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<title>Simplizialkomplex</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Simplizialkomplex</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>Simplizialkomplex</b> ist ein Begriff der <a href="Algebraische_Topologie" title="Algebraische Topologie">algebraischen Topologie</a>. Bei einem Simplizialkomplex handelt es sich um ein rein <a href="Kombinatorik" title="Kombinatorik">kombinatorisch</a> beschreibbares Objekt, mit dessen Hilfe die entscheidenden Eigenschaften von bestimmten, als <a href="Triangulierung_(Topologie)" title="Triangulierung (Topologie)">triangulierbar</a> bezeichneten <a href="Topologischer_Raum" title="Topologischer Raum">topologischen Räumen</a> algebraisch charakterisiert werden können. Insbesondere werden Simplizialkomplexe dazu verwendet, für den zugrundeliegenden topologischen Raum <a href="Invariante_(Mathematik)" title="Invariante (Mathematik)">Invarianten</a> zu definieren.

</p><p>Die Idee des Simplizialkomplexes besteht darin, einen topologischen Raum dadurch zu untersuchen, dass – sofern möglich – durch Zusammenfügen von <a href="Simplex_(Mathematik)" title="Simplex (Mathematik)">Simplizes</a> eine Menge im <i>d</i>-dimensionalen <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> konstruiert wird, die <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">homöomorph</a> ist zum gegebenen topologischen Raum. Die „Anleitung zum Zusammenbau“ der Simplizes, das heißt die Angaben darüber, wie die Simplizes zusammengefügt sind, wird dann in Form einer Sequenz von <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismen</a> rein algebraisch charakterisiert.

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

<p>Der formalen Definition eines Simplizialkomplexes liegt die Idee zugrunde, dass bestimmte Teilmengen des <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span>-dimensionalen euklidischen Raums zerlegt werden können in <a href="Punkt_(Mathematik)" class="mw-redirect" title="Punkt (Mathematik)">Punkte</a>, <a href="Strecke_(Geometrie)" title="Strecke (Geometrie)">Strecken</a>, <a href="Dreieck" title="Dreieck">Dreiecke</a>, <a href="Tetraeder" title="Tetraeder">Tetraeder</a> und so weiter. Da es sich bei den vier aufgezählten geometrischen Objekten um die einfachsten <a href="Polytop_(Geometrie)" title="Polytop (Geometrie)">Polytope</a> der jeweiligen <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</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 n=0,1,2,3}">

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<annotation encoding="application/x-tex">{\displaystyle n=0,1,2,3}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c982bb3cb58ec4fcccddb605c8ef7086684d567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.245ex; height:2.509ex;" alt="{\displaystyle n=0,1,2,3}" loading="lazy"></span> handelt, die als <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 n}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-<a href="Simplex_(Mathematik)" title="Simplex (Mathematik)">Simplex</a> bezeichnet werden, handelt es sich allgemein um Zerlegungen 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 n}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Simplizes: Punkt (0-Simplex), Gerade (1-Simplex), Dreieck (2-Simplizes), Tetraeder (3-Simplex), <a href="Pentachoron" title="Pentachoron">Pentachoron</a> (4-Simplex), 5-Simplex und so weiter.

</p><p>Bei der formalen Beschreibung einer solchen Zerlegung im Rahmen eines Simplizialkomplexes werden die Kantenlängen ausgeblendet. Maßgeblich ist nur die Art des „Zusammenbaus“, d. h. die Information darüber, wie die <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 n}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Simplizes aneinandergefügt sind. Diese Informationen dienen dann dazu, die zerlegte <a href="Punktmenge" class="mw-redirect" title="Punktmenge">Punktmenge</a> zu charakterisieren.

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

<div class="mw-heading mw-heading3"><h3 id="Abstrakter_Simplizialkomplex">Abstrakter Simplizialkomplex</h3></div>

<p>Ein <i>abstraktes Simplex</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 \sigma }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> ist eine endliche nichtleere Menge. Ein Element eines abstrakten Simplexes nennt man <i>Ecke</i> von <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 }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span>, eine nichtleere Teilmenge von <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 }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> ist wieder ein abstraktes Simplex und wird <i>Facette</i> (oder <i>Seite</i>) von <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 }">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> genannt.

</p><p>Ein <i>abstrakter</i> oder auch <i>kombinatorischer Simplizialkomplex</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 {\mathcal {K}}}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> ist eine Menge von Simplizes mit der Eigenschaft, dass jede Facette <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 '\subseteq \sigma }">

<msup>

</msup>

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7868d69d96f3705623c1aebe6003132ff11b6250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.443ex; height:2.676ex;" alt="{\displaystyle \sigma '\subseteq \sigma }" loading="lazy"></span> eines Simplexes <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}}}">

</mrow>

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99e2bc7bd864fd07cb7b9d0b07871041888c9cca.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> wieder zu <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}}}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> gehört, also <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}}}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14cfba1ad4c91733cb98a6b22fb7374cef06c822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.627ex; height:2.509ex;" alt="{\displaystyle \sigma '\in {\mathcal {K}}}" loading="lazy"></span>. Die Vereinigungsmenge aller Ecken von Simplizes des Simplizialkomplexes <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}}}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> wird <i>Eckenmenge</i> oder <i>Eckpunktbereich</i> genannt und mit <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 V({\mathcal {K}})}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60cef9f762a6b31fc889e5c858209f24baffaf16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.368ex; height:2.843ex;" alt="{\displaystyle V({\mathcal {K}})}" loading="lazy"></span> bezeichnet.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Die Dimension eines abstrakten Simplex, das <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 k+1}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/552a558062ed4c0486297b5b5531c5ee044dbd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k+1}" loading="lazy"></span> Ecken enthält, ist definiert als <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 k}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>, und die Dimension des Simplizialkomplexes <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> ist definiert als das Maximum der Dimension aller Simplizes. Falls die Dimension der Simplizes nicht beschränkt ist, dann heißt <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> unendlichdimensional.

</p><p>Der Simplizialkomplex <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> heißt endlich, falls er eine endliche Menge ist, und lokal endlich, falls jede Ecke nur zu endlich vielen Simplizes gehört.

</p><p>Das <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 n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Skelett <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}}_{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c31a97386689c8c2bbfdffe680e71c4cbf0ccf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.989ex; height:2.509ex;" alt="{\displaystyle {\mathcal {K}}_{n}}" loading="lazy"></span> eines Simplizialkomplexes <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> ist die Menge aller seiner Simplizes der 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 \leq n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61fbd99c9375a5553cc47f02b7da90cb8becd4ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.848ex; height:2.176ex;" alt="{\displaystyle \leq n}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Geometrischer_Simplizialkomplex">Geometrischer Simplizialkomplex</h3></div>

<p>Ein <i>geometrischer Simplizialkomplex</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 {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> ist eine Menge von <a href="Simplex_(Mathematik)" title="Simplex (Mathematik)">Simplizes</a> in einem euklidischen Raum <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 \mathbb {R} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a713426956296f1668fce772df3c60b9dde8a685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}" loading="lazy"></span> mit der Eigenschaft, dass jede Facette <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 '\subseteq \sigma }">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7868d69d96f3705623c1aebe6003132ff11b6250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.443ex; height:2.676ex;" alt="{\displaystyle \sigma '\subseteq \sigma }" loading="lazy"></span> eines Simplexes <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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a2c5f5aa5c1d2278b3bb1bfd89c46207dbf8ca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.663ex; height:2.176ex;" alt="{\displaystyle \sigma \in {\mathcal {S}}}" loading="lazy"></span> wieder zu <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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> gehört und dass für alle Simplizes <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 ,\tau \in {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd96c9a8e7a7164777fdcbe24dac1fb14b1293ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.898ex; height:2.509ex;" alt="{\displaystyle \sigma ,\tau \in {\mathcal {S}}}" loading="lazy"></span> der <a href="Schnittmenge" class="mw-redirect" title="Schnittmenge">Durchschnitt</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 \sigma \cap \tau }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1b5fbc85a391ea8ded8940802c28265a131830b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.114ex; height:2.009ex;" alt="{\displaystyle \sigma \cap \tau }" loading="lazy"></span> entweder <a href="Leere_Menge" title="Leere Menge">leer</a> oder eine gemeinsame Facette von <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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> und <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 \tau }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span> ist. Mit <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 {S}}|}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/227447c0b154dba74dd6cb91f65cdefb3b0c2ada.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.786ex; height:2.843ex;" alt="{\displaystyle |{\mathcal {S}}|}" loading="lazy"></span> wird die Vereinigung aller Simplizes des geometrischen Komplexes bezeichnet.

</p>

<div class="mw-heading mw-heading3"><h3 id="Geometrische_Realisierung">Geometrische Realisierung</h3></div>

<p>Ein geometrischer Simplizialkomplex <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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>, dessen Ecken einem gegebenen abstrakten Simplizialkomplex <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> entsprechen, heißt <i>geometrische Realisierung</i> des Simplizialkomplexes <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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>. Sie wird mit <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 \vert {\mathcal {K}}\vert }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84251362a67d2a641cc52dba9dd6fd9a82cd3f4a.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 \vert {\mathcal {K}}\vert }" loading="lazy"></span> bezeichnet. Alle geometrischen Realisierungen eines abstrakten Simplizialkomplexes sind zueinander <a href="Hom%C3%B6omorph" class="mw-redirect" title="Homöomorph">homöomorph</a>.

</p><p>Zu einem Punkt <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 \vert {\mathcal {K}}\vert }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5542d3cb92050a5506fbe4faa7db9dac55c11d0.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 \vert {\mathcal {K}}\vert }" loading="lazy"></span> gibt es einen eindeutigen Simplex aus <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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>, in dessen Innerem <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.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> liegt. Dieser Simplex wird als <i>Trägersimplex</i> von <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.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> bezeichnet.

</p><p>Ein <i>simplizialer</i> <i>Teilkomplex</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 {\mathcal {S}}'\subseteq S}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0fd9a71705dc9197d7b2cb451fb30dd0c05e005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.8ex; height:2.676ex;" alt="{\displaystyle {\mathcal {S}}'\subseteq S}" loading="lazy"></span> ist eine Menge von Simplizes 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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> derart, dass die Vereinigung der Simplizes 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 {S}}'}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e577a88611d41be7e2cd935d34000ca976f9d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}'}" loading="lazy"></span>einen simplizialen Komplex bildet.<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>

<div class="mw-heading mw-heading3"><h3 id="Triangulierung">Triangulierung</h3></div>

<p>Ein topologischer Raum heißt <a href="Triangulierung_(Topologie)" title="Triangulierung (Topologie)">triangulierbar</a>, wenn er homöomorph zu einem geometrischen Simplizialkomplex ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Abschluss,_Stern_und_Link"><span id="Abschluss.2C_Stern_und_Link"></span>Abschluss, Stern und Link</h2></div><p>

Sei <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> eine Menge von Simplizes in einem geometrischen Simplizialkomplex <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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>. Man kann nun durch drei Konstruktionen <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> zu einem Teilkomplex von <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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> machen, wobei der <i><a href="Stern_(Topologie)" title="Stern (Topologie)">Stern</a></i> von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> beim Beweis des <a href="Simpliziale_Approximation" title="Simpliziale Approximation">simplizialen Approximationssatzes</a> gebraucht wird.</p><ul class="center gallery mw-gallery-traditional">

<div class="gallerytext">Zwei Simplizes (gelb) und deren Abschluss (grün).</div>

</li>

<div class="gallerytext">Ein Simplex und dessen Stern.</div>

</li>

<div class="gallerytext">Ein Simplex und dessen Link.</div>

</li>

</ul>

<div class="mw-heading mw-heading3"><h3 id="Abschluss">Abschluss</h3></div>

<p>Der Abschluss <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 {cl} (M)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} (M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16513de910cadf6238f6be7ee101c52485761d63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.931ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} (M)}" loading="lazy"></span> von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> ist der kleinste simpliziale Teilkomplex von <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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>, der jedes 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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> enthält. Man definiert <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 {cl} (M)=\{\sigma \in {\mathcal {S}}:\sigma {\text{ ist eine Seite eines Simplex in }}M\}}">

</mrow>

<mtext> ist eine Seite eines Simplex in </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} (M)=\{\sigma \in {\mathcal {S}}:\sigma {\text{ ist eine Seite eines Simplex in }}M\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1cd5d7af11dade55a5566b8eadccc34700d93d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.105ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} (M)=\{\sigma \in {\mathcal {S}}:\sigma {\text{ ist eine Seite eines Simplex in }}M\}}" loading="lazy"></span>. Der Abschluss entsteht, indem man zu jedem 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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> all seine Seiten (<i>Facetten</i>) hinzufügt.

</p>

<div class="mw-heading mw-heading3"><h3 id="Stern">Stern</h3></div>

<p>Der <a href="Stern_(Topologie)" title="Stern (Topologie)">Stern</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 \operatorname {st} (M)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {st} (M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa02731abde0c306c9fa15ca8b2df0a460b28f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \operatorname {st} (M)}" loading="lazy"></span> von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> ist der Abschluss aller Simplizes, die eine Seite 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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> besitzen. Man definiert <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 {st} (M)=\operatorname {cl} (\{\sigma \in {\mathcal {S}}:\sigma {\text{ besitzt eine Seite in }}M\})}">

</mrow>

<mtext> besitzt eine Seite in </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {st} (M)=\operatorname {cl} (\{\sigma \in {\mathcal {S}}:\sigma {\text{ besitzt eine Seite in }}M\})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08a68019b83f3620a95d69419ff77a3bca362f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.841ex; height:2.843ex;" alt="{\displaystyle \operatorname {st} (M)=\operatorname {cl} (\{\sigma \in {\mathcal {S}}:\sigma {\text{ besitzt eine Seite in }}M\})}" loading="lazy"></span>. Den Stern kann man verstehen als die <i>kleinste simpliziale Umgebung</i> von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> 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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>. Weiterhin bildet <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 {\overset {\circ }{\operatorname {st} }}(\operatorname {cl} (M))}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overset {\circ }{\operatorname {st} }}(\operatorname {cl} (M))}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52a79ba397f3ffb34f92a11263219875a6cff3b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.561ex; height:3.843ex;" alt="{\displaystyle {\overset {\circ }{\operatorname {st} }}(\operatorname {cl} (M))}" loading="lazy"></span> eine offene simpliziale Umgebung von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> 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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>.

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {lk} (M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4f2f0a1f32a70b5bb74d31aa85a5bcabb9190ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.126ex; height:2.843ex;" alt="{\displaystyle \operatorname {lk} (M)}" loading="lazy"></span> besteht aus allen Simplizes im Stern von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, die kein Simplex von <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> treffen. Man definiert: <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 {lk} (M)=\{\sigma \in \operatorname {st} (M):\sigma \cap \tau =\emptyset {\text{ für alle }}\tau \in M\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {lk} (M)=\{\sigma \in \operatorname {st} (M):\sigma \cap \tau =\emptyset {\text{ für alle }}\tau \in M\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/763901f9079b9e9b0a4e6c78a5e5276eecc24e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:47.86ex; height:3.343ex;" alt="{\displaystyle \operatorname {lk} (M)=\{\sigma \in \operatorname {st} (M):\sigma \cap \tau =\emptyset {\text{ für alle }}\tau \in M\}}" loading="lazy"></span>. Den Link kann man als den <a href="Rand_(Topologie)" title="Rand (Topologie)">topologischen Rand</a> der simplizialen Umgebung auffassen.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Simpliziale_Abbildungen">Simpliziale Abbildungen</h2></div>

<p>Eine simpliziale Abbildung <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\colon {\mathcal {K}}\to {\mathcal {L}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37761ecfd742bf260a827deb4eb5d0db8ba05b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.301ex; height:2.509ex;" alt="{\displaystyle f\colon {\mathcal {K}}\to {\mathcal {L}}}" loading="lazy"></span> ist eine Abbildung zwischen den Eckenmengen <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\colon V({\mathcal {K}})\to V({\mathcal {L}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon V({\mathcal {K}})\to V({\mathcal {L}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5fa9eb2304b1f1757e936b2412bd946be7aca99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.494ex; height:2.843ex;" alt="{\displaystyle f\colon V({\mathcal {K}})\to V({\mathcal {L}})}" loading="lazy"></span>, bei der für jedes Simplex aus <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> dessen Ecken unter der Abbildung <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> auf die Ecken eines 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 {L}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"></span> abgebildet werden.<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><p>Eine simpliziale Abbildung <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\colon {\mathcal {K}}\to {\mathcal {L}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37761ecfd742bf260a827deb4eb5d0db8ba05b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.301ex; height:2.509ex;" alt="{\displaystyle f\colon {\mathcal {K}}\to {\mathcal {L}}}" loading="lazy"></span> induziert eine stetige Abbildung <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 \vert f\vert \colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vert f\vert \colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190b681ef77d09e083f1983fefeb815e1048f745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.182ex; height:2.843ex;" alt="{\displaystyle \vert f\vert \colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }" loading="lazy"></span>. Dazu wird im Inneren jedes geometrischen Simplex eine <a href="Affine_Abbildung" title="Affine Abbildung">affin lineare</a> Fortsetzung konstruiert.

</p><p>Umgekehrt lässt sich eine stetige Abbildung <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 g\colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0173eb79582ab925bf098ff502d450c9145ba10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.726ex; height:2.843ex;" alt="{\displaystyle g\colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }" loading="lazy"></span> nach endlich vielen <a href="Baryzentrische_Unterteilung" title="Baryzentrische Unterteilung">baryzentrischen Unterteilungen</a> durch eine simpliziale Abbildung <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\colon \operatorname {Bd} ^{m}({\mathcal {K}})\to {\mathcal {L}}}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \operatorname {Bd} ^{m}({\mathcal {K}})\to {\mathcal {L}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f35aec9dc5d36c30590a75338df581c48c9e5ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.724ex; height:2.843ex;" alt="{\displaystyle f\colon \operatorname {Bd} ^{m}({\mathcal {K}})\to {\mathcal {L}}}" loading="lazy"></span> approximieren, siehe <a href="Simplizialer_Approximationssatz" class="mw-redirect" title="Simplizialer Approximationssatz">simplizialer Approximationssatz</a>. Hierbei steht <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 {Bd} }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edfbbce06432bed414c6997b0e3cc0479161e375.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.938ex; height:2.176ex;" alt="{\displaystyle \operatorname {Bd} }" loading="lazy"></span> für die baryzentrische Unterteilung.

</p><p>Eine simpliziale Abbildung, die <a href="Bijektion" class="mw-redirect" title="Bijektion">bijektiv</a> ist, das heißt, die <a href="Umkehrabbildung" class="mw-redirect" title="Umkehrabbildung">Umkehrabbildung</a> ist auch eine simpliziale Abbildung, nennt man einen simplizialen <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Der_Simplizialkomplex_als_Kettenkomplex">Der Simplizialkomplex als Kettenkomplex</h2></div>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> ein endlicher Simplizialkomplex. Die <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 p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-te simpliziale Gruppe von <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> ist die <a href="Freie_abelsche_Gruppe" title="Freie abelsche Gruppe">freie abelsche Gruppe</a>, die von der Menge der Simplizes mit 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 p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> erzeugt wird, sie wird mit <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 C_{p}^{\Delta }({\mathcal {K}})}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{p}^{\Delta }({\mathcal {K}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9c4e56f0c63b8440dfb313072877857a818c52d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.979ex; height:3.176ex;" alt="{\displaystyle C_{p}^{\Delta }({\mathcal {K}})}" loading="lazy"></span> notiert. Die Elemente der Gruppe heißen simpliziale <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 p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-Ketten. Wählt man eine <a href="Totale_Ordnung" class="mw-redirect" title="Totale Ordnung">totale Ordnung</a> für alle Ecken, die in irgendeinem Simplex von <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}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.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> liegen, so erhält man durch Einschränkung auch eine Ordnung für jedes einzelne <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 p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-Simplex. Ein <a href="Randoperator" class="mw-redirect" title="Randoperator">Randoperator</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 \partial \colon C_{p}^{\Delta }({\mathcal {K}})\to C_{p-1}^{\Delta }({\mathcal {K}})}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial \colon C_{p}^{\Delta }({\mathcal {K}})\to C_{p-1}^{\Delta }({\mathcal {K}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e3c7bfb138a852fbe42b02ebe57274090a24e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.347ex; height:3.509ex;" alt="{\displaystyle \partial \colon C_{p}^{\Delta }({\mathcal {K}})\to C_{p-1}^{\Delta }({\mathcal {K}})}" loading="lazy"></span> wird dann definiert durch

</p>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial (\langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle ):=\sum _{i=0}^{p}(-1)^{i}\langle v_{k_{0}},\ldots ,v_{k_{i-1}},v_{k_{i+1}},\ldots ,v_{k_{p}}\rangle ,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9120cd2589896535a69f5570a571977ed5b792f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.934ex; height:7.009ex;" alt="{\displaystyle \partial (\langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle ):=\sum _{i=0}^{p}(-1)^{i}\langle v_{k_{0}},\ldots ,v_{k_{i-1}},v_{k_{i+1}},\ldots ,v_{k_{p}}\rangle ,}" loading="lazy"></span></dd></dl>

<p>wobei <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 \langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle }">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/300cb060cc453d57ab639198b69eaebca0984298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.087ex; height:3.176ex;" alt="{\displaystyle \langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle }" loading="lazy"></span> das aus den Ecken erzeugte Gruppenelement meint. Für den Randoperator gilt <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 \partial (\partial c)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial (\partial c)=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/817b47ca1db3c9437f61b2bfec8c02d6c5b0f102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.713ex; height:2.843ex;" alt="{\displaystyle \partial (\partial c)=0}" loading="lazy"></span> für alle simplizialen <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 p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-Ketten <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 c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>. Daher ist <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 (C_{p}^{\Delta }({\mathcal {K}}),\partial )}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (C_{p}^{\Delta }({\mathcal {K}}),\partial )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9282bd0dd6aec94402538228e5b8026c70db39f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.141ex; height:3.176ex;" alt="{\displaystyle (C_{p}^{\Delta }({\mathcal {K}}),\partial )}" loading="lazy"></span> ein <a href="Kettenkomplex" title="Kettenkomplex">Kettenkomplex</a> und man kann auf gewohnte Weise auf diesem eine <a href="Homologie_(Mathematik)" class="mw-redirect" title="Homologie (Mathematik)">Homologie</a> erklären. Diese Homologie wird <a href="Simpliziale_Homologie" title="Simpliziale Homologie">simpliziale Homologie</a> genannt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Anwendung_in_der_Graphentheorie">Anwendung in der Graphentheorie</h2></div>

<p>Man kann einem <a href="Graphentheorie" title="Graphentheorie">Graphen</a> Simplizialkomplexe zuweisen, um so untere Schranken an die <a href="F%C3%A4rbung_(Graphentheorie)" title="Färbung (Graphentheorie)">chromatische Zahl</a> zu beweisen. Wahrscheinlich am bekanntesten sind die Nachbarschaftskomplexe von <a href="L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a>.

</p>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Topologische_Kombinatorik#Anwendungen_von_Methoden_aus_der_Topologie_auf_Probleme_der_diskreten_Mathematik" title="Topologische Kombinatorik">Topologische_Kombinatorik#Anwendungen von Methoden aus der Topologie auf Probleme der diskreten Mathematik</a></i></div>

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

<p>Triangulierungen und ein in Matrixschreibweise formuliertes Äquivalent zu dem daraus gebildeten Kettenkomplex wurden von <a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> gegen Ende des neunzehnten Jahrhunderts untersucht. Simplizale Abbildungen wurde erstmals 1912 von <a href="Luitzen_Egbertus_Jan_Brouwer" title="Luitzen Egbertus Jan Brouwer">Brouwer</a> verwendet. In den 1920er-Jahren entstand dann die Sichtweise, die zum Begriff des Kettenkomplexes führte.<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>

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Simpliziales_Polytop" title="Simpliziales Polytop">Simpliziales Polytop</a></li>

<li><a href="Simpliziale_Menge" title="Simpliziale Menge">Simpliziale Menge</a></li>

<li><a href="Simpliziale_Homologie" title="Simpliziale Homologie">Simpliziale Homologie</a></li>

<li><a href="CW-Komplex" class="mw-redirect" title="CW-Komplex">CW-Komplex</a> – ein allgemeinerer Begriff, der Simplizialkomplexe als Spezialfall umfasst.</li>

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

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Heinz_Hopf" title="Heinz Hopf">H. Hopf</a>, <a href="Pawel_Sergejewitsch_Alexandrow" title="Pawel Sergejewitsch Alexandrow">P. Alexandroff</a>: <i>Topologie</i>, Berlin, 1935, S. 158 (<a rel="nofollow" class="external text" href="https://resolver.sub.uni-goettingen.de/purl?PPN379157292">online</a>)</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Herbert Seifert, William Threlfall: <cite style="font-style:italic">Lehrbuch der Topologie</cite>. Hrsg.: AMS Chelsea Publ. 2004, ISBN 978-0-8218-3595-1, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>47</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Simplizialkomplex&rft.au=Herbert+Seifert%2C+William+Threlfall&rft.btitle=Lehrbuch+der+Topologie&rft.date=2004-01-20&rft.genre=book&rft.isbn=9780821835951&rft.pages=47" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Fridtjof Toenniessen: <cite style="font-style:italic">Topologie: Ein Lesebuch von den elementaren Grundlagen bis zur Homologie und Kohomologie</cite>. 1. Auflage. Springer Spektrum, 2017, ISBN 978-3-662-54963-6, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>163–164</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Simplizialkomplex&rft.au=Fridtjof+Toenniessen&rft.btitle=Topologie%3A+Ein+Lesebuch+von+den+elementaren+Grundlagen+bis+zur+Homologie+und+Kohomologie&rft.date=2017-08&rft.edition=1&rft.genre=book&rft.isbn=9783662549636&rft.pages=163-164&rft.pub=Springer+Spektrum" style="display:none"> </span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">H. Hopf, P. Alexandroff: <i>Topologie</i>, Berlin, 1935, S. 172 (<a rel="nofollow" class="external text" href="https://resolver.sub.uni-goettingen.de/purl?PPN379157292">online</a>)</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Jean Dieudonné: <i>A History of Algebraic and Differential Topology 1900-1960</i>, S. 4–6, Boston 1989, Reprint 2009, ISBN 978-0-8176-4906-7, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-0-8176-4907-4">10.1007/978-0-8176-4907-4</a></span></span>

</li>

</ol>

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

<ul><li>John M. Lee: <i>Introduction to Topological Manifolds.</i> Springer-Verlag, New York NY u. a. 2000, ISBN 0-387-98759-2 (<i>Graduate Texts in Mathematics</i> 202), Seiten 96, 323–324</li>

<li>S. N. Malygin & M. M. Postnikov: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Simplicial complex</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Simplicial_complex">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Simplizialkomplex&rft.atitle=Simplicial+complex&rft.au=S.+N.+Malygin+%26+M.+M.+Postnikov&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul>

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

<ul><li><a href="J%C3%B6rg_Bewersdorff" title="Jörg Bewersdorff">Jörg Bewersdorff</a>: <a rel="nofollow" class="external text" href="http://www.galois-theorie.de/pdf/algebraische-topologie-fixpunkte.pdf">Algebraische Topologie und Fixpunkte</a>. Einführender Überblicksartikel (PDF-Datei; 179 kB).</li>

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</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304074658/http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/References/WuJie.pdf">Lecture Notes on Algebraic Topology: Simplicial Complexes (PDF; 713 kB)</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 4. März 2016 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>).</li></ul></div><div><div class="zim-footer">

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