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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Dieser_Artikel"><div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" id="bksicon" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Dieser Artikel behandelt den topologischen Begriff Homotopie. Für die daraus entstandene Begriffsbildung der homologischen Algebra siehe <a href="/wiki/Homotopie_(homologische_Algebra)" class="mw-redirect" title="Homotopie (homologische Algebra)">Homotopie (homologische Algebra)</a>. Zu homotopen Gruppen in der <a href="/wiki/Chemie" title="Chemie">Chemie</a> siehe <a href="/wiki/Topizit%C3%A4t" title="Topizität">Topizität</a>.</div> </div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Mug_and_Torus_morph.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Mug_and_Torus_morph.gif/200px-Mug_and_Torus_morph.gif" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gif 1.5x" data-file-width="240" data-file-height="240" /></a><figcaption>Eine Homotopie, die eine Kaffeetasse in einen Donut (einen <a href="/wiki/Volltorus" title="Volltorus">Volltorus</a>) überführt.</figcaption></figure> In der <a href="/wiki/Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a> ist eine Homotopie (von <a href="/wiki/Neugriechische_Sprache" title="Neugriechische Sprache">griechisch</a> ὁμός <style data-mw-deduplicate="TemplateStyles:r183723573">.mw-parser-output .Latn{font-family:"Akzidenz Grotesk","Arial","Avant Garde Gothic","Calibri","Futura","Geneva","Gill Sans","Helvetica","Lucida Grande","Lucida Sans Unicode","Lucida Grande","Stone Sans","Tahoma","Trebuchet","Univers","Verdana"}</style>homos ‚gleich‘ und τόπος tópos ‚Ort‘, ‚Platz‘) eine <a href="/wiki/Stetige_Funktion" title="Stetige Funktion">stetige</a> Deformation zwischen zwei <a href="/wiki/Abbildung_(Mathematik)" class="mw-redirect" title="Abbildung (Mathematik)">Abbildungen</a> von einem <a href="/wiki/Topologischer_Raum" title="Topologischer Raum">topologischen Raum</a> in einen anderen, beispielsweise die Deformation einer <a href="/wiki/Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurve</a> in eine andere Kurve. Eine Anwendung von Homotopie ist die Definition der <a href="/wiki/Homotopiegruppe" title="Homotopiegruppe">Homotopiegruppen</a>, welche wichtige <a href="/wiki/Invariante_(Mathematik)" title="Invariante (Mathematik)">Invarianten</a> in der <a href="/wiki/Algebraische_Topologie" title="Algebraische Topologie">algebraischen Topologie</a> sind. Der Begriff „Homotopie“ bezeichnet sowohl die Eigenschaft zweier Abbildungen, zueinander homotop (präferiert) zu sein, als auch die Abbildung („stetige Deformation“), die diese Eigenschaft vermittelt. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition">1 Definition</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Eigenschaften">1.1 Eigenschaften</a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#Beispiel">2 Beispiel</a></li> <li class="toclevel-1 tocsection-4"><a href="#Relative_Homotopie">3 Relative Homotopie</a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Beispiel:_Die_Fundamentalgruppe">3.1 Beispiel: Die Fundamentalgruppe</a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Homotopieäquivalenz">4 Homotopieäquivalenz</a></li> <li class="toclevel-1 tocsection-7"><a href="#Isotopie">5 Isotopie</a> <ul> <li class="toclevel-2 tocsection-8"><a href="#Definition_2">5.1 Definition</a></li> <li class="toclevel-2 tocsection-9"><a href="#Beispiele">5.2 Beispiele</a></li> <li class="toclevel-2 tocsection-10"><a href="#Unterschied_zur_Homotopie">5.3 Unterschied zur Homotopie</a></li> <li class="toclevel-2 tocsection-11"><a href="#Anwendungen">5.4 Anwendungen</a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Kettenhomotopie">6 Kettenhomotopie</a></li> <li class="toclevel-1 tocsection-13"><a href="#Punktierte_Homotopie">7 Punktierte Homotopie</a></li> <li class="toclevel-1 tocsection-14"><a href="#Literatur">8 Literatur</a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise">9 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=1" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Eine Homotopie zwischen zwei <a href="/wiki/Stetige_Funktion" title="Stetige Funktion">stetigen Abbildungen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99c3354a919e8e246796bdc329e13c6604e95a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.83ex; height:2.509ex;" alt="{\displaystyle f,g\colon X\to Y}"> ist eine stetige Abbildung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\colon X\times {[0,1]}\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\colon X\times {[0,1]}\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd0ab72cf69e6757265eafe0b2788b32a877460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.958ex; height:2.843ex;" alt="{\displaystyle H\colon X\times {[0,1]}\to Y}"></dd></dl> mit der Eigenschaft für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"> gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,0)=f(x)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,0)=f(x)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209662245d78c6a066454cd50eee20aa0abb82e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.238ex; height:2.843ex;" alt="{\displaystyle H(x,0)=f(x)\quad }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad H(x,1)=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad H(x,1)=g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc95d14ab152091d12de57eb76ff38c57770667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.075ex; height:2.843ex;" alt="{\displaystyle \quad H(x,1)=g(x)}"></dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"> das <a href="/wiki/Intervall_(Mathematik)#Abgeschlossenes_Intervall" title="Intervall (Mathematik)">Einheitsintervall</a> ist. Der erste Parameter entspricht also dem der ursprünglichen Abbildungen und der zweite gibt den Grad der <a href="/wiki/Verformung" title="Verformung">Deformation</a> an. Eine Homotopie definiert eine ein-parametrige Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (H_{t}(x))_{0\leq t\leq 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (H_{t}(x))_{0\leq t\leq 1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfdacb15e17b4fd22276496c15ba5531edd01dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.732ex; height:2.843ex;" alt="{\displaystyle (H_{t}(x))_{0\leq t\leq 1}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{t}(x):=H(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{t}(x):=H(x,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2953c28b467449ba2139801015caab0ec6599342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.718ex; height:2.843ex;" alt="{\displaystyle H_{t}(x):=H(x,t)}">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}(x)=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c03cbed27293611463ad886d5fc3f8ff304fa45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.64ex; height:2.843ex;" alt="{\displaystyle H_{0}(x)=f(x)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}(x)=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}(x)=g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705f501c5d1e2463a227ac3a2ec6217a653dbe51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.478ex; height:2.843ex;" alt="{\displaystyle H_{1}(x)=g(x)}">. Besonders anschaulich wird die Definition, wenn man sich den zweiten Parameter als „Zeit“ vorstellt (vgl. Bild). Äquivalent kann man eine Homotopie definieren als einen (stetigen) <a href="/wiki/Weg_(Mathematik)" title="Weg (Mathematik)">Weg</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> im Raum der stetigen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(X,Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc34e8f08e76555abf1181aae10f5cddd68fdd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.363ex; height:2.843ex;" alt="{\displaystyle C(X,Y)}"> mit der <a href="/wiki/Kompakt-offene_Topologie" class="mw-redirect" title="Kompakt-offene Topologie">kompakt-offenen Topologie</a>. Man sagt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> sei homotop zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> und schreibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\sim g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∼</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\sim g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc50b415ab7e2ffa27d032cdf150ab0984a44170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.493ex; height:2.509ex;" alt="{\displaystyle f\sim g}">. Homotopie ist eine <a href="/wiki/%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> auf der <a href="/wiki/Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> der stetigen Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290b16963d52e4a7995aae01ee854b97a6ea10c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\to Y}">, die zugehörigen <a href="/wiki/%C3%84quivalenzrelation#Äquivalenzklassen" title="Äquivalenzrelation">Äquivalenzklassen</a> heißen Homotopieklassen, die Menge dieser Äquivalenzklassen wird häufig mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94470b44d283fde62130212956058ca6b727da37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle [X,Y]}"> bezeichnet. Eine stetige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"> heißt nullhomotop, wenn sie homotop zu einer <a href="/wiki/Konstante_Funktion" title="Konstante Funktion">konstanten Abbildung</a> ist. <div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=2" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> <ul><li>Homotopierelationen bleiben unter <a href="/wiki/Komposition_(Mathematik)" title="Komposition (Mathematik)">Kompositionen</a> erhalten, das heißt wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\,f_{2}\colon Y\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},\,f_{2}\colon Y\to Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0687eee698408e2f83dcd629b6928863349aac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.91ex; height:2.509ex;" alt="{\displaystyle f_{1},\,f_{2}\colon Y\to Z}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},\,g_{2}\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1},\,g_{2}\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2749810c7d4e745de9e7d55ae3ada44b71fac502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.149ex; height:2.509ex;" alt="{\displaystyle g_{1},\,g_{2}\colon X\to Y}"> stetige Funktionen sind und</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\sim f_{2};\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}\sim f_{2};\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4163ca53dc610e5d0703dec8e92885e9a8d9af5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.455ex; height:2.509ex;" alt="{\displaystyle f_{1}\sim f_{2};\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}\sim g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}\sim g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f60a79d3b9385e47d059d98ddfb629e06de8132a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.425ex; height:2.009ex;" alt="{\displaystyle g_{1}\sim g_{2}}"></dd></dl></dd> <dd>gilt, dann gilt auch <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\circ g_{1}\sim f_{2}\circ g_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}\circ g_{1}\sim f_{2}\circ g_{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52db020724b73f87d08a7035affbe721edf0cdfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.848ex; height:2.509ex;" alt="{\displaystyle f_{1}\circ g_{1}\sim f_{2}\circ g_{2}.}"><a href="#cite_note-1">[1]</a></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=3" title="Abschnitt bearbeiten: Beispiel" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Beispiel">Quelltext bearbeiten</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Homotopie_Bsp.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Homotopie_Bsp.png/150px-Homotopie_Bsp.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Homotopie_Bsp.png/225px-Homotopie_Bsp.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Homotopie_Bsp.png/300px-Homotopie_Bsp.png 2x" data-file-width="406" data-file-height="406" /></a><figcaption>Homotopie eines Kreises in R² auf einen Punkt</figcaption></figure> Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=S^{1}\subset \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=S^{1}\subset \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5094ce24c991686f1e49d1f3a72df4d9c51f590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.485ex; height:2.676ex;" alt="{\displaystyle X=S^{1}\subset \mathbb {R} ^{2}}"> der <a href="/wiki/Einheitskreis" title="Einheitskreis">Einheitskreis</a> in der Ebene und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1e27cdeee04fafff6bee93ae6147534f1606bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.604ex; height:2.676ex;" alt="{\displaystyle Y=\mathbb {R} ^{2}}"> die ganze Ebene. Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> sei die <a href="/wiki/Einbettung_(Mathematik)" title="Einbettung (Mathematik)">Einbettung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}">, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> sei die Abbildung, die ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> auf den <a href="/wiki/Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Ursprung</a> abbildet, also <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f690285952308aa49e3c6aac892df31cad6d1b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle f(x)=x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d818269c4f3ae277985d810b77037d965e6994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.517ex; height:2.509ex;" alt="{\displaystyle g\colon X\to Y}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9bb64a9a8234edd8fb1c622dddfab478719776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.843ex;" alt="{\displaystyle g(x)=0}">.</dd></dl> Dann sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> zueinander homotop. Denn <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\colon X\times [0,1]\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\colon X\times [0,1]\to \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381acb4ab217a845d31e78b02a437016ac4f4912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.917ex; height:3.176ex;" alt="{\displaystyle H\colon X\times [0,1]\to \mathbb {R} ^{2}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,t)=(1-t)\cdot f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,t)=(1-t)\cdot f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e401e8a017f7670c3b3755d18f6ad582f514d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.923ex; height:2.843ex;" alt="{\displaystyle H(x,t)=(1-t)\cdot f(x)}"></dd></dl> ist stetig und erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,0)=1\cdot f(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,0)=1\cdot f(x)=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70ebb8ec78c56866ae474d32803a067059f7e46a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.273ex; height:2.843ex;" alt="{\displaystyle H(x,0)=1\cdot f(x)=f(x)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,1)=0\cdot f(x)=0=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,1)=0\cdot f(x)=0=g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ec45e2a55a771645139025faa5eeda46cd2d14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.371ex; height:2.843ex;" alt="{\displaystyle H(x,1)=0\cdot f(x)=0=g(x)}">. <div class="mw-heading mw-heading2"><h2 id="Relative_Homotopie">Relative Homotopie</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=4" title="Abschnitt bearbeiten: Relative Homotopie" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Relative Homotopie">Quelltext bearbeiten</a>]</div> Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> eine Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}">, und stimmen zwei stetige Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99c3354a919e8e246796bdc329e13c6604e95a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.83ex; height:2.509ex;" alt="{\displaystyle f,g\colon X\to Y}"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> überein, so heißen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> homotop relativ zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">, wenn es eine Homotopie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\colon f\sim g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mi>f</mi> <mo>∼</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\colon f\sim g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02328396bad6fabfeb6998df4fd4ee4c9c243ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.591ex; height:2.509ex;" alt="{\displaystyle H\colon f\sim g}"> gibt, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(e,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(e,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1f94e419babf6590e05a0fedc9c6e2b03f4bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.83ex; height:2.843ex;" alt="{\displaystyle H(e,t)}"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34778736d9c6d607a4da3d25594b38dd3e8c82ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}"> unabhängig von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> ist. <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Homotopy_curves.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Homotopy_curves.svg/240px-Homotopy_curves.svg.png" decoding="async" width="240" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Homotopy_curves.svg/360px-Homotopy_curves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Homotopy_curves.svg/480px-Homotopy_curves.svg.png 2x" data-file-width="377" data-file-height="245" /></a><figcaption>Homotopie zweier Kurven</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:HomotopySmall.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7e/HomotopySmall.gif" decoding="async" width="220" height="161" class="mw-file-element" data-file-width="220" data-file-height="161" /></a><figcaption>Die beiden hier gezeigten gestrichelten Wege sind relativ zu ihren Endpunkten homotop. Die Animation repräsentiert eine mögliche Homotopie.</figcaption></figure> Ein wichtiger Spezialfall ist die Homotopie von Wegen relativ der Endpunkte: Ein Weg ist eine stetige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon [0,1]\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \colon [0,1]\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ca01f43292d9201ba7b9d04bfff1dc493670d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.543ex; height:2.843ex;" alt="{\displaystyle \gamma \colon [0,1]\to X}">; dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"> das Einheitsintervall. Zwei Wege heißen homotop relativ der Endpunkte, wenn sie homotop relativ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28de5781698336d21c9c560fb1cbb3fb406923eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{0,1\}}"> sind, d. h. wenn die Homotopie die Anfangs- und Endpunkte festhält. (Sonst wären Wege in der gleichen <a href="/wiki/Zusammenh%C3%A4ngender_Raum" title="Zusammenhängender Raum">Wegzusammenhangskomponente</a> immer homotop.) Sind also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407f2fbc64e21bb432f397fdb816de0f5083d900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2d6da217f4d6e937887bad1c90b371314830f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{1}}"> zwei Wege in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}(0)=\gamma _{1}(0)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}(0)=\gamma _{1}(0)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4903722fafaf3b600524a1f05867cce0404a2aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.987ex; height:2.843ex;" alt="{\displaystyle \gamma _{0}(0)=\gamma _{1}(0)=x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}(1)=\gamma _{1}(1)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}(1)=\gamma _{1}(1)=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b7af9923ae39994a50cfba33fadf632ef6a806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.813ex; height:2.843ex;" alt="{\displaystyle \gamma _{0}(1)=\gamma _{1}(1)=y}">, so ist eine Homotopie relativ der Endpunkte zwischen ihnen eine stetige Abbildung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:[0,1]\times [0,1]\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H:[0,1]\times [0,1]\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8815746134663607d6871cf0682b6ae486ac69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.534ex; height:2.843ex;" alt="{\displaystyle H:[0,1]\times [0,1]\to Y}"></dd></dl> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(t,0)=\gamma _{0}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(t,0)=\gamma _{0}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0374607d0c6c14be535d38412ee79b4ee2fc6022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.915ex; height:2.843ex;" alt="{\displaystyle H(t,0)=\gamma _{0}(t)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(t,1)=\gamma _{1}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(t,1)=\gamma _{1}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebd4418970ad8b361d7884f1c409419b49b66c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.915ex; height:2.843ex;" alt="{\displaystyle H(t,1)=\gamma _{1}(t)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(0,s)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(0,s)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aaa5a4591db71505c04cf6a86ff71cf3c0d4359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.588ex; height:2.843ex;" alt="{\displaystyle H(0,s)=x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(1,s)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(1,s)=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb2379a97b24c039dbd6441ac7547ec76ec7c4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.414ex; height:2.843ex;" alt="{\displaystyle H(1,s)=y}">. Ein Weg heißt nullhomotop genau dann, wenn er homotop zum konstanten Weg <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (t)=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (t)=x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df9f839b608d6efe2fff59e6a05600831476673b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.394ex; height:2.843ex;" alt="{\displaystyle \gamma (t)=x_{0}}"> ist. Der andere häufig auftretende Fall ist die Homotopie von Abbildungen zwischen <a href="/wiki/Punktierter_topologischer_Raum" title="Punktierter topologischer Raum">punktierten Räumen</a>. Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,x_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e4e34222493782d38d182b5e25bccd49b9fc3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (X,x_{0})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y,y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,y_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5ddd57c24e692df974226e85938d4f0224ca41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.81ex; height:2.843ex;" alt="{\displaystyle (Y,y_{0})}"> punktierte Räume, so sind zwei stetige Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd4d4df56ea3c15674f3f837c750d98568a87f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.094ex; height:2.843ex;" alt="{\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}"> homotop als Abbildungen von punktierten Räumen, wenn sie relativ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> homotop sind. <div class="mw-heading mw-heading3"><h3 id="Beispiel:_Die_Fundamentalgruppe">Beispiel: Die Fundamentalgruppe</h3>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=5" title="Abschnitt bearbeiten: Beispiel: Die Fundamentalgruppe" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Beispiel: Die Fundamentalgruppe">Quelltext bearbeiten</a>]</div> Die Menge der Homotopieklassen von Abbildungen punktierter Räume von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S^{1},*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S^{1},*)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15eb9fd11d980416780d37210c8ccc7f2c786d4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.582ex; height:3.176ex;" alt="{\displaystyle (S^{1},*)}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,x_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e4e34222493782d38d182b5e25bccd49b9fc3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (X,x_{0})}"> ist die <a href="/wiki/Fundamentalgruppe" title="Fundamentalgruppe">Fundamentalgruppe</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> zum Basispunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}">. Ist zum Beispiel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,x_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e4e34222493782d38d182b5e25bccd49b9fc3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (X,x_{0})}"> ein Kreis mit einem beliebigen ausgewählten Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}">, dann ist der Weg, der durch einmaliges Umrunden des Kreises beschrieben wird, nicht homotop zum Weg, den man durch Stillstehen am Ausgangspunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> erhält. <div class="mw-heading mw-heading2"><h2 id="Homotopieäquivalenz">Homotopieäquivalenz</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=6" title="Abschnitt bearbeiten: Homotopieäquivalenz" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Homotopieäquivalenz">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Homotopie%C3%A4quivalenz" title="Homotopieäquivalenz">Homotopieäquivalenz</a></div> Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> zwei topologische Räume und sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263df9f4ac00972d999b70dafb0a2f485531fa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.517ex; height:2.509ex;" alt="{\displaystyle g\colon Y\to X}"> stetige Abbildungen. Dann sind die Verknüpfungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"> jeweils stetige Abbildungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> auf sich selbst, und man kann versuchen, diese zur Identität auf X bzw. Y zu homotopieren. Falls es solche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> gibt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> homotop zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623b238853010f9319f926446bc6cb2253e53e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.572ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{X}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"> homotop zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{Y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f280c0425786426a2cac3d1df83e73a5cdb4d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.426ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{Y}}"> ist, so nennt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> homotopieäquivalent oder vom gleichen Homotopietyp. Die Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> heißen dann Homotopieäquivalenzen. Homotopieäquivalente Räume haben die meisten topologischen Eigenschaften gemeinsam. Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> homotopieäquivalent sind, so gilt <ul><li>falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> <a href="/wiki/Zusammenh%C3%A4ngender_Raum#Wegzusammenhängend" title="Zusammenhängender Raum">wegzusammenhängend</a>, so auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}">.</li> <li>falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> wegzusammenhängend, so sind die <a href="/wiki/Fundamentalgruppe" title="Fundamentalgruppe">Fundamentalgruppen</a> und die höheren <a href="/wiki/Homotopiegruppe" title="Homotopiegruppe">Homotopiegruppen</a> isomorph.</li> <li>die <a href="/wiki/Axiomatische_Homologie" title="Axiomatische Homologie">Homologie</a>- und <a href="/wiki/Kohomologie" title="Kohomologie">Kohomologiegruppen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> sind gleich.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> sind <a href="/wiki/Retrakt#Deformationsretrakt" class="mw-redirect" title="Retrakt">Deformationsretrakte</a> eines topologischen Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}">.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Isotopie"> Isotopie</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=7" title="Abschnitt bearbeiten: Isotopie" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Isotopie">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=8" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Wenn zwei gegebene homotope Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d818269c4f3ae277985d810b77037d965e6994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.517ex; height:2.509ex;" alt="{\displaystyle g\colon X\to Y}"> zu einer bestimmten <a href="/wiki/Regularit%C3%A4tsklasse" class="mw-redirect" title="Regularitätsklasse">Regularitätsklasse</a> gehören oder andere zusätzliche Eigenschaften besitzen, kann man sich fragen, ob die beiden innerhalb dieser Klasse durch einen Weg miteinander verbunden werden können. Dies führt zum Konzept der Isotopie. Eine Isotopie ist eine Homotopie <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\colon X\times [0,1]\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\colon X\times [0,1]\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92c51bbfba9192c5ddfe1c6fef25ffc5365a99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.958ex; height:2.843ex;" alt="{\displaystyle H\colon X\times [0,1]\to Y}"></dd></dl> wie oben, wobei alle Zwischenabbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{t}:=H(\cdot ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{t}:=H(\cdot ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e1d058991ffec354e2c3d573c392a5c67a8b4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.896ex; height:2.843ex;" alt="{\displaystyle H_{t}:=H(\cdot ,t)}"> (für festes t) ebenfalls die geforderten Zusatzeigenschaften besitzen sollen. Die zugehörigen Äquivalenzklassen heißen Isotopieklassen. <div class="mw-heading mw-heading3"><h3 id="Beispiele">Beispiele</h3>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=9" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Beispiele">Quelltext bearbeiten</a>]</div> Zwei <a href="/wiki/Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismen</a> sind also isotop, wenn eine Homotopie existiert, so dass alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e33c7739bb90406c5b8dd53d3f9cb82b071ac2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.757ex; height:2.509ex;" alt="{\displaystyle H_{t}}"> Homöomorphismen sind. Zwei <a href="/wiki/Diffeomorphismus" title="Diffeomorphismus">Diffeomorphismen</a> sind isotop, wenn alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e33c7739bb90406c5b8dd53d3f9cb82b071ac2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.757ex; height:2.509ex;" alt="{\displaystyle H_{t}}"> selbst Diffeomorphismen sind. (Man bezeichnet sie dann auch als diffeotop.) Zwei <a href="/wiki/Einbettung_(Mathematik)" title="Einbettung (Mathematik)">Einbettungen</a> sind isotop, wenn alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e33c7739bb90406c5b8dd53d3f9cb82b071ac2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.757ex; height:2.509ex;" alt="{\displaystyle H_{t}}"> Einbettungen sind. <div class="mw-heading mw-heading3"><h3 id="Unterschied_zur_Homotopie">Unterschied zur Homotopie</h3>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=10" title="Abschnitt bearbeiten: Unterschied zur Homotopie" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Unterschied zur Homotopie">Quelltext bearbeiten</a>]</div> Zu verlangen, dass zwei Abbildungen isotop sind, kann tatsächlich eine stärkere Anforderung sein, als zu verlangen, dass sie homotop sind. Zum Beispiel ist der Homöomorphismus der <a href="/wiki/Einheitskreis" title="Einheitskreis">Einheitskreisscheibe</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}">, der durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=(-x,-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=(-x,-y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0881f76b6074592883e9b3c019b3ef769f4ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.65ex; height:2.843ex;" alt="{\displaystyle f(x,y)=(-x,-y)}"> definiert ist, dasselbe wie eine 180-Grad-Drehung um den Nullpunkt, darum sind die Identitätsabbildung und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> isotop, denn sie können durch Drehungen miteinander verbunden werden. Im Gegensatz dazu ist die Abbildung auf dem Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-1,1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-1,1\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79566f857ac1fcd0ef0f62226298a4ed15b796ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle \left[-1,1\right]}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=-x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a57f7495c03ae2b21e25aa228e60247b3f8f54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.654ex; height:2.843ex;" alt="{\displaystyle f(x)=-x}"> nicht isotop zur Identität. Das liegt daran, dass jede Homotopie der beiden Abbildungen zu einem bestimmten Zeitpunkt die beiden Endpunkte miteinander vertauschen muss; zu diesem Zeitpunkt werden sie auf denselben Punkt abgebildet und die entsprechende Abbildung ist kein Homöomorphismus. Hingegen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> homotop zur Identität, zum Beispiel durch die Homotopie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H''\colon \left[-1,1\right]\times \left[0,1\right]\to \left[-1,1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>″</mo> </msup> <mo>:</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>×</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H''\colon \left[-1,1\right]\times \left[0,1\right]\to \left[-1,1\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46388d26ccd800bb239972878e55eaace1902ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.303ex; height:3.009ex;" alt="{\displaystyle H''\colon \left[-1,1\right]\times \left[0,1\right]\to \left[-1,1\right]}">, gegeben durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,t)=2tx-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>t</mi> <mi>x</mi> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,t)=2tx-x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce90bbf1223c3bf6f8144d365a2bab3fd35183e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.676ex; height:2.843ex;" alt="{\displaystyle H(x,t)=2tx-x}">. <div class="mw-heading mw-heading3"><h3 id="Anwendungen">Anwendungen</h3>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=11" title="Abschnitt bearbeiten: Anwendungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Anwendungen">Quelltext bearbeiten</a>]</div> In der <a href="/wiki/Geometrische_Topologie" title="Geometrische Topologie">Geometrischen Topologie</a> werden Isotopien benutzt, um Äquivalenzrelationen herzustellen. Zum Beispiel in der <a href="/wiki/Knotentheorie" title="Knotentheorie">Knotentheorie</a> – wann sind zwei Knoten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e1b324cf5b68f2729a8634ff76e396b634b75d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{2}}"> als gleich zu betrachten? Die intuitive Idee, den einen Knoten in den anderen zu deformieren, führt dazu, dass man einen Weg von Homöomorphismen verlangt: Eine Isotopie, die mit der Identität des <a href="/wiki/Dreidimensionaler_Raum" class="mw-redirect" title="Dreidimensionaler Raum">dreidimensionalen Raumes</a> beginnt und bei einem Homöomorphismus h endet, so dass h den Knoten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}"> in den Knoten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e1b324cf5b68f2729a8634ff76e396b634b75d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{2}}"> überführt. Eine solche Isotopie des umgebenden Raumes wird ambiente Isotopie<a href="#cite_note-2">[2]</a> oder Umgebungsisotopie genannt. Eine andere wichtige Anwendung ist die Definition der <a href="/wiki/Abbildungsklassengruppe" title="Abbildungsklassengruppe">Abbildungsklassengruppe</a> Mod(M) einer <a href="/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeit</a> M. Man betrachtet Diffeomorphismen von M „bis auf Isotopie“, das heißt, dass Mod(M) die (<a href="/wiki/Diskretheit" class="mw-redirect" title="Diskretheit">diskrete</a>) <a href="/wiki/Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> der Diffeomorphismen von M ist, <a href="/wiki/Quotientengruppe" class="mw-redirect" title="Quotientengruppe">modulo</a> der Gruppe der Diffeomorphismen, die isotop zur Identität sind. Homotopie kann in der <a href="/wiki/Numerische_Mathematik" title="Numerische Mathematik">numerischen Mathematik</a> für eine robuste <a href="/wiki/Initialisierung" class="mw-disambig" title="Initialisierung">Initialisierung</a> zur Lösung von <a href="/wiki/Differential-algebraische_Gleichung" title="Differential-algebraische Gleichung">differential-algebraischen Gleichungen</a> eingesetzt werden (siehe <a href="/wiki/Homotopieverfahren" title="Homotopieverfahren">Homotopieverfahren</a>). <div class="mw-heading mw-heading2"><h2 id="Kettenhomotopie">Kettenhomotopie</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=12" title="Abschnitt bearbeiten: Kettenhomotopie" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Kettenhomotopie">Quelltext bearbeiten</a>]</div> Zwei <a href="/wiki/Kettenkomplex#Kettenhomomorphismus" title="Kettenkomplex">Kettenhomomorphismen</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\bullet },g_{\bullet }\colon (A_{\bullet },d_{A,\bullet })\to (B_{\bullet },d_{B,\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>,</mo> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>,</mo> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\bullet },g_{\bullet }\colon (A_{\bullet },d_{A,\bullet })\to (B_{\bullet },d_{B,\bullet })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79965de7d809421492eefb100c227d1e44f152eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.262ex; height:3.009ex;" alt="{\displaystyle f_{\bullet },g_{\bullet }\colon (A_{\bullet },d_{A,\bullet })\to (B_{\bullet },d_{B,\bullet })}"></dd></dl> zwischen <a href="/wiki/Kettenkomplex" title="Kettenkomplex">Kettenkomplexen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{\bullet },d_{A,\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>,</mo> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A_{\bullet },d_{A,\bullet })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e07c9b325b2fdfd57849b6c294975343619200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.594ex; height:3.009ex;" alt="{\displaystyle (A_{\bullet },d_{A,\bullet })}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B_{\bullet },d_{B,\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>,</mo> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B_{\bullet },d_{B,\bullet })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e756e5c016e30ac7d0087e0b60790471dd52b35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.629ex; height:3.009ex;" alt="{\displaystyle (B_{\bullet },d_{B,\bullet })}"> heißen kettenhomotop, wenn es einen Homomorphismus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\bullet }\colon (A_{\bullet })\to (B_{\bullet +1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\bullet }\colon (A_{\bullet })\to (B_{\bullet +1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d89c2565fcfba21a159ebf705a765e913faf1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.01ex; height:2.843ex;" alt="{\displaystyle K_{\bullet }\colon (A_{\bullet })\to (B_{\bullet +1})}"></dd></dl> mit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{B,{\bullet +1}}K_{\bullet }+K_{\bullet -1}d_{A,{\bullet }}=f_{\bullet }-g_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> <mo>+</mo> <mn>1</mn> </mrow> </mrow> </msub> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{B,{\bullet +1}}K_{\bullet }+K_{\bullet -1}d_{A,{\bullet }}=f_{\bullet }-g_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd5ce9dd6fc2aa9f81ee4629d5e37a6ed53e728d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.313ex; height:2.843ex;" alt="{\displaystyle d_{B,{\bullet +1}}K_{\bullet }+K_{\bullet -1}d_{A,{\bullet }}=f_{\bullet }-g_{\bullet }}"></dd></dl> gibt. Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99c3354a919e8e246796bdc329e13c6604e95a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.83ex; height:2.509ex;" alt="{\displaystyle f,g\colon X\to Y}"> homotope Abbildungen zwischen topologischen Räumen sind, dann sind die induzierten Abbildungen der <a href="/wiki/Singul%C3%A4rer_Kettenkomplex" class="mw-redirect" title="Singulärer Kettenkomplex">singulären Kettenkomplexe</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\bullet },g_{\bullet }\colon (C_{\bullet }(X),d_{\bullet })\to (C_{\bullet }(Y),d_{\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\bullet },g_{\bullet }\colon (C_{\bullet }(X),d_{\bullet })\to (C_{\bullet }(Y),d_{\bullet })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a2ad2be81f13b7d229e63b5d6d408fae92ad80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.055ex; height:2.843ex;" alt="{\displaystyle f_{\bullet },g_{\bullet }\colon (C_{\bullet }(X),d_{\bullet })\to (C_{\bullet }(Y),d_{\bullet })}"></dd></dl> kettenhomotop. <div class="mw-heading mw-heading2"><h2 id="Punktierte_Homotopie">Punktierte Homotopie</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=13" title="Abschnitt bearbeiten: Punktierte Homotopie" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Punktierte Homotopie">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Punktierter_topologischer_Raum" title="Punktierter topologischer Raum">Punktierter topologischer Raum</a></div> Zwei punktierte Abbildungen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd4d4df56ea3c15674f3f837c750d98568a87f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.094ex; height:2.843ex;" alt="{\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}"></dd></dl> heißen homotop, wenn es eine stetige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\colon X\times \left[0,1\right]\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\colon X\times \left[0,1\right]\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48afd513813e0305e9f63e3f0bb2138edaa35a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.958ex; height:2.843ex;" alt="{\displaystyle H\colon X\times \left[0,1\right]\to Y}"> mit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,0)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,0)=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/555928778a222ea2127bf985b8d9b06fba46c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.915ex; height:2.843ex;" alt="{\displaystyle H(x,0)=f(x)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,1)=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,1)=g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ee15cbc3c622c4c355ac0d49a859aa4d321705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.752ex; height:2.843ex;" alt="{\displaystyle H(x,1)=g(x)}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x_{0},t)=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x_{0},t)=y_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/438e412ee655e8e537f8029dbfd13ad229477fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.422ex; height:2.843ex;" alt="{\displaystyle H(x_{0},t)=y_{0}}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \left[0,1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \left[0,1\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd270c3bd356bcd89e081db8a147db4ac9552d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:2.843ex;" alt="{\displaystyle t\in \left[0,1\right]}"></dd></dl> gibt. Die Menge der Homotopieklassen punktierter Abbildungen wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[X,Y\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[X,Y\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec156cb9ce0e02cf471b32b77dadfa62679adb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle \left[X,Y\right]}"> bezeichnet. <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=14" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li>Brayton Gray: <cite style="font-style:italic">Homotopy theory. An introduction to algebraic topology</cite> (= <cite style="font-style:italic">Pure and Applied Mathematics</cite>. Nr. 64). Academic Press, New York u. a. 1975, <a href="/wiki/Spezial:ISBN-Suche/0122960505" class="internal mw-magiclink-isbn">ISBN 0-12-296050-5</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:Homotopie&rft.au=Brayton+Gray&rft.btitle=Homotopy+theory.+An+introduction+to+algebraic+topology&rft.date=1975&rft.genre=book&rft.isbn=0122960505&rft.place=New+York+u.+a.&rft.pub=Academic+Press&rft.series=Pure+and+Applied+Mathematics" style="display:none"> </li> <li>Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, Cambridge 2002, <a href="/wiki/Spezial:ISBN-Suche/0521795400" class="internal mw-magiclink-isbn">ISBN 0-521-79540-0</a> (<a rel="nofollow" class="external text" href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">cornell.edu</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:Homotopie&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2002&rft.genre=book&rft.isbn=0521795400&rft.place=Cambridge&rft.pub=Cambridge+University+Press" style="display:none"> </li> <li>John McCleary (Hrsg.): <cite style="font-style:italic">Higher Homotopy Structures in Topology and Mathematical Physics</cite>. Proceedings of an international Conference, June 13 – 15, 1996 at Vassar College, Poughkeepsie, New York, to Honor the sixtieth Birthday of Jim Stasheff (= <cite style="font-style:italic">Contemporary Mathematics</cite>. Band 227). American Mathematical Society, Providence RI 1999, <a href="/wiki/Spezial:ISBN-Suche/082180913X" class="internal mw-magiclink-isbn">ISBN 0-8218-0913-X</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:Homotopie&rft.btitle=Higher+Homotopy+Structures+in+Topology+and+Mathematical+Physics&rft.date=1999&rft.genre=book&rft.isbn=082180913X&rft.place=Providence+RI&rft.pub=American+Mathematical+Society&rft.series=Contemporary+Mathematics" style="display:none"> </li> <li>George W. Whitehead: <cite style="font-style:italic">Elements of Homotopy Theory</cite>. Corrected 3rd Printing (= <cite style="font-style:italic">Graduate Texts in Mathematics</cite>. Band 61). Springer, New York u. a. 1995, <a href="/wiki/Spezial:ISBN-Suche/0387903364" class="internal mw-magiclink-isbn">ISBN 0-387-90336-4</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:Homotopie&rft.au=George+W.+Whitehead&rft.btitle=Elements+of+Homotopy+Theory&rft.date=1995&rft.genre=book&rft.isbn=0387903364&rft.place=New+York+u.+a.&rft.pub=Springer&rft.series=Graduate+Texts+in+Mathematics" style="display:none"> </li> <li>M. Sielemann, F. Casella, M. Otter, C. Claus, J. Eborn, S. E. Mattsson, H. Olsson: <cite style="font-style:italic">Robust Initialization of Differential-Algebraic Equations Using Homotopy</cite>. International Modelica Conference, Dresden 2011, <a href="/wiki/Spezial:ISBN-Suche/9789173930963" class="internal mw-magiclink-isbn">ISBN 978-91-7393-096-3</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:Homotopie&rft.au=M.+Sielemann%2C+F.+Casella%2C+M.+Otter%2C+...&rft.btitle=Robust+Initialization+of+Differential-Algebraic+Equations+Using+Homotopy&rft.date=2011&rft.genre=book&rft.isbn=9789173930963&rft.place=Dresden&rft.pub=International+Modelica+Conference" style="display:none"> </li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Homotopie&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Homotopie&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> John M. Lee: <cite style="font-style:italic">Introduction to Smooth Manifolds</cite>. Hrsg.: Springer. 2. Auflage. S. 74–75.<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:Homotopie&rft.au=John+M.+Lee&rft.btitle=Introduction+to+Smooth+Manifolds&rft.edition=2&rft.genre=book&rft.pages=74-75" style="display:none">  </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="/wiki/Tammo_tom_Dieck" title="Tammo tom Dieck">Tammo tom Dieck</a>: <cite style="font-style:italic">Topologie</cite>. 2. Auflage. de Gruyter, Berlin 2000, <a href="/wiki/Spezial:ISBN-Suche/3110162369" class="internal mw-magiclink-isbn">ISBN 3-11-016236-9</a>, S. 277.<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:Homotopie&rft.au=Tammo+tom+Dieck&rft.btitle=Topologie&rft.date=2000&rft.edition=2&rft.genre=book&rft.isbn=3110162369&rft.pages=277&rft.place=Berlin&rft.pub=de+Gruyter" style="display:none">  </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Homotopie&oldid=232636412">https://de.wikipedia.org/w/index.php?title=Homotopie&oldid=232636412</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Algebraische_Topologie" title="Kategorie:Algebraische Topologie">Algebraische Topologie</a></li><li><a href="/wiki/Kategorie:Homotopietheorie" title="Kategorie:Homotopietheorie">Homotopietheorie</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Homotopie" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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mw-list-item"><a href="https://ast.wikipedia.org/wiki/Homotop%C3%ADa" title="Homotopía – Asturisch" lang="ast" hreflang="ast" data-title="Homotopía" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Homotopia" title="Homotopia – Katalanisch" lang="ca" hreflang="ca" data-title="Homotopia" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Homotopie" title="Homotopie – Tschechisch" lang="cs" hreflang="cs" data-title="Homotopie" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BC%CE%BF%CF%84%CE%BF%CF%80%CE%AF%CE%B1" title="Ομοτοπία – Griechisch" lang="el" hreflang="el" data-title="Ομοτοπία" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Homotopy" title="Homotopy – Englisch" lang="en" hreflang="en" data-title="Homotopy" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Homotopio" title="Homotopio – Esperanto" lang="eo" hreflang="eo" data-title="Homotopio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Homotop%C3%ADa" title="Homotopía – Spanisch" lang="es" hreflang="es" data-title="Homotopía" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Homotopia" title="Homotopia – Baskisch" lang="eu" hreflang="eu" data-title="Homotopia" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%85%D9%88%D8%AA%D9%88%D9%BE%DB%8C" title="هموتوپی – Persisch" lang="fa" hreflang="fa" data-title="هموتوپی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Homotopia" title="Homotopia – Finnisch" lang="fi" hreflang="fi" data-title="Homotopia" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Homotopie" title="Homotopie – Französisch" lang="fr" hreflang="fr" data-title="Homotopie" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Homotop%C3%ADa" title="Homotopía – Galicisch" lang="gl" hreflang="gl" data-title="Homotopía" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%95%D7%9E%D7%95%D7%98%D7%95%D7%A4%D7%99%D7%94" title="הומוטופיה – Hebräisch" lang="he" hreflang="he" data-title="הומוטופיה" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Homotopija" title="Homotopija – Kroatisch" lang="hr" hreflang="hr" data-title="Homotopija" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Omotopia" title="Omotopia – Italienisch" lang="it" hreflang="it" data-title="Omotopia" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9B%E3%83%A2%E3%83%88%E3%83%94%E3%83%BC" title="ホモトピー – Japanisch" lang="ja" hreflang="ja" data-title="ホモトピー" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%98%B8%EB%AA%A8%ED%86%A0%ED%94%BC" title="호모토피 – Koreanisch" lang="ko" hreflang="ko" data-title="호모토피" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Homotopi" title="Homotopi – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Homotopi" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Homotopia" title="Homotopia – Polnisch" lang="pl" hreflang="pl" data-title="Homotopia" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Homotopia" title="Homotopia – Portugiesisch" lang="pt" hreflang="pt" data-title="Homotopia" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%BE%D0%BC%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D1%8F" title="Гомотопия – Russisch" lang="ru" hreflang="ru" data-title="Гомотопия" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Homotopija" title="Homotopija – Serbokroatisch" lang="sh" hreflang="sh" data-title="Homotopija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Homotopy" title="Homotopy – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Homotopy" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%BE%D0%BC%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D1%98%D0%B0" title="Хомотопија – Serbisch" lang="sr" hreflang="sr" data-title="Хомотопија" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Homotopi" title="Homotopi – Schwedisch" lang="sv" hreflang="sv" data-title="Homotopi" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Homotopi" title="Homotopi – Türkisch" lang="tr" hreflang="tr" data-title="Homotopi" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%BE%D0%BC%D0%BE%D1%82%D0%BE%D0%BF%D1%96%D1%8F" title="Гомотопія – Ukrainisch" lang="uk" hreflang="uk" data-title="Гомотопія" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%93ng_lu%C3%A2n" title="Đồng luân – Vietnamesisch" lang="vi" hreflang="vi" data-title="Đồng luân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%8C%E4%BC%A6" title="同伦 – Wu" lang="wuu" hreflang="wuu" data-title="同伦" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%8C%E5%80%AB" title="同倫 – Chinesisch" lang="zh" hreflang="zh" data-title="同倫" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%8C%E5%80%AB" title="同倫 – Kantonesisch" lang="yue" hreflang="yue" data-title="同倫" data-language-autonym="粵語" data-language-local-name="Kantonesisch" 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