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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hopf-Faserung</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Hopf-Faserung</b> (nach <a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a>) ist eine bestimmte <a href="Abbildung_(Mathematik)" class="mw-redirect" title="Abbildung (Mathematik)">Abbildung</a> im <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a>. Es handelt sich um eine Abbildung der <a href="Topologische_Sph%C3%A4re" title="Topologische Sphäre">3-Sphäre</a>, die man sich als den dreidimensionalen Raum zusammen mit einem unendlich fernen Punkt vorstellen kann, in die 2-Sphäre, also eine Kugeloberfläche:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \colon S^{3}\to S^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>η<!-- η --></mi>
<mo>:<!-- : --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \eta \colon S^{3}\to S^{2}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458065383aea4d0ae1954e97cbdf750892febfe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.616ex; height:3.176ex;" alt="{\displaystyle \eta \colon S^{3}\to S^{2}.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Beschreibung_der_Abbildung">Beschreibung der Abbildung</h2></div>
<p>Man erhält sie wie folgt:
Zuerst wird die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01e57c690f890937838c10ba57853ff21bf30ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}" loading="lazy"></span> als <a href="Einheitssph%C3%A4re" class="mw-redirect" title="Einheitssphäre">Einheitssphäre</a> in den <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}" loading="lazy"></span> eingebettet.
Durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z_{1},z_{2})\mapsto (z_{1}/z_{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (z_{1},z_{2})\mapsto (z_{1}/z_{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c7ab5892bc490c07eee5d8aaba3be7d5ae592e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.971ex; height:2.843ex;" alt="{\displaystyle (z_{1},z_{2})\mapsto (z_{1}/z_{2})}" loading="lazy"></span> werden Paare komplexer Zahlen auf ihren Quotienten in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \cup \infty =\mathbb {R} ^{2}\cup \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>∪<!-- ∪ --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></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>
<mo>∪<!-- ∪ --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \cup \infty =\mathbb {R} ^{2}\cup \infty }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49725dcbe3477599b3238140dc22a4bcfbafc088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.322ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} \cup \infty =\mathbb {R} ^{2}\cup \infty }" loading="lazy"></span> abgebildet.
Danach bildet man den Bildpunkt mit der inversen <a href="Stereografische_Projektion" title="Stereografische Projektion">stereographischen Projektion</a> bzgl. des Nordpoles auf die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b6401d5d0155afb1406770d1eb80badce4e08ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}" loading="lazy"></span> ab.
Um die Abbildung konkret in Formeln anzugeben, gibt es verschiedene Möglichkeiten.
</p>
<div class="mw-heading mw-heading3"><h3 id="Mit_reellen_Zahlen">Mit reellen Zahlen</h3></div>
<p>Die Abbildung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}\to \mathbb {R} ^{3},\quad (x_{1},x_{2},x_{3},x_{4})\mapsto (y_{1},y_{2},y_{3})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>,</mo>
<mspace width="1em"></mspace>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}\to \mathbb {R} ^{3},\quad (x_{1},x_{2},x_{3},x_{4})\mapsto (y_{1},y_{2},y_{3})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ce7e251b34764600325b4a57365eb0ef0d47f7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.954ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{4}\to \mathbb {R} ^{3},\quad (x_{1},x_{2},x_{3},x_{4})\mapsto (y_{1},y_{2},y_{3})}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}=2(x_{1}x_{3}+x_{2}x_{4})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{1}=2(x_{1}x_{3}+x_{2}x_{4})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/989037b7ca0d667914043f149757d9fc75761077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.64ex; height:2.843ex;" alt="{\displaystyle y_{1}=2(x_{1}x_{3}+x_{2}x_{4})}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2}=2(x_{2}x_{3}-x_{1}x_{4})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{2}=2(x_{2}x_{3}-x_{1}x_{4})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/137b73d37cb8c9c9446560f7da63e4dd82daff75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.64ex; height:2.843ex;" alt="{\displaystyle y_{2}=2(x_{2}x_{3}-x_{1}x_{4})}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{3}=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{3}=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b3ee7b159841e7b6b419db86d5e0837ebf7385e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.349ex; height:3.176ex;" alt="{\displaystyle y_{3}=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}}" loading="lazy"></span></dd></dl>
<p>bildet die 3-Sphäre <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in \mathbb {R} ^{4}\mid x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} ^{4}\mid x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63fe57c2992c84da7e07279d62e8b78b487c4dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.483ex; height:3.343ex;" alt="{\displaystyle \{x\in \mathbb {R} ^{4}\mid x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\}}" loading="lazy"></span> auf die 2-Sphäre <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y\in \mathbb {R} ^{3}\mid y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>y</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{y\in \mathbb {R} ^{3}\mid y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6cf4e36ed12c2b333d2faec0937505eb20452ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.576ex; height:3.343ex;" alt="{\displaystyle \{y\in \mathbb {R} ^{3}\mid y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1\}}" loading="lazy"></span> ab. Diese Einschränkung ist die Hopf-Abbildung.
</p>
<div class="mw-heading mw-heading3"><h3 id="Mit_komplexen_Zahlen">Mit komplexen Zahlen</h3></div>
<p>Die 3-Sphäre werde als die Teilmenge
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(z,w)\in \mathbb {C} ^{2}\mid |z|^{2}+|w|^{2}=1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>z</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>w</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{(z,w)\in \mathbb {C} ^{2}\mid |z|^{2}+|w|^{2}=1\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f85e196619baf802026c7ae61e902c2eac6626fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.98ex; height:3.343ex;" alt="{\displaystyle \{(z,w)\in \mathbb {C} ^{2}\mid |z|^{2}+|w|^{2}=1\}}" loading="lazy"></span></dd></dl>
<p>des zweidimensionalen komplexen Raums aufgefasst, die 2-Sphäre als <a href="Riemannsche_Zahlenkugel" title="Riemannsche Zahlenkugel">riemannsche Zahlenkugel</a>. Dann ist die Hopf-Abbildung durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z,w)\mapsto {\frac {z}{w}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>z</mi>
<mi>w</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (z,w)\mapsto {\frac {z}{w}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bf4ca85f8b4e50c8d9edb851140ca14b48435c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.71ex; height:4.676ex;" alt="{\displaystyle (z,w)\mapsto {\frac {z}{w}}}" loading="lazy"></span></dd></dl>
<p>gegeben. Fasst man die riemannsche Zahlenkugel als <a href="Projektive_Gerade" title="Projektive Gerade">projektive Gerade</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} P^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<msup>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} P^{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a64b4437008df5a326cf123af0fe2460b27bb1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.554ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} P^{1}}" loading="lazy"></span> auf, so kann man die Abbildung unter Verwendung <a href="Homogene_Koordinaten" title="Homogene Koordinaten">homogener Koordinaten</a> auch als
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z,w)\mapsto [z:w]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">[</mo>
<mi>z</mi>
<mo>:</mo>
<mi>w</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (z,w)\mapsto [z:w]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b8542c57903b1de110d089b62dccfc92c94df3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.193ex; height:2.843ex;" alt="{\displaystyle (z,w)\mapsto [z:w]}" loading="lazy"></span></dd></dl>
<p>schreiben.
</p>
<div class="mw-heading mw-heading3"><h3 id="Mit_Lie-Gruppen">Mit Lie-Gruppen</h3></div>
<p>Die 3-Sphäre ist <a href="Diffeomorphismus" title="Diffeomorphismus">diffeomorph</a> zur <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a> Spin(3), die als <a href="%C3%9Cberlagerung_(Topologie)" title="Überlagerung (Topologie)">Überlagerung</a> der <a href="Drehgruppe" title="Drehgruppe">Drehgruppe</a> SO(3) auf der 2-Sphäre <a href="Gruppenoperation" title="Gruppenoperation">operiert</a>. Durch diese Operation erhält man Identifikationen
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}=SO(3)/SO(2)=Spin(3)/Spin(2)=S^{3}/S^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mi>S</mi>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>S</mi>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>S</mi>
<mi>p</mi>
<mi>i</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>S</mi>
<mi>p</mi>
<mi>i</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{2}=SO(3)/SO(2)=Spin(3)/Spin(2)=S^{3}/S^{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a07bf3bbe9d15bfe0cb3390c85c9ff053011f5c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.674ex; height:3.176ex;" alt="{\displaystyle S^{2}=SO(3)/SO(2)=Spin(3)/Spin(2)=S^{3}/S^{1}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Beispiele_aus_der_Physik">Beispiele aus der Physik</h3></div>
<p>Als natürliche Anschauung der Hopf-Faserung lassen sich <a href="Quantenzustand" class="mw-redirect" title="Quantenzustand">Quantenzustände</a> nicht relativistischer Elektronen auf der <a href="Einheitssph%C3%A4re" class="mw-redirect" title="Einheitssphäre">Einheitssphäre</a> darstellen.
</p><p>Hierbei ist der Zustandsvektor:<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1b1095591010881376ae0b508e7fc0216694a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.352ex; height:6.176ex;" alt="{\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1},\psi _{2}\in \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{1},\psi _{2}\in \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0406d0eca24e3f2315575d975806596a1b0304f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.687ex; height:2.509ex;" alt="{\displaystyle \psi _{1},\psi _{2}\in \mathbb {C} }" loading="lazy"></span> gegeben.
Ferner sei die Gestalt der Einheitssphäre des 2-dimensionalen <a href="Hilbertraum" title="Hilbertraum">Hilbertraums</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbb {C} ^{2})=\{\psi \in \mathbb {C} ^{2}:||\psi ||=1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>ψ<!-- ψ --></mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S(\mathbb {C} ^{2})=\{\psi \in \mathbb {C} ^{2}:||\psi ||=1\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9eb1c3530059d57dc3e3fc31ec2f673b6afb2a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.849ex; height:3.176ex;" alt="{\displaystyle S(\mathbb {C} ^{2})=\{\psi \in \mathbb {C} ^{2}:||\psi ||=1\}}" loading="lazy"></span></dd></dl>
<p>Aus dem Skalarprodukt des Quantenzustands
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle ={\begin{pmatrix}\alpha _{1}+i\beta _{1}\\\alpha _{2}+i\beta _{2}\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>ψ<!-- ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>i</mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>i</mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle ={\begin{pmatrix}\alpha _{1}+i\beta _{1}\\\alpha _{2}+i\beta _{2}\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebfae95daa5d3958c1bda5bad612adb31b612b28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.89ex; height:6.176ex;" alt="{\displaystyle |\psi \rangle ={\begin{pmatrix}\alpha _{1}+i\beta _{1}\\\alpha _{2}+i\beta _{2}\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>folgt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi |\psi \rangle =\alpha _{1}^{2}+\beta _{1}^{2}+\alpha _{2}^{2}+\beta _{2}^{2}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>ψ<!-- ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<msubsup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle \psi |\psi \rangle =\alpha _{1}^{2}+\beta _{1}^{2}+\alpha _{2}^{2}+\beta _{2}^{2}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88cd32e2219590671626072d0afdbdacfb1bb175.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.229ex; height:3.176ex;" alt="{\displaystyle \langle \psi |\psi \rangle =\alpha _{1}^{2}+\beta _{1}^{2}+\alpha _{2}^{2}+\beta _{2}^{2}=1}" loading="lazy"></span></dd></dl>
<p>Dieses entspricht der 3-Sphäre.
</p><p>Zwei Quantenzustände <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{a},\psi _{b}\in S(\mathbb {C} ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>S</mi>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{a},\psi _{b}\in S(\mathbb {C} ^{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/769d6054e2aae40b8b17eb2601daee807d22eba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.981ex; height:3.176ex;" alt="{\displaystyle \psi _{a},\psi _{b}\in S(\mathbb {C} ^{2})}" loading="lazy"></span> sind äquivalent wenn es eine komplexe Zahl bzw. ein Repräsentant der <a href="Unit%C3%A4re_Gruppe" title="Unitäre Gruppe">unitären Gruppe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in U(1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mi>U</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda \in U(1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e7220c02a69819490a1fd5f0aa40f8beaf7f048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.95ex; height:2.843ex;" alt="{\displaystyle \lambda \in U(1)}" loading="lazy"></span> gibt, welcher die Forderung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{a}=\lambda \psi _{b}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{a}=\lambda \psi _{b}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/583fc5860b0e94a447daa04254a426b25cb9e58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.52ex; height:2.509ex;" alt="{\displaystyle \psi _{a}=\lambda \psi _{b}}" loading="lazy"></span> erfüllt.
Betrachtet man die gesamte Vereinigungsmenge der <a href="%C3%84quivalenzklasse#Äquivalenzklassen" class="mw-redirect" title="Äquivalenzklasse">Äquivalenzklasse</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\psi ]:=\{\lambda \psi :\lambda \in U(1),|\lambda |=1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">]</mo>
<mo>:=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>λ<!-- λ --></mi>
<mi>ψ<!-- ψ --></mi>
<mo>:</mo>
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mi>U</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\psi ]:=\{\lambda \psi :\lambda \in U(1),|\lambda |=1\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c3b16d1aee852a51aa73b6eeb8a9aeaefeaf1cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.577ex; height:2.843ex;" alt="{\displaystyle [\psi ]:=\{\lambda \psi :\lambda \in U(1),|\lambda |=1\}}" loading="lazy"></span></dd></dl>
<p>auf der Sphäre
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbb {C} ^{2})=\bigcup _{S(\psi \in \mathbb {C} ^{2})}[\psi ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>⋃<!-- ⋃ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>ψ<!-- ψ --></mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</munder>
<mo stretchy="false">[</mo>
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S(\mathbb {C} ^{2})=\bigcup _{S(\psi \in \mathbb {C} ^{2})}[\psi ]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d19cb8cd363d054c5fe324c2a1f0a8b22fb594c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:18.47ex; height:6.343ex;" alt="{\displaystyle S(\mathbb {C} ^{2})=\bigcup _{S(\psi \in \mathbb {C} ^{2})}[\psi ]}" loading="lazy"></span></dd></dl>
<p>so operiert die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e62b00d74ee0cefb86cc052365625abff56d43e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" loading="lazy"></span> Gruppe auf der Einheitssphäre.
Die Mengen der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\psi ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\psi ]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ea1b5cb99c910c356116b097444a3257efc991e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.807ex; height:2.843ex;" alt="{\displaystyle [\psi ]}" loading="lazy"></span> werden auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e62b00d74ee0cefb86cc052365625abff56d43e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" loading="lazy"></span>-Faser genannt.
Dargestellt wird diese Menge der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e62b00d74ee0cefb86cc052365625abff56d43e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" loading="lazy"></span>-Faser wie folgt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbb {C} ^{2})/U(1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>U</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S(\mathbb {C} ^{2})/U(1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38737d7225220dd097315bfe243bae5996588a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.958ex; height:3.176ex;" alt="{\displaystyle S(\mathbb {C} ^{2})/U(1)}" loading="lazy"></span></dd></dl>
<p>Die Hopf-Faserung (als <i>„Hopfion“</i>) wurde in vielen Bereichen der Physik als mögliche topologische Textur in unterschiedlichen zugrundeliegenden physikalischen Feldern diskutiert, ähnlich dem <a href="Skyrmion" title="Skyrmion">Skyrmion</a>, allerdings wurden sie im Gegensatz zu diesem bisher (2021) nicht in der Natur nachgewiesen. Das reicht von magnetischen Strukturen in Festkörpern, <a href="Ferroelektrikum" title="Ferroelektrikum">Ferroelektrika</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Teilchenphysik, Supraflüsisigkeiten bis zur Biologie. Diese und ähnliche solche topologischen Strukturen stellen teilchenartige, durch ihre Topologie geschützte bzw. stabilisierte, „verwirbelte“ Feldanregungen (verbunden mit ganzzahligen topologischen Quantenzahlen, in diesem Fall die Hopf-Invariante) dar und sind insbesondere in der Festkörperphysik ein aktuelles Forschungsgebiet (2022). Das Hopfion wurde zwar bisher nicht in der Natur beobachtet, aber 2021 über ihre Projektion aus vier Dimensionen 2021 in Form eines Lichtfeldes mit quantenoptischen Methoden künstlich erzeugt (<a href="Cornelia_Denz" title="Cornelia Denz">Cornelia Denz</a> u. a. 2021).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Die Textur selbst wurde dabei in der Phasen- und Polarisationsstruktur des Lichtfeldes abgebildet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<ul><li>Die Hopf-Abbildung ist ein <a href="Faserb%C3%BCndel" title="Faserbündel">Faserbündel</a> mit Faser <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"></span> (sogar ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"></span>-<a href="U(1)-Hauptfaserb%C3%BCndel" title="U(1)-Hauptfaserbündel">Hauptfaserbündel</a>).</li>
<li>Je zwei Fasern bilden eine <a href="Hopf-Verschlingung" title="Hopf-Verschlingung">Hopf-Verschlingung</a>.</li>
<li>Die Hopf-Abbildung erzeugt die <a href="Homotopiegruppe" title="Homotopiegruppe">Homotopiegruppe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}(S^{2})\cong \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>≅<!-- ≅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{3}(S^{2})\cong \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f41d019d828e1c05832f078880f4afbeca34951.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.413ex; height:3.176ex;" alt="{\displaystyle \pi _{3}(S^{2})\cong \mathbb {Z} }" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2></div>
<p>Die oben angegebene Beschreibung mithilfe komplexer Zahlen kann analog auch mit <a href="Quaternion" title="Quaternion">Quaternionen</a> oder mit <a href="Cayley-Zahl" class="mw-redirect" title="Cayley-Zahl">Cayley-Zahlen</a> durchgeführt werden; man erhält dann Faserungen
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}\to S^{7}\to S^{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{3}\to S^{7}\to S^{4}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e3f75160fa420b73c4c87e92ad70ba5ac4ae6f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.956ex; height:2.676ex;" alt="{\displaystyle S^{3}\to S^{7}\to S^{4}}" loading="lazy"></span> bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{7}\to S^{15}\to S^{8}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle S^{7}\to S^{15}\to S^{8}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d97e3e1ea8204744ccffc825cab5993026e5349b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.778ex; height:2.676ex;" alt="{\displaystyle S^{7}\to S^{15}\to S^{8}}" loading="lazy"></span>,</dd></dl>
<p>die ebenfalls als Hopf-Faserungen bezeichnet werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2></div>
<p><a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a> gab diese Abbildung <a href="1931" title="1931">1931</a> in seiner Arbeit <i>Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche</i> an und zeigte, dass sie nicht <a href="Nullhomotop" class="mw-redirect" title="Nullhomotop">nullhomotop</a> ist (genauer: dass ihre <a href="Hopf-Invariante" class="mw-redirect" title="Hopf-Invariante">Hopf-Invariante</a> gleich 1 ist).
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a>: <i>Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche.</i> <i>Math. Ann.</i> 104 (1931), 637–665 (<a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0104&DMDID=dmdlog46">PDF</a>)</li>
<li><a href="Eberhard_Zeidler_(Mathematiker)" title="Eberhard Zeidler (Mathematiker)">Eberhard Zeidler</a>: <i>Quantum Field theory I - Basics in Mathematics and Physics.</i> Springer Verlag, 2006, ISBN 3-540-34762-3, S. 269 ff.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">I. Lukyanchuk, V. M. Vinokur u. a.: <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41diese">Hopfions emerge in ferroelectrics</a>, Nature Communications, Band 11, 2020, S. 2433, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1907.03866">Arxiv</a></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41467-021-26171-5">Danica Sugic, Cornelia Denz, Mark Denner u. a., Particle-like topologies in light</a>, Nature Communications, Band 12, 2021, Nr. 6785.</span>
</li>
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