction to Topological manifolds</i>, New York/Berlin/Heidelberg: Springer Verlag, p. 92, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98759-2</bdi></cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFR._C._KirbyL._C._Siebenmann1977" class="citation cs2">R. C. Kirby; L. C. Siebenmann (1977-12-31), "Annex B. On The Triangulation of Manifolds and the Hauptvermutung", <i>Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88)</i>, Princeton University Press, pp. 299–306</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFAkbulutMcCarthy2016" class="citation book cs1">Akbulut, Selman; McCarthy, John D. (19 April 2016). "Chapter IV: Casson's Invariant for Oriented Homology 3-spheres". <i>Casson's Invariant for Oriented Homology Three-Spheres | Princeton University Press</i>. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780691636085</bdi>.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFJ._H._C._Whitehead1940" class="citation cs2">J. H. C. Whitehead (1940), "On C1-Complexes", <i>Annals of Mathematics</i>, vol. 41, no. 4, pp. 809–824, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968861">10.2307/1968861</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968861">1968861</a></cite></span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><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"><i>Classical Topology and Combinatorial Group Theory</i></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>.</span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFPoonen2014" class="citation arxiv cs1">Poonen, Bjorn (2014-10-25). "Undecidable problems: a sampler". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1204.0299">1204.0299</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite></span>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFSpanier1966" class="citation cs2"><a href="Edwin_Spanier" title="Edwin Spanier">Spanier, Edwin H.</a> (1966), <i>Algebraic Topology</i>, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94426-5</bdi></cite></li>

<li><cite id="CITEREFMaunder1996" class="citation cs2">Maunder, Charles R.F. (1996), <i>Algebraic Topology</i> (Reprint of the 1980 ed.), Mineola, NY: Dover, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-69131-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1402473">1402473</a></cite></li>

<li><cite id="CITEREFHiltonWylie1967" class="citation cs2"><a href="Peter_Hilton" title="Peter Hilton">Hilton, Peter J.</a>; <a href="Shaun_Wylie" title="Shaun Wylie">Wylie, Shaun</a> (1967), <i>Homology Theory</i>, New York: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-09422-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0115161">0115161</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Simplicial_complex"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SimplicialComplex.html">"Simplicial complex"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>

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</style><div id="Topology1035" style="font-size:114%;margin:0 4em"><a href="Topology" title="Topology">Topology</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_topology" title="General topology">General (point-set)</a></li>

<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a></li>

<li><a href="Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li>

<li><a href="Continuum_(topology)" title="Continuum (topology)">Continuum</a></li>

<li><a href="Differential_topology" title="Differential topology">Differential</a></li>

<li><a href="Geometric_topology" title="Geometric topology">Geometric</a>

<ul><li><a href="Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li>

<li><a href="Homology_(mathematics)" title="Homology (mathematics)">Homology</a>

<ul><li><a href="Cohomology" title="Cohomology">cohomology</a></li></ul></li>

<li><a href="Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>

<li><a href="Digital_topology" title="Digital topology">Digital</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="Klein_bottle" title="Klein bottle"></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Open_set" title="Open set">Open set</a> / <a href="Closed_set" title="Closed set">Closed set</a></li>

<li><a href="Interior_(topology)" title="Interior (topology)">Interior</a></li>

<li><a href="Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li>

<li><a href="Topological_space" title="Topological space">Space</a>

<ul><li><a href="Compact_space" title="Compact space">compact</a></li>

<li><a href="Connected_space" title="Connected space">connected</a></li>

<li><a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a></li>

<li><a href="Metric_space" title="Metric space">metric</a></li>

<li><a href="Uniform_space" title="Uniform space">uniform</a></li></ul></li>

<li><a href="Homotopy" title="Homotopy">Homotopy</a>

<ul><li><a href="Homotopy_group" title="Homotopy group">homotopy group</a></li>

<li><a href="Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li>

<li><a href="CW_complex" title="CW complex">CW complex</a></li>

<li><a href="Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li>

<li><a href="Second-countable_space" title="Second-countable space">Second-countable space</a></li>

<li><a href="Cobordism" title="Cobordism">Cobordism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Metrics and properties</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li>

<li><a href="Betti_number" title="Betti number">Betti number</a></li>

<li><a href="Winding_number" title="Winding number">Winding number</a></li>

<li><a href="Chern_class" title="Chern class">Chern number</a></li>

<li><a href="Orientability" title="Orientability">Orientability</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li>

<li><a href="Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li>

<li><a href="Tychonoff's_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li>

<li><a href="Urysohn's_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li>

<li><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li>

<li><span class="noviewer" typeof="mw:File"></span> <a href="https://en.wikibooks.org/wiki/Topology" class="extiw external" title="wikibooks:Topology">Wikibook</a></li>

<li><span class="noviewer" typeof="mw:File"></span> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw external" title="wikiversity:Topology">Wikiversity</a></li>

<li><span class="noviewer" typeof="mw:File"><span title="List-Class article"></span></span> <a href="List_of_topology_topics" title="List of topology topics">Topics</a>

<ul><li><a href="List_of_general_topology_topics" title="List of general topology topics">general</a></li>

<li><a href="List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li>

<li><a href="List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li>

<li><span class="noviewer" typeof="mw:File"><span title="List-Class article"></span></span> <a href="List_of_important_publications_in_mathematics" class="mw-redirect" title="List of important publications in mathematics">Publications</a></li></ul>

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