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<title>Faserbündel</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Faserbündel</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">In der <a href="Algebraische_Topologie" title="Algebraische Topologie">algebraischen Topologie</a>, einem <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">Teilgebiet</a> der Mathematik, ist ein Faserbündel ein <a href="Topologischer_Raum" title="Topologischer Raum">topologischer Raum</a>, der lokal als <a href="Kartesisches_Produkt" title="Kartesisches Produkt">kartesisches Produkt</a> zweier topologischer Räume dargestellt werden kann, zusammen mit einer Abbildung, die diese Ähnlichkeit wiedergibt.

Faserbündel spielen eine wichtige Rolle in der <a href="Homotopietheorie" class="mw-redirect" title="Homotopietheorie">Homotopietheorie</a>, <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> und <a href="Differentialtopologie" title="Differentialtopologie">Differentialtopologie</a>.

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

Das Konzept eines Faserbündels kam erstmals im Zusammenhang mit der Topologie und Geometrie von <a href="Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeiten</a> auf.<a href="#cite_note-1">[1]</a> <a href="Herbert_Seifert" title="Herbert Seifert">Herbert Seifert</a> führte im Jahr 1933 die Begriffe Faser und gefaserter Raum ein.<a href="#cite_note-2">[2]</a>

Die erste Definition eines Faserbündels gab <a href="Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a> im Jahr 1935 unter dem Namen Sphären-Raum (engl. sphere space). In den Jahren von 1935 bis 1940 wurden Faserbündel in der Mathematik ein eigenes Forschungsgebiet. Die Arbeiten von Whitney, <a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a> und <a href="Eduard_Stiefel" title="Eduard Stiefel">Eduard Stiefel</a> gaben Ausblicke auf die Bedeutung von Faserbündeln in Topologie und Differentialgeometrie.<a href="#cite_note-3">[3]</a>

Bis zum Jahr 1950 wurde die Definition eines Faserbündels klar notiert und die Theorie über Homotopieklassifikation und Charakteristikklassen von Faserbündeln von mehreren Mathematikern, darunter <a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Shiing-Shen Chern</a>, <a href="Lew_Semjonowitsch_Pontrjagin" title="Lew Semjonowitsch Pontrjagin">Lew Pontrjagin</a>, Stiefel und Whitney, vorangetrieben. In den Jahren von 1950 bis 1955 konnte <a href="Friedrich_Hirzebruch" title="Friedrich Hirzebruch">Friedrich Hirzebruch</a> unter Verwendung der Charakteristikklassen von Faserbündeln den <a href="Satz_von_Hirzebruch-Riemann-Roch" title="Satz von Hirzebruch-Riemann-Roch">Satz von Hirzebruch-Riemann-Roch</a> beweisen. <a href="John_Willard_Milnor" title="John Willard Milnor">John Milnor</a> gab im Jahr 1955 eine Konstruktion eines universellen Faserbündels für beliebige topologische Gruppen an. In den frühen 1960ern entwickelten <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a>, <a href="Michael_Francis_Atiyah" title="Michael Francis Atiyah">Michael Atiyah</a> und Hirzebruch eine verallgemeinerte <a href="Kohomologie" title="Kohomologie">Kohomologietheorie</a>, die <a href="K-Theorie" title="K-Theorie">K-Theorie</a>, mit Hilfe von Stabilitätsklassen von Vektorbündeln.<a href="#cite_note-4">[4]</a>

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

Ein Faserbündel ist ein Quadrupel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,B,\pi ,F)}">

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<annotation encoding="application/x-tex">{\displaystyle (E,B,\pi ,F)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/363fe17ec08d0de5e0e563fcbf103e619a4c0ec3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.524ex; height:2.843ex;" alt="{\displaystyle (E,B,\pi ,F)}" loading="lazy"> bestehend aus topologischen Räumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" 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}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> und einer <a href="Stetige_Funktion" title="Stetige Funktion">stetigen</a> <a href="Surjektive_Funktion" title="Surjektive Funktion">surjektiven</a> Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy">, wobei für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aac01724708de4e1d41423bc64b35e9d94c9009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.934ex; height:2.176ex;" alt="{\displaystyle x\in B}" loading="lazy"> eine offene Umgebung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/858638a6c9ba36e8aa68ffaf90eb9b38c4b20fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.645ex; height:2.343ex;" alt="{\displaystyle U\subseteq B}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und ein <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon \pi ^{-1}(U)\to U\times F}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \colon \pi ^{-1}(U)\to U\times F}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7461a1e41947a3f765045f790faf8ef8b724ec4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.791ex; height:3.176ex;" alt="{\displaystyle \varphi \colon \pi ^{-1}(U)\to U\times F}" loading="lazy"> existieren, sodass das folgende Diagramm <a href="Kommutatives_Diagramm" title="Kommutatives Diagramm">kommutiert</a>:

Hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} _{1}\colon U\times F\to U}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {proj} _{1}\colon U\times F\to U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f972625f58c9c5e87c6e27d851df643f2f539c38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.927ex; height:2.676ex;" alt="{\displaystyle \operatorname {proj} _{1}\colon U\times F\to U}" loading="lazy"> die natürliche Projektion. Ein solcher Homöomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> wird lokale Trivialisierung des Bündels und die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> Projektion genannt. Der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> heißt der Basisraum des Bündels, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" 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}" loading="lazy"> der Totalraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> die Faser.

Der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\times F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9f06bc8785fae8deb989c1a480d8084d092f66c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.364ex; height:2.176ex;" alt="{\displaystyle U\times F}" loading="lazy"> ist mit der <a href="Produkttopologie" title="Produkttopologie">Produkttopologie</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\times F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9f06bc8785fae8deb989c1a480d8084d092f66c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.364ex; height:2.176ex;" alt="{\displaystyle U\times F}" loading="lazy"> versehen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(U)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52414f17b55ee456b0b6cabc0da7f5c141981c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.259ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(U)}" loading="lazy"> mit der <a href="Teilraumtopologie" title="Teilraumtopologie">Teilraumtopologie</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" 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}" loading="lazy">.

Um zusätzlich die Faser des Bündels zu nennen, wird auch die Notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\hookrightarrow E\to B}">

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<annotation encoding="application/x-tex">{\displaystyle F\hookrightarrow E\to B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73c1117380315b9e5d1729f4ae8231597ca7afdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.801ex; height:2.176ex;" alt="{\displaystyle F\hookrightarrow E\to B}" loading="lazy"> für ein Faserbündel verwendet. Hierbei ist die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\hookrightarrow E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f1998c7efdb3af49beb68554370f6d67401d86c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.423ex; height:2.176ex;" alt="{\displaystyle F\hookrightarrow E}" loading="lazy"> die <a href="Inklusionsabbildung" title="Inklusionsabbildung">Inklusion</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{b}=\pi ^{-1}(\{b\})}">

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<annotation encoding="application/x-tex">{\displaystyle F_{b}=\pi ^{-1}(\{b\})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9582fdabc034798be0f621ee3c0f226c10be952.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.329ex; height:3.176ex;" alt="{\displaystyle F_{b}=\pi ^{-1}(\{b\})}" loading="lazy">, der Faser über einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61dbfba9ff608c8700a30596649d98dcc6147d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}" loading="lazy">, identifiziert.<a href="#cite_note-5">[5]</a>

Jedes Faserbündel ist eine <a href="Serre-Faserung" class="mw-redirect" title="Serre-Faserung">Serre-Faserung</a>.<a href="#cite_note-6">[6]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Triviales_Bündel">Triviales Bündel</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=B\times F}">

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<annotation encoding="application/x-tex">{\displaystyle E=B\times F}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8aa627b0a478850e41c73b0eb70cadd831feac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.219ex; height:2.176ex;" alt="{\displaystyle E=B\times F}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> die Projektion auf den ersten Faktor, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" 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}" loading="lazy"> nicht nur lokal ein Produkt, sondern auch global. Ein solches Faserbündel heißt triviales Bündel oder Produktbündel.<a href="#cite_note-7">[7]</a>

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

Ein Faserbündel mit <a href="Diskrete_Topologie" title="Diskrete Topologie">diskreter</a> Faser ist eine <a href="%C3%9Cberlagerung_(Topologie)" title="Überlagerung (Topologie)">Überlagerung</a>. Ebenso ist jede Überlagerung, deren Fasern alle die gleiche <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Kardinalität</a> haben, ein Faserbündel mit diskreter Faser. Insbesondere ist eine Überlagerung über einem <a href="Zusammenh%C3%A4ngender_Raum" title="Zusammenhängender Raum">zusammenhängenden</a> Basisraum ein Faserbündel.<a href="#cite_note-8">[8]</a>

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

Das <a href="M%C3%B6biusband" title="Möbiusband">Möbiusband</a> ist ein anschauliches Beispiel für ein nichttriviales Faserbündel. Der Basisraum ist die Kreislinie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

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

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</math><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">, die mittig des Bandes verläuft. Die Faser ist durch ein abgeschlossenes Intervall gegeben, z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1].}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,1].}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ca7e11ebbb9225cdba1609c129e46d0ec5101a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.108ex; height:2.843ex;" alt="{\displaystyle [-1,1].}" loading="lazy">

Der Totalraum ist gegeben durch den Quotientenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=([0,1]\times [-1,1])/\sim }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=([0,1]\times [-1,1])/\sim }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33c93637e1b40249c1a4d1680781559b0e68beb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.253ex; height:2.843ex;" alt="{\displaystyle E=([0,1]\times [-1,1])/\sim }" loading="lazy"> mit der <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy"> gegeben durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,a)\sim (1,-a).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,a)\sim (1,-a).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c0f743b6fa047fdeb24c54d54f809f09a482c6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.024ex; height:2.843ex;" alt="{\displaystyle (0,a)\sim (1,-a).}" loading="lazy"> Die Bündelprojektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9208c6cd87a2cae5c0a843cabbd703b4e2c24a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.332ex; height:2.676ex;" alt="{\displaystyle \pi \colon E\to S^{1}}" loading="lazy"> ist die von der Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} \colon [0,1]\times [-1,1]\to [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {proj} \colon [0,1]\times [-1,1]\to [0,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4272d2d3667198e7b955b02fbcfb964b3f4e1ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.333ex; height:2.843ex;" alt="{\displaystyle \operatorname {proj} \colon [0,1]\times [-1,1]\to [0,1]}" loading="lazy"> induzierten Abbildung, d. h., eine Äquivalenzklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(x,y)]\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [(x,y)]\in E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93f87d702fdfa3d1b2d70e7ca6a12bb386eb0f2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:2.843ex;" alt="{\displaystyle [(x,y)]\in E}" loading="lazy"> wird unter der Bündelprojektion auf die Äquivalenzklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07548563c21e128890501e14eb7c80ee2d6fda4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle [x]}" loading="lazy"> abgebildet, wobei die Äquivalenzrelation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><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"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\sim 1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0\sim 1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b72da290e5324c6828ed459f05f88be0f8e4794c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:2.843ex;" alt="{\displaystyle (0\sim 1)}" loading="lazy"> gegeben ist.

Das entsprechende triviale Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\times [-1,1]}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}\times [-1,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e108a673362453f0d2bac504f6619cefee22cd0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.877ex; height:3.176ex;" alt="{\displaystyle S^{1}\times [-1,1]}" loading="lazy"> ist ein <a href="Zylinder_(Geometrie)" title="Zylinder (Geometrie)">Zylinder</a>. Möbiusband und Zylinder unterscheiden sich durch eine Verdrehung der Faser. Diese Verdrehung ist nur global sichtbar, lokal sind Möbiusband und Zylinder identisch.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Kleinsche_Flasche">Kleinsche Flasche</h3></div>

Ein weiteres nichttriviales Faserbündel ist die <a href="Kleinsche_Flasche" title="Kleinsche Flasche">Kleinsche Flasche</a>. Der Basisraum und die Faser sind durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><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"> und der Totalraum durch den Quotientenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=([0,1]\times [0,1])/\sim }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=([0,1]\times [0,1])/\sim }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c9d5fad347c76d845156094189c03e648e6894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.445ex; height:2.843ex;" alt="{\displaystyle E=([0,1]\times [0,1])/\sim }" loading="lazy"> gegeben, wobei die Äquivalenzrelation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,y)\sim (1,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,y)\sim (1,y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba8bcb3a7d7fc74a58c850593696b1609b9e6411.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.421ex; height:2.843ex;" alt="{\displaystyle (0,y)\sim (1,y)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,0)\sim (1-x,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,0)\sim (1-x,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf92adf882171962ea250e999c4729b490b20844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.772ex; height:2.843ex;" alt="{\displaystyle (x,0)\sim (1-x,1)}" loading="lazy"> gegeben ist. Die Bündel-Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9208c6cd87a2cae5c0a843cabbd703b4e2c24a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.332ex; height:2.676ex;" alt="{\displaystyle \pi \colon E\to S^{1}}" loading="lazy"> bildet ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(a,b)]\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [(a,b)]\in E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60816bf3cdf050331c0167e1f6cf859bb4ca7ed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.981ex; height:2.843ex;" alt="{\displaystyle [(a,b)]\in E}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ([(a,b)])=[b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ([(a,b)])=[b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e72dcdc106caff24bce7631c30693eb71c658cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.895ex; height:2.843ex;" alt="{\displaystyle \pi ([(a,b)])=[b]}" loading="lazy"> mit der Äquivalenzrelation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\sim 1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0\sim 1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b72da290e5324c6828ed459f05f88be0f8e4794c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:2.843ex;" alt="{\displaystyle (0\sim 1)}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ab.

Das entsprechende triviale Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\times S^{1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}\times S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/120ffc4a9b57e92089797fa472099abd3e1099db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.992ex; height:2.676ex;" alt="{\displaystyle S^{1}\times S^{1}}" loading="lazy"> ist ein <a href="Torus" title="Torus">Torus</a>, der lokal von der Kleinschen Flasche nicht unterscheidbar ist.<a href="#cite_note-10">[10]</a>

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

Das <a href="Hopf-Faserung" title="Hopf-Faserung">Hopf-Bündel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b654c5cce87980c31bc0034a7a7e274429887af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.248ex; height:2.676ex;" alt="{\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2}}" loading="lazy"> hat als Faser, Totalraum und Basisraum Sphären und ist eines der ersten entdeckten nicht trivialen Faserbündel. Es ist ein Spezialfall für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"> des Faserbündels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\hookrightarrow S^{2n+1}\to \mathbb {C} P^{n}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}\hookrightarrow S^{2n+1}\to \mathbb {C} P^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c94118cf71189c1d72f2bd6fe95f755a70e14e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.477ex; height:2.676ex;" alt="{\displaystyle S^{1}\hookrightarrow S^{2n+1}\to \mathbb {C} P^{n}}" loading="lazy"> über dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen</a> <a href="Projektiver_Raum" title="Projektiver Raum">projektiven Raum</a>. Weitere Hopf-Bündel, auch verallgemeinerte Hopf-Bündel genannt, lassen sich durch Ersetzen der komplexen Zahlen durch die <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>, die <a href="Quaternion" title="Quaternion">Quaternionen</a> und die <a href="Oktave_(Mathematik)" title="Oktave (Mathematik)">Oktonionen</a> herleiten:

<ul><li>Die Überlagerung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{0}\hookrightarrow S^{n}\to \mathbb {R} P^{n}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{0}\hookrightarrow S^{n}\to \mathbb {R} P^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/845ebb0b8dd2b27bc141e6742bc45c5604078995.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.555ex; height:2.676ex;" alt="{\displaystyle S^{0}\hookrightarrow S^{n}\to \mathbb {R} P^{n}}" loading="lazy"> über dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen projektiven Raum ergibt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"> das reelle Hopf-Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{0}\hookrightarrow S^{1}\to S^{1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{0}\hookrightarrow S^{1}\to S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0528626e308560d4c1e2453db69658d951098cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.248ex; height:2.676ex;" alt="{\displaystyle S^{0}\hookrightarrow S^{1}\to S^{1}}" loading="lazy">.</li></ul>

<ul><li>Für die Quaternionen ergibt sich das Hopf-Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4}\cong \mathbb {H} P^{1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4}\cong \mathbb {H} P^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3402ec9890a7a1488f1a093b1d0ec9efb98996da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.031ex; height:2.676ex;" alt="{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4}\cong \mathbb {H} P^{1}}" loading="lazy">.</li></ul>

<ul><li>Für die Oktionen ergibt sich das Hopf-Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{7}\hookrightarrow S^{15}\to S^{8}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{7}\hookrightarrow S^{15}\to S^{8}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/707f5c05b4f0b640b074e30f8350c75298e64889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.07ex; height:2.676ex;" alt="{\displaystyle S^{7}\hookrightarrow S^{15}\to S^{8}}" loading="lazy">.</li></ul>

Weitere Faserbündel, deren Faser, Totalraum und Basisraum Sphären sind, existieren nicht. Dies ist eine Folgerung aus dem Satz von <a href="John_Frank_Adams" title="John Frank Adams">Adam</a>, welcher das Problem von Hopf über die Anzahl der Abbildungen zwischen Sphären mit <a href="Hopf-Invariante" class="mw-redirect" title="Hopf-Invariante">Hopf-Invariante</a> 1 löst.<a href="#cite_note-11">[11]</a>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Schnitt_(Faserb%C3%BCndel)" title="Schnitt (Faserbündel)">Schnitt (Faserbündel)</a></div>

Ein globaler Schnitt eines Faserbündels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,B,\pi ,F)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,B,\pi ,F)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/363fe17ec08d0de5e0e563fcbf103e619a4c0ec3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.524ex; height:2.843ex;" alt="{\displaystyle (E,B,\pi ,F)}" loading="lazy"> ist eine stetige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\colon B\to E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\colon B\to E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00d7e1952a54b33a127c625f18cfb735da040de5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.278ex; height:2.176ex;" alt="{\displaystyle s\colon B\to E}" loading="lazy">, die zur Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> rechtsinvers ist. Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61dbfba9ff608c8700a30596649d98dcc6147d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}" loading="lazy"> gilt also, dass die Verknüpfung von Projektion und Schnitt gleich der Identität ist: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi \circ s)(b)=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\pi \circ s)(b)=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3774fed9eabee23ed2b1f792391338171df595a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.329ex; height:2.843ex;" alt="{\displaystyle (\pi \circ s)(b)=b}" loading="lazy">. Anders ausgedrückt liegt für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61dbfba9ff608c8700a30596649d98dcc6147d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}" loading="lazy"> das Bild des Schnitts in der Faser über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(b)\in \pi ^{-1}(\{b\})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(b)\in \pi ^{-1}(\{b\})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b27819fdb7c8196231d19dfb110f0e0a3aa7e3ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.537ex; height:3.176ex;" alt="{\displaystyle s(b)\in \pi ^{-1}(\{b\})}" loading="lazy">.

Ein lokaler Schnitt eines Faserbündels ist eine stetige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\colon V\to E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\colon V\to E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88df3f4bb158369ad50774065475871507e45879.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.301ex; height:2.176ex;" alt="{\displaystyle s\colon V\to E}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d49ccce47b5922a67dc021734ab42b5dde1d240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.65ex; height:2.343ex;" alt="{\displaystyle V\subseteq B}" loading="lazy"> eine offene Teilmenge ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi \circ s)(b)=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\pi \circ s)(b)=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3774fed9eabee23ed2b1f792391338171df595a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.329ex; height:2.843ex;" alt="{\displaystyle (\pi \circ s)(b)=b}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/636817e811899cea6f0f94afded0eb675248dab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.625ex; height:2.176ex;" alt="{\displaystyle b\in V}" loading="lazy"> gilt.<a href="#cite_note-12">[12]</a>

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

Ein Bündelmorphismus (auch Bündelabbildung genannt) zwischen zwei Faserbündeln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{1},B_{1},\pi _{1},F_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E_{1},B_{1},\pi _{1},F_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8af3d0b5b55202bdd43be3d4e9addb2093bc2269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.427ex; height:2.843ex;" alt="{\displaystyle (E_{1},B_{1},\pi _{1},F_{1})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{2},B_{2},\pi _{2},F_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E_{2},B_{2},\pi _{2},F_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf322acabbe15b3727f154778b408b4ad01ffce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.427ex; height:2.843ex;" alt="{\displaystyle (E_{2},B_{2},\pi _{2},F_{2})}" loading="lazy"> ist eine Abbildung, die die Bündelstruktur erhält; in gewissem Sinne ist er eine Faser-erhaltende Abbildung. Genauer ist ein Bündelmorphismus durch ein Tupel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,f)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c03d3ab7499e605e576a34f64424415039ebcbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.452ex; height:2.843ex;" alt="{\displaystyle (u,f)}" loading="lazy"> von zwei Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\colon E_{1}\to E_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\colon E_{1}\to E_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/192c4be07a37dcf2779394d39225b44b1d9c7ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.517ex; height:2.509ex;" alt="{\displaystyle u\colon E_{1}\to E_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon B_{1}\to B_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon B_{1}\to B_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e32c769886fdffe94bee7ed90c1b53a4237c92d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.563ex; height:2.509ex;" alt="{\displaystyle f\colon B_{1}\to B_{2}}" loading="lazy"> gegeben, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}\circ u=f\circ \pi _{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{2}\circ u=f\circ \pi _{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61e2c692378414578387c65c7cef79391cd6da09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.855ex; height:2.509ex;" alt="{\displaystyle \pi _{2}\circ u=f\circ \pi _{1}}" loading="lazy"> gilt. Die Situation wird durch das folgende kommutative Diagramm verdeutlicht:

Eine Faser über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in B_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fabca529f7de00d37a5bf9697932a9c058885a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.656ex; height:2.509ex;" alt="{\displaystyle b\in B_{1}}" loading="lazy"> wird unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> auf eine Faser über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(b)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adbef9aaf500c30a3f107e5d7efb7a8627d3bba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.085ex; height:2.843ex;" alt="{\displaystyle f(b)}" loading="lazy"> abgebildet; dies wird durch die Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\pi _{1}^{-1}(b))\subseteq \pi _{2}^{-1}(f(b))}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(\pi _{1}^{-1}(b))\subseteq \pi _{2}^{-1}(f(b))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d52c52f6fa0b8ad3b126689f4c88dbae97b050d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.273ex; height:3.343ex;" alt="{\displaystyle u(\pi _{1}^{-1}(b))\subseteq \pi _{2}^{-1}(f(b))}" loading="lazy"> dargestellt.

Sind die Basisräume identisch, so ist der Bündelmorphismus durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,\operatorname {id} _{B})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u,\operatorname {id} _{B})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e66208f20ff7967e21d7d6eb78b3faa8836c40f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.592ex; height:2.843ex;" alt="{\displaystyle (u,\operatorname {id} _{B})}" loading="lazy"> gegeben und man spricht von einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">-Morphismus oder einem Bündelmorphismus über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=B_{1}=B_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B=B_{1}=B_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a3e0bc30069840a6818d46667709eb911ccdf66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.597ex; height:2.509ex;" alt="{\displaystyle B=B_{1}=B_{2}}" loading="lazy"> gilt. Die Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}=\pi _{2}\circ u}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{1}=\pi _{2}\circ u}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/910d9fb56442ef798316045ebe9b3d7c597d61d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.381ex; height:2.009ex;" alt="{\displaystyle \pi _{1}=\pi _{2}\circ u}" loading="lazy"> ist durch das folgende Diagramm gegeben:

Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61dbfba9ff608c8700a30596649d98dcc6147d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}" loading="lazy"> gilt die Bedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\pi _{1}^{-1}(\{b\}))\subseteq \pi _{2}^{-1}(\{b\})}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(\pi _{1}^{-1}(\{b\}))\subseteq \pi _{2}^{-1}(\{b\})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/804b4d948a5eb5dc141766e31910e9a992c052a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.835ex; height:3.343ex;" alt="{\displaystyle u(\pi _{1}^{-1}(\{b\}))\subseteq \pi _{2}^{-1}(\{b\})}" loading="lazy">, weshalb <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> auch Faser-erhaltend genannt wird.<a href="#cite_note-13">[13]</a>

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

Für jeden Basisraum eines Faserbündels existiert ein <a href="Atlas_(Mathematik)" title="Atlas (Mathematik)">Atlas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{i},h_{U_{i}}^{-1})|i\in I\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(U_{i},h_{U_{i}}^{-1})|i\in I\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96a90a3649ae1b049f07192cf96f543f359b053a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.689ex; height:3.676ex;" alt="{\displaystyle \{(U_{i},h_{U_{i}}^{-1})|i\in I\}}" loading="lazy"> von Karten, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}\subseteq B}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{i}\subseteq B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd278824e593d78aa1c3ea9cb5bd5ca2d29e91d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.25ex; height:2.509ex;" alt="{\displaystyle U_{i}\subseteq B}" loading="lazy"> offene Teilmengen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{U_{i}}}">

<msub>

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</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a476f1b747900eb52d91a6c810b2ac558efd052.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.319ex; height:2.843ex;" alt="{\displaystyle h_{U_{i}}}" loading="lazy"> lokale Trivialisierungen des Faserbündels sind. Zwei Karten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U_{i},h_{U_{i}}^{-1})}">

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<annotation encoding="application/x-tex">{\displaystyle (U_{i},h_{U_{i}}^{-1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bce92fba2d49ba096ac98ef59fa3afba5a016175.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.902ex; height:3.676ex;" alt="{\displaystyle (U_{i},h_{U_{i}}^{-1})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U_{j},h_{U_{j}}^{-1})}">

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<annotation encoding="application/x-tex">{\displaystyle (U_{j},h_{U_{j}}^{-1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/564ab56ca318b31df053a4b88ef7ad2b98c81303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.012ex; height:3.843ex;" alt="{\displaystyle (U_{j},h_{U_{j}}^{-1})}" loading="lazy"> können mittels stetiger Kartenwechsel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{i,j}\colon (U_{i}\cap U_{j})\to \operatorname {Aut} (F)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{i,j}\colon (U_{i}\cap U_{j})\to \operatorname {Aut} (F)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbde83f1f0e31274510a948ce6bd3d1ef2630026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.027ex; height:3.009ex;" alt="{\displaystyle \Phi _{i,j}\colon (U_{i}\cap U_{j})\to \operatorname {Aut} (F)}" loading="lazy"> gewechselt werden. Die Kartenwechsel geben Auskunft darüber, welche Symmetrien der Fasern beim Übergang benutzt werden, weshalb sie auch Übergangsfunktionen genannt werden. Für ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in U_{i}\cap U_{j}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in U_{i}\cap U_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5227c06543d22907b447377dd437bb08d69438d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.305ex; height:2.843ex;" alt="{\displaystyle b\in U_{i}\cap U_{j}}" loading="lazy"> ist die Übergangsfunktion durch den Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{i,j}(b)=(pr_{F}\circ h_{U_{i}}\circ h_{U_{j}}^{-1})(b,\cdot )\colon F\to F}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{i,j}(b)=(pr_{F}\circ h_{U_{i}}\circ h_{U_{j}}^{-1})(b,\cdot )\colon F\to F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/308160d498c44860b6b176a92d83507da1ab96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:39.006ex; height:3.843ex;" alt="{\displaystyle \Phi _{i,j}(b)=(pr_{F}\circ h_{U_{i}}\circ h_{U_{j}}^{-1})(b,\cdot )\colon F\to F}" loading="lazy"> gegeben. Das folgende Diagramm verdeutlicht die Situation:

In der ersten Zeile ist die erste Komponente durch die Identität und die zweite Komponente durch die Übergangsfunktion gegeben.<a href="#cite_note-14">[14]</a>

Eine topologische Transformationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eines topologischen Raumes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> relativ zu einer Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \colon G\times F\to F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta \colon G\times F\to F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/273c3cfc326e88f6cfbb02ed8eb13202a6acd471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.966ex; height:2.676ex;" alt="{\displaystyle \eta \colon G\times F\to F}" loading="lazy"> ist eine topologische Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> sodass:

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> stetig ist</li>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta (e,f)=f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccab34933bd164010e5d5cbbd6bfdd6dfa110f9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.752ex; height:2.843ex;" alt="{\displaystyle \eta (e,f)=f}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> die Identität von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist und</li>

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<annotation encoding="application/x-tex">{\displaystyle \eta (g_{1},g_{2},f)=\eta (g_{1},\eta (g_{2},f))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a180ef4779a546eb66a9aec2ff90dc9f3745aa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.381ex; height:2.843ex;" alt="{\displaystyle \eta (g_{1},g_{2},f)=\eta (g_{1},\eta (g_{2},f))}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},g_{2}\in G}">

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<annotation encoding="application/x-tex">{\displaystyle g_{1},g_{2}\in G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fed2caced34a1a8d895166a3a67a011c3351ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.028ex; height:2.509ex;" alt="{\displaystyle g_{1},g_{2}\in G}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in F.}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff73d7f8227fb95af6d72639a6ec93cd3ee1a3a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.507ex; height:2.509ex;" alt="{\displaystyle f\in F.}" loading="lazy"></li></ul>

Oft betrachtet man mehr als nur eine solche Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> und ersetzt deshalb <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (g,f)}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta (g,f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78756f351b4dbe5e67b79d4f214b36760bc9bff8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.407ex; height:2.843ex;" alt="{\displaystyle \eta (g,f)}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\cdot f.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/671d07f678089ac02070d5316512788e2db9c64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.721ex; height:2.509ex;" alt="{\displaystyle g\cdot f.}" loading="lazy"><a href="#cite_note-15">[15]</a>

Ein Koordinatenbündel ist ein Faserbündel zusammen mit einer effektiven topologischen Transformationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> sodass die folgenden zwei Bedingungen gelten:

<ul><li>Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in U_{i}\cap U_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle b\in U_{i}\cap U_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5227c06543d22907b447377dd437bb08d69438d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.305ex; height:2.843ex;" alt="{\displaystyle b\in U_{i}\cap U_{j}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in I}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle i,j\in I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7596675a16b3250fc6b02be961d315d3ed467af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.509ex;" alt="{\displaystyle i,j\in I}" loading="lazy"> entspricht der Homöomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h_{U_{j}}\circ h_{U_{i}}^{-1})(b,\cdot )\colon F\to F}">

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<annotation encoding="application/x-tex">{\displaystyle (h_{U_{j}}\circ h_{U_{i}}^{-1})(b,\cdot )\colon F\to F}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e59873942174b47a80df22fad411d49de46bf545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.701ex; height:3.676ex;" alt="{\displaystyle (h_{U_{j}}\circ h_{U_{i}}^{-1})(b,\cdot )\colon F\to F}" loading="lazy"> der Operation eines Gruppenelements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></li>

<li>für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i,j\in I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7596675a16b3250fc6b02be961d315d3ed467af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.509ex;" alt="{\displaystyle i,j\in I}" loading="lazy"> ist die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{j,i}\colon U_{i}\cap U_{j}\to G}">

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<annotation encoding="application/x-tex">{\displaystyle \tau _{j,i}\colon U_{i}\cap U_{j}\to G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2beb89bf570fa7b41f980f60527c70ba29c947bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.892ex; height:2.843ex;" alt="{\displaystyle \tau _{j,i}\colon U_{i}\cap U_{j}\to G}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{j,i}(b)=(h_{U_{j}}\circ h_{U_{i}}^{-1})(b,\cdot )}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{j,i}(b)=(h_{U_{j}}\circ h_{U_{i}}^{-1})(b,\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50c2ac12701ee8c976a139191453963c423e5339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.427ex; height:3.676ex;" alt="{\displaystyle \tau _{j,i}(b)=(h_{U_{j}}\circ h_{U_{i}}^{-1})(b,\cdot )}" loading="lazy"> stetig.</li></ul>

Die Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{j,i}}">

<msub>

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</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a8e22bff5906c0091e9c7df27565da00183ab91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.951ex; height:2.343ex;" alt="{\displaystyle \tau _{j,i}}" loading="lazy"> heißen Koordinaten-Übergangsfunktionen (teilweise auch nur Übergangsfunktionen genannt<a href="#cite_note-16">[16]</a>) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> heißt die Strukturgruppe des Bündels. Die Koordinaten-Übergangsfunktionen haben die folgenden drei Eigenschaften:

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<annotation encoding="application/x-tex">{\displaystyle \tau _{k,j}(b)\tau _{j,i}(b)=\tau _{k,i}(b)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3fda33881f42989696cc59869f2224a7abf00ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.839ex; height:3.009ex;" alt="{\displaystyle \tau _{k,j}(b)\tau _{j,i}(b)=\tau _{k,i}(b)}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j,k\in I}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle i,j,k\in I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f143b895ae4879313b60afdd5cc651b75b3eecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.052ex; height:2.509ex;" alt="{\displaystyle i,j,k\in I}" loading="lazy"> und jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in U_{i}\cap U_{j}\cap U_{k}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in U_{i}\cap U_{j}\cap U_{k}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d9cd1510706582f107c3c3ad1427f43dbc756b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.211ex; height:2.843ex;" alt="{\displaystyle b\in U_{i}\cap U_{j}\cap U_{k}.}" loading="lazy"></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{i,i}(b)=id_{G}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5567f2bf917738100f860f7905a25e765cdd3848.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.281ex; height:3.009ex;" alt="{\displaystyle \tau _{i,i}(b)=id_{G}}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in U_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in U_{i}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52b6b8b3b1931ca8532f3d3080e3ef397f7abe5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.872ex; height:2.509ex;" alt="{\displaystyle b\in U_{i}.}" loading="lazy"></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{j,k}(b)=(\tau _{k,j}(b))^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cea90a0414a1aa1538b8c39da3480200c278adb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.334ex; height:3.343ex;" alt="{\displaystyle \tau _{j,k}(b)=(\tau _{k,j}(b))^{-1}}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in U_{j}\cap U_{k}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in U_{j}\cap U_{k}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53e55e28bd57a839791123f95df509dc03ec69e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.241ex; height:2.843ex;" alt="{\displaystyle b\in U_{j}\cap U_{k}.}" loading="lazy"></li></ol>

Zwei Koordinatenbündel mit selbem Basisraum und Totalraum, gleicher Faser, Projektion und Strukturgruppe heißen äquivalent, wenn die Atlanten<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{i},h_{U_{i}})|i\in I\}}">

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(U_{i},h_{U_{i}})|i\in I\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f6492885cbd6a83597327a84c43423fcd4393f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.336ex; height:3.009ex;" alt="{\displaystyle \{(U_{i},h_{U_{i}})|i\in I\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{j}^{\prime },h_{U_{j}^{\prime }}^{\prime })|j\in J\}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(U_{j}^{\prime },h_{U_{j}^{\prime }}^{\prime })|j\in J\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fbe7b97cfef336f169d1b5eb280675a68c89abe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.019ex; height:4.009ex;" alt="{\displaystyle \{(U_{j}^{\prime },h_{U_{j}^{\prime }}^{\prime })|j\in J\}}" loading="lazy"> für zwei Indexmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> die folgenden zwei Bedingungen erfüllen:

<ul><li>Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in U_{i}\cap U_{k}^{\prime }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in U_{i}\cap U_{k}^{\prime }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/934ae5563447b9a57bc4b02b2dbd1f8ac586c124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.484ex; height:2.843ex;" alt="{\displaystyle b\in U_{i}\cap U_{k}^{\prime }}" loading="lazy"> stimmt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\tau }}_{k,i}(b)=(h_{U_{k}^{\prime }}^{\prime }\circ h_{U_{i}}^{-1})(b,\cdot )}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\tau }}_{k,i}(b)=(h_{U_{k}^{\prime }}^{\prime }\circ h_{U_{i}}^{-1})(b,\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06c80e3f12d19b019b5268087c1d1d515e6a9990.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.103ex; height:4.176ex;" alt="{\displaystyle {\tilde {\tau }}_{k,i}(b)=(h_{U_{k}^{\prime }}^{\prime }\circ h_{U_{i}}^{-1})(b,\cdot )}" loading="lazy"> mit der Operation eines Gruppenelements überein und</li>

<li>die so definierten Koordinaten-Übergangsfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\tau }}_{k,i}\colon U_{i}\cap U_{k}^{\prime }\to G}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\tau }}_{k,i}\colon U_{i}\cap U_{k}^{\prime }\to G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd4d77a7f0064fab31e3568db9e19a78e4ce6a42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.602ex; height:2.843ex;" alt="{\displaystyle {\tilde {\tau }}_{k,i}\colon U_{i}\cap U_{k}^{\prime }\to G}" loading="lazy"> sind stetig.</li></ul>

Ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel ist eine Äquivalenzklasse von Koordinatenbündeln. Häufig wird ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel auch als maximales Koordinatenbündel definiert.<a href="#cite_note-17">[17]</a>

Der Bündelkonstruktionssatz liefert Bedingungen, unter welchen die Existenz eines Koordinatenbündels garantiert ist:

Für jede topologische Transformationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> von einem Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> und System von Übergangsfunktionen in einem Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, das heißt eine Überdeckung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{i}|i\in I\}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{U_{i}|i\in I\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe4da2f6be7527694e3eabb5cf36bff11fd4665a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.174ex; height:2.843ex;" alt="{\displaystyle \{U_{i}|i\in I\}}" loading="lazy"> und eine Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\tau _{j,i}|i,j\in I\}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\tau _{j,i}|i,j\in I\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b394b7737a63db3d2e3fd65ef972c49502c3a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.729ex; height:3.009ex;" alt="{\displaystyle \{\tau _{j,i}|i,j\in I\}}" loading="lazy"> von stetigen Abbildungen mit den drei oben genannten Eigenschaften für Koordinaten-Übergangsfunktionen, existiert ein Koordinatenbündel mit Basisraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/075d661417b8ca5a991a2a7bd4991cc1ab856d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle B,}" loading="lazy"> Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed179a16e860b290a2c25c573d3fc0b2360f0fc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.388ex; height:2.509ex;" alt="{\displaystyle F,}" loading="lazy"> Strukturgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> und Übergangsfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{j,i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{j,i}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5e598aadbc7466c13add8c59e3d640e8e99866f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.597ex; height:2.343ex;" alt="{\displaystyle \tau _{j,i}.}" loading="lazy"><a href="#cite_note-18">[18]</a>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Hauptfaserb%C3%BCndel" title="Hauptfaserbündel">Hauptfaserbündel</a></div>

Ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Hauptfaserbündel ist ein Faserbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> mit Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> und einer Strukturgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> die auf der Faser durch Linksmultiplikation operiert. Die Strukturgruppe operiert frei auf dem Totalraum durch Rechtsmultiplikation mit Bahnenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0eccf5bca7cdc1fa4439af2d31831db6bde00473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}" loading="lazy"><a href="#cite_note-19">[19]</a>

Eine offene <a href="%C3%9Cberdeckung_(Mathematik)" title="Überdeckung (Mathematik)">Überdeckung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U_{i})_{i\in I}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U_{i})_{i\in I}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e19fc9e6317dca261a077df0afcfa1dbf5cfe73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.921ex; height:2.843ex;" alt="{\displaystyle (U_{i})_{i\in I}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> wird abzählbar genannt, falls eine lokal endliche Zerlegung der Eins existiert:

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=\sum _{i\in I}u_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d25f8c7bba3da2fe0ea1226efc09647f3d255fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.132ex; height:5.676ex;" alt="{\displaystyle 1=\sum _{i\in I}u_{i}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {supp} (u_{i})\subseteq U_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {supp} (u_{i})\subseteq U_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90e4763d6fe839ef3bbf594191448c66be2e14de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.218ex; height:2.843ex;" alt="{\displaystyle \operatorname {supp} (u_{i})\subseteq U_{i}}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\in I.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3627df37846ed8181157cbb3195735ec0988baa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.462ex; height:2.176ex;" alt="{\displaystyle i\in I.}" loading="lazy"></dd></dl>

Ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Hauptfaserbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> heißt abzählbar, falls eine abzählbare Überdeckung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U_{i})_{i\in I}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U_{i})_{i\in I}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e19fc9e6317dca261a077df0afcfa1dbf5cfe73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.921ex; height:2.843ex;" alt="{\displaystyle (U_{i})_{i\in I}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> existiert, sodass die eingeschränkten Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}=p|_{p^{-1}(U_{i})}\colon p^{-1}(U_{i})\to U_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{i}=p|_{p^{-1}(U_{i})}\colon p^{-1}(U_{i})\to U_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa5b72997754d7f8ac8dbd111e39fd7c61345038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:27.663ex; height:3.676ex;" alt="{\displaystyle p_{i}=p|_{p^{-1}(U_{i})}\colon p^{-1}(U_{i})\to U_{i}}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d740fe587228ce31b71c9628e089d1a9b37c6be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}" loading="lazy"> triviale Bündel sind. Ein abzählbares <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Hauptfaserbündel heißt <a href="Klassifizierender_Raum" title="Klassifizierender Raum">universelles Bündel</a>, falls für jeden Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon [X,B]\to Prin_{G}(X)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \colon [X,B]\to Prin_{G}(X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b67be415e7d1d24a96cdad59188143f64ec2d32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.226ex; height:2.843ex;" alt="{\displaystyle \rho \colon [X,B]\to Prin_{G}(X)}" loading="lazy"> von der Menge der Homotopieklassen von Abbildungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> in die Menge der Isomorphieklassen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Hauptfaserbündeln eine Bijektion ist. Bei einem universellen Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> wird der Basisraum <a href="Klassifizierender_Raum" title="Klassifizierender Raum">klassifizierender Raum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> genannt.<a href="#cite_note-20">[20]</a>

Hauptfaserbündel spielen eine wichtige Rolle bei der Klassifikation von Bündeln. Zudem kann jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel mit einem Hauptfaserbündel assoziiert werden und umgekehrt jedes Hauptfaserbündel mit einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel.

<div class="mw-heading mw-heading3"><h3 id="Assoziierte_Hauptfaserbündel">Assoziierte Hauptfaserbündel</h3></div>

Für ein gegebenes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel lässt sich ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Hauptfaserbündel konstruieren. Die Existenz ist durch den Bündelkonstruktionssatz gegeben, wobei die Faser als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> gewählt wird und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> zusätzlich auf sich selbst durch Linksmultiplikation operiert. Der Basisraum und das System von Übergangsfunktionen werden identisch mit denen des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündels gewählt.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading3"><h3 id="Assoziierte_G-Faserbündel">Assoziierte G-Faserbündel</h3></div>

Für ein gegebenes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Hauptfaserbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> und einen links <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> lässt sich ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel konstruieren:

Auf dem Produktraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\times F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea8736b40a0533aa9bc44eb0e8525b39459bdc2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.357ex; height:2.176ex;" alt="{\displaystyle E\times F}" loading="lazy"> ist eine rechts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Raum Struktur durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((x,b),g)=(gx,g^{-1}b)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((x,b),g)=(gx,g^{-1}b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f1947b3053423d43e076ea36fe507af6368d6b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.966ex; height:3.176ex;" alt="{\displaystyle ((x,b),g)=(gx,g^{-1}b)}" loading="lazy"> definiert. Das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Faserbündel ist durch die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{F}\colon (E\times F)/G\to B}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{F}\colon (E\times F)/G\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b17624b7ba6df000f232ae00c510faa1776c9d48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.356ex; height:2.843ex;" alt="{\displaystyle \pi _{F}\colon (E\times F)/G\to B}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{F}((x,b)G)=\pi (x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{F}((x,b)G)=\pi (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0707c90440aa96da9a4d31c522f00bc6034d4cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.164ex; height:2.843ex;" alt="{\displaystyle \pi _{F}((x,b)G)=\pi (x)}" loading="lazy"> und der Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> gegeben.<a href="#cite_note-22">[22]</a>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Vektorb%C3%BCndel" title="Vektorbündel">Vektorbündel</a></div>

Ein Vektorbündel vom Rang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> über einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy"> ist ein Faserbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to E\xrightarrow {\pi } B,}">

<mover>

</mpadded>

</mover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to E\xrightarrow {\pi } B,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa74737445b67d38f1bf7dcd46f42f560b049f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.409ex; width:13.202ex; height:3.676ex;" alt="{\displaystyle V\to E\xrightarrow {\pi } B,}" loading="lazy"> dessen Fasern die Struktur eines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy">-<a href="Vektorraum" title="Vektorraum">Vektorraumes</a> haben und zusätzlich jede lokale Trivialisierung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon \pi ^{-1}(U)\to U\times V,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon \pi ^{-1}(U)\to U\times V,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7299dd88306496e79ca6a3d0b50986b852f5b086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.484ex; height:3.176ex;" alt="{\displaystyle \varphi \colon \pi ^{-1}(U)\to U\times V,}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq B,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\subseteq B,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1a960e59fef1642aa22bd535ea2a0ec5991658f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.292ex; height:2.509ex;" alt="{\displaystyle U\subseteq B,}" loading="lazy"> einen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy">-linearen <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a> auf den einzelnen Fasern induziert. Das bedeutet, dass die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> eingeschränkt auf ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy"> ein Isomorphismus ist und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(x)\cong \{x\}\times V}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(x)\cong \{x\}\times V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c44c3cb47d0866200a27286d8842edaf6537ab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.187ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(x)\cong \{x\}\times V}" loading="lazy"> gilt. Häufig betrachtet man reelle oder komplexe Vektorbündel, bei denen der Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy"> durch die <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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} }" loading="lazy"> bzw. durch die <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" 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 {C} }" loading="lazy"> gegeben sind.

Es gibt eine natürliche Bijektion zwischen den Isomorphieklassen von Vektorbündeln mit Rang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> von <a href="Parakompakter_Raum" title="Parakompakter Raum">parakompakten</a> Räumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> und der Menge der Homotopieklassen von Abbildungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> in die <a href="Gra%C3%9Fmann-Mannigfaltigkeit" title="Graßmann-Mannigfaltigkeit">Graßmann-Mannigfaltigkeit</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-dimensionalen Unterräumen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\infty }:}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223cb3a694230e73ef5876a14640b05b28b5f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.845ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\infty }:}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [B,G_{n}(\mathbb {R} ^{\infty })]\cong Vect^{n}(B).}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [B,G_{n}(\mathbb {R} ^{\infty })]\cong Vect^{n}(B).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f33799eee25c4d0e3c4cd0ebb316b233e67d76f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.754ex; height:2.843ex;" alt="{\displaystyle [B,G_{n}(\mathbb {R} ^{\infty })]\cong Vect^{n}(B).}" loading="lazy"><a href="#cite_note-23">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Beispiele_2">Beispiele</h3></div>

<ul><li>Das <a href="Tangentialb%C3%BCndel" title="Tangentialbündel">Tangentialbündel</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}\subseteq \mathbb {R} ^{n+1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{n}\subseteq \mathbb {R} ^{n+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bfaa7cbad8044f274e5090fbee1a9f035273986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.836ex; height:2.843ex;" alt="{\displaystyle S^{n}\subseteq \mathbb {R} ^{n+1}}" loading="lazy"> mit Totalraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\{(x,v)\in S^{n}\times \mathbb {R} ^{n+1}|x\perp v=0\}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\{(x,v)\in S^{n}\times \mathbb {R} ^{n+1}|x\perp v=0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a8c21f5dc1acc79cc597ca505595280f44ae292.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.381ex; height:3.176ex;" alt="{\displaystyle E=\{(x,v)\in S^{n}\times \mathbb {R} ^{n+1}|x\perp v=0\}}" loading="lazy"> und Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\to S^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\colon E\to S^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b87915f45a47e8e0cebbfdd793e8dc407d4f2bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.423ex; height:2.676ex;" alt="{\displaystyle p\colon E\to S^{n}}" loading="lazy"> ist ein Vektorbündel mit Fasern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(b)\cong \mathbb {R} ^{n}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{-1}(b)\cong \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cdc7f69e28acbd590eeefcf410498c73df7c88b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.393ex; height:3.176ex;" alt="{\displaystyle p^{-1}(b)\cong \mathbb {R} ^{n}}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in S^{n}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in S^{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/032e0f6c3a44db736c01df84839a0bff1a81d75e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.225ex; height:2.343ex;" alt="{\displaystyle b\in S^{n}.}" loading="lazy"></li>

<li>Das kanonische Vektorbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{k}^{n}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/891c907cfbd1cbc77949ca15d00f5a352c5711fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.498ex; height:2.843ex;" alt="{\displaystyle \gamma _{k}^{n}}" loading="lazy"> mit Rang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> der Graßmann-Mannigfaltigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{k}(\mathbb {R} ^{n})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{k}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f6f9b528f8f51bc0f5de67db5e561b0d0ee3484.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.621ex; height:2.843ex;" alt="{\displaystyle G_{k}(\mathbb {R} ^{n})}" loading="lazy"> ist durch den Totalraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\{(V,x)\in G_{k}(\mathbb {R} ^{n})\times \mathbb {R} ^{n}|x\in V\}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\{(V,x)\in G_{k}(\mathbb {R} ^{n})\times \mathbb {R} ^{n}|x\in V\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e34c6e18e3ef828c3491ef1315a835cd2370091c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.962ex; height:2.843ex;" alt="{\displaystyle E=\{(V,x)\in G_{k}(\mathbb {R} ^{n})\times \mathbb {R} ^{n}|x\in V\}}" loading="lazy"> und die Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{k}^{n}\colon E\to G_{k}(\mathbb {R} ^{n})}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{k}^{n}\colon E\to G_{k}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74e3768b8cf3d40af57c0e42987296ef82398bde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.543ex; height:3.009ex;" alt="{\displaystyle \gamma _{k}^{n}\colon E\to G_{k}(\mathbb {R} ^{n})}" loading="lazy"> gegeben.<a href="#cite_note-24">[24]</a></li></ul>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Sph%C3%A4renb%C3%BCndel" title="Sphärenbündel">Sphärenbündel</a></div>

Ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Sphärenbündel ist ein Faserbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> mit der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Sphäre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee006452a59bf1eb29983b4412348b66517a2d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}" loading="lazy"> als Faser. Oft ist ein Sphärenbündel zusammen mit der <a href="Orthogonale_Gruppe" title="Orthogonale Gruppe">orthogonalen Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0852b83035449116b712c15ebeead2a3d8998117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.98ex; height:2.843ex;" alt="{\displaystyle O(n+1)}" loading="lazy"> als Strukturgruppe gegeben.<a href="#cite_note-25">[25]</a>

Ein Sphärenbündel wird orientierbar genannt, falls die Strukturgruppe durch die <a href="Drehgruppe" title="Drehgruppe">Drehgruppe</a> gegeben ist.<a href="#cite_note-26">[26]</a>

Die Kohomologie von Sphärenbündeln kann mittels der <a href="Gysin-Sequenz" title="Gysin-Sequenz">Gysin-Sequenz</a> berechnet werden.

<div class="mw-heading mw-heading2"><h2 id="Kohomologie_von_Faserbündeln">Kohomologie von Faserbündeln</h2></div>

Die Bestimmung der Kohomologiegruppen von Faserbündeln ist deutlich schwieriger, als die Bestimmung der Homotopiegruppen. Die Homotopiegruppen sind durch eine lange <a href="Exakte_Sequenz" title="Exakte Sequenz">exakte Sequenz</a> gegeben, die Kohomologiegruppen haben dagegen nur unter bestimmten Voraussetzungen eine lange exakte Sequenz.

Für ein triviales Bündel ist die Beziehung der Kohomologiegruppen durch die <a href="Satz_von_K%C3%BCnneth" title="Satz von Künneth">Künneth-Formel</a> gegeben. Für beliebige Faserbündel werden Hilfsmittel, wie <a href="Spektralsequenz" title="Spektralsequenz">Spektralsequenzen</a> benötigt.

Der Satz von Leray-Hirsch liefert ausreichende Bedingungen an ein Faserbündel, sodass die Struktur der Kohomologiegruppen der eines trivialen Bündels sehr ähnlich ist.

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df88c6333caaf6471cf277f24b802ff9931b133e.svg" 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 (n-1)}" loading="lazy">-Sphärenbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\to B,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\colon E\to B,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/572c154e775e332484cd5a05cabe21d57509bcbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B,}" loading="lazy"> die zusätzlich eine Orientierbarkeitsbedingung erfüllen, existiert eine lange exakte Sequenz von Kohomologiegruppen. Die Sequenz ist unter dem Namen <a href="Gysin-Sequenz" title="Gysin-Sequenz">Gysin-Sequenz</a> bekannt:

<msup>

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</msup>

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</mpadded>

</mpadded>

</munderover>

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<msup>

</mrow>

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</mrow>

</mpadded>

<msup>

</mrow>

</msup>

</mpadded>

</munderover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdots \to H^{i-n}(B;R){\xrightarrow[{}]{\smile e}}H^{i}(B;R){\xrightarrow[{}]{p^{*}}}H^{i}(E;R)\to H^{i-n+1}(B;R)\to \cdots .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eec39b6b6ee83d35570257c5e89ac01e656a7882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.654ex; margin-top: -0.288ex; margin-bottom: -0.518ex; width:67.777ex; height:5.676ex;" alt="{\displaystyle \cdots \to H^{i-n}(B;R){\xrightarrow[{}]{\smile e}}H^{i}(B;R){\xrightarrow[{}]{p^{*}}}H^{i}(E;R)\to H^{i-n+1}(B;R)\to \cdots .}" loading="lazy"></dd></dl>

Hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> eine bestimmte <a href="Euler-Klasse" title="Euler-Klasse">Eulerklasse</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{n}(B;R).}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{n}(B;R).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f76d44b858d69474a907e43913b242d1e24d0d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.34ex; height:2.843ex;" alt="{\displaystyle H^{n}(B;R).}" loading="lazy"><a href="#cite_note-27">[27]</a>

<div class="mw-heading mw-heading3"><h3 id="Beispiele_3">Beispiele</h3></div>

<ul><li>Das Hopf-Bündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\to S^{3}\to S^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}\to S^{3}\to S^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37021ba44d812157a508dfaf044ce53466dfd7ab.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^{1}\to S^{3}\to S^{2}}" loading="lazy"> hat nicht die Kohomologiestruktur eines trivialen Bündels, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(S^{3})\not \approx H^{*}(S^{2})\otimes H^{*}(S^{1})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{*}(S^{3})\not \approx H^{*}(S^{2})\otimes H^{*}(S^{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/005284081f2b3f82b86437c357754989321c7eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.569ex; height:3.176ex;" alt="{\displaystyle H^{*}(S^{3})\not \approx H^{*}(S^{2})\otimes H^{*}(S^{1})}" loading="lazy"> gilt.<a href="#cite_note-28">[28]</a></li>

<li>Für das Faserbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(n-1)\hookrightarrow U(n)\to S^{2n-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(n-1)\hookrightarrow U(n)\to S^{2n-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0feff415e745578c5449332766306fc05d4b6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.159ex; height:3.176ex;" alt="{\displaystyle U(n-1)\hookrightarrow U(n)\to S^{2n-1}}" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(U(n);\mathbb {Z} )\approx \Lambda _{\mathbb {Z} }[x_{1},x_{3},\cdots ,x_{2n-1}].}">

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{*}(U(n);\mathbb {Z} )\approx \Lambda _{\mathbb {Z} }[x_{1},x_{3},\cdots ,x_{2n-1}].}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cc070a75ef57cfd2a325874befd831974b0b481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.969ex; height:2.843ex;" alt="{\displaystyle H^{*}(U(n);\mathbb {Z} )\approx \Lambda _{\mathbb {Z} }[x_{1},x_{3},\cdots ,x_{2n-1}].}" loading="lazy"><a href="#cite_note-29">[29]</a></li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Herbert Seifert: <cite style="font-style:italic">Topologie Dreidimensionaler Gefaserter Räume</cite>. Band 60. Acta Mathematica, 1933, S. 147–238, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF02398271">10.1007/BF02398271</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:Faserb%C3%BCndel&rft.au=Herbert+Seifert&rft.btitle=Topologie+Dreidimensionaler+Gefaserter+R%C3%A4ume&rft.date=1933&rft.doi=10.1007%2FBF02398271&rft.genre=book&rft.pages=147-238&rft.pub=Acta+Mathematica&rft.volume=60" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Hassler Whitney: <cite style="font-style:italic">Sphere-Spaces</cite>. Band 21, Nr. 7. Proceedings of the National Academy of Science of the United States of America, 12. Juni 1935, S. 464–468, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073/pnas.21.7.464">10.1073/pnas.21.7.464</a>, <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076627/">PMC 1076627</a> (freier Volltext).<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:Faserb%C3%BCndel&rft.au=Hassler+Whitney&rft.btitle=Sphere-Spaces&rft.date=1935-06-12&rft.doi=10.1073%2Fpnas.21.7.464&rft.genre=book&rft.issue=7&rft.pages=464-468&rft.pmc=1076627&rft.pub=Proceedings+of+the+National+Academy+of+Science+of+the+United+States+of+America&rft.volume=21" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">  Preface

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">  Preface

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 376–377.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=376-377&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 379.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=379&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 3.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=3&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 377.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=377&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 377.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=377&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 4.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=4&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 377–379.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=377-379&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1, S. 11.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.pages=11&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1, S. 14.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.pages=14&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Gerd Laures, Markus Szymik: <cite style="font-style:italic">Grundkurs Topologie</cite>. 2. Auflage. Springer Spektrum, Berlin / Heidelberg 2014, ISBN 978-3-662-45952-2, S. 184, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-45953-9">10.1007/978-3-662-45953-9</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:Faserb%C3%BCndel&rft.au=Gerd+Laures%2C+Markus+Szymik&rft.btitle=Grundkurs+Topologie&rft.date=2014&rft.doi=10.1007%2F978-3-662-45953-9&rft.edition=2.&rft.genre=book&rft.isbn=9783662459522&rft.pages=184&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer+Spektrum" style="display:none">

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 7.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=7&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> James F. Davis, Paul Kirk: <cite style="font-style:italic">Lecture Notes in Algebraic Topology</cite>. 1991, S. 77–80.<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:Faserb%C3%BCndel&rft.au=James+F.+Davis%2C+Paul+Kirk&rft.btitle=Lecture+Notes+in+Algebraic+Topology&rft.date=1991&rft.genre=book&rft.pages=77-80" style="display:none">

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 6–9.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=6-9&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 14.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=14&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> James F. Davis, Paul Kirk: <cite style="font-style:italic">Lecture Notes in Algebraic Topology</cite>. 1991, S. 84.<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:Faserb%C3%BCndel&rft.au=James+F.+Davis%2C+Paul+Kirk&rft.btitle=Lecture+Notes+in+Algebraic+Topology&rft.date=1991&rft.genre=book&rft.pages=84" style="display:none">

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1, S. 48–50.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.pages=48-50&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">

</li>

<li id="cite_note-21"><a href="#cite_ref-21">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 36.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=36&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-22"><a href="#cite_ref-22">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1, S. 43–44.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.pages=43-44&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">

</li>

<li id="cite_note-23"><a href="#cite_ref-23">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1, S. 23.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.pages=23&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">

</li>

<li id="cite_note-24"><a href="#cite_ref-24">↑</a> Dale Husemoller: <cite style="font-style:italic">Fibre Bundles</cite>. Springer Verlag, Princeton NJ 1994, ISBN 0-387-94087-1, S. 12–13.<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:Faserb%C3%BCndel&rft.au=Dale+Husemoller&rft.btitle=Fibre+Bundles&rft.date=1994&rft.genre=book&rft.isbn=0387940871&rft.pages=12-13&rft.place=Princeton+NJ&rft.pub=Springer+Verlag" style="display:none">

</li>

<li id="cite_note-25"><a href="#cite_ref-25">↑</a> Edwin H. Spanier: <cite style="font-style:italic">Algebraic Topology</cite>. Springer Science & Business Media, ISBN 978-0-387-94426-5, S. 91, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4684-9322-1">10.1007/978-1-4684-9322-1</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:Faserb%C3%BCndel&rft.au=Edwin+H.+Spanier&rft.btitle=Algebraic+Topology&rft.doi=10.1007%2F978-1-4684-9322-1&rft.genre=book&rft.isbn=9780387944265&rft.pages=91&rft.pub=Springer+Science+%26+Business+Media" style="display:none">

</li>

<li id="cite_note-26"><a href="#cite_ref-26">↑</a> Norman Steenrod: <cite style="font-style:italic">The Topology of Fibre Bundles</cite>. Princeton University Press, Princeton NJ 1951, ISBN 0-691-08055-0, S. 34.<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:Faserb%C3%BCndel&rft.au=Norman+Steenrod&rft.btitle=The+Topology+of+Fibre+Bundles&rft.date=1951&rft.genre=book&rft.isbn=0691080550&rft.pages=34&rft.place=Princeton+NJ&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-27"><a href="#cite_ref-27">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 438.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=438&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-28"><a href="#cite_ref-28">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 432.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=432&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-29"><a href="#cite_ref-29">↑</a> Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X, S. 434.<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:Faserb%C3%BCndel&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.pages=434&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none">

</li>

</ol>

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<ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/encyclopedia/FiberBundle.html">PlanetMath: Fiber Bundle</a></li>

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