English+GermanWikipedia - /page/entry.mu

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<title>Faserung</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Faserung</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">Der Begriff der Faserung verallgemeinert den Begriff eines <a href="Faserb%C3%BCndel" title="Faserbündel">Faserbündels</a> und spielt 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 eine wichtige Rolle.

Anwendung finden Faserungen zum Beispiel in <a href="Postnikow-Turm" title="Postnikow-Turm">Postnikow-Systemen</a> oder der <a href="Obstruktionstheorie" title="Obstruktionstheorie">Obstruktionstheorie</a>.

In diesem Artikel sind alle Abbildungen <a href="Stetige_Funktion" title="Stetige Funktion">stetige</a> Abbildungen zwischen <a href="Topologischer_Raum" title="Topologischer Raum">topologischen Räumen</a>.

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

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

Eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\to B}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> erfüllt die Homotopie-Hochhebungseigenschaft für einen Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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

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</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">, falls:

<ul><li>für jede <a href="Homotopie" title="Homotopie">Homotopie</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon X\times [0,1]\to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd9099735a17a7378fbcf1112b509cef6e8fd60e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.224ex; height:2.843ex;" alt="{\displaystyle h\colon X\times [0,1]\to B}" loading="lazy"> und</li>

<li>für jede Abbildung (auch Lift genannt) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {h}}_{0}\colon X\to E,}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}_{0}\colon X\to E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f0c4074646848bf495ee8f764404144442a092b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.444ex; height:3.009ex;" alt="{\displaystyle {\tilde {h}}_{0}\colon X\to E,}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}=h|_{X\times {0}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h_{0}=h|_{X\times {0}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d71e6903d7679f5b63e2ee037ffd3549cfc7ee65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.21ex; height:3.009ex;" alt="{\displaystyle h_{0}=h|_{X\times {0}}}" loading="lazy"> hochhebt (bzw. liftet) (d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}=p\circ {\tilde {h}}_{0}}">

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</msub>

<msub>

<mover>

</mover>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle h_{0}=p\circ {\tilde {h}}_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52b3dfc30934e278054d49adb865117df33eb631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.249ex; height:3.009ex;" alt="{\displaystyle h_{0}=p\circ {\tilde {h}}_{0}}" loading="lazy">),</li></ul>

existiert eine Homotopie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {h}}\colon X\times [0,1]\to E,}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}\colon X\times [0,1]\to E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f973c19c2051a0f9cb863fa34e2837b74b4f245c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.882ex; height:3.176ex;" alt="{\displaystyle {\tilde {h}}\colon X\times [0,1]\to E,}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> hochhebt (d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=p\circ {\tilde {h}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe20127acba60acae6e948ad61dd699ea80c4e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.141ex; height:3.009ex;" alt="{\displaystyle h=p\circ {\tilde {h}}}" loading="lazy">) mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {h}}_{0}={\tilde {h}}|_{X\times {0}}.}">

<msub>

<mover>

</mover>

</mrow>

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</msub>

<mover>

</mover>

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<msub>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}_{0}={\tilde {h}}|_{X\times {0}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebc5579255184c81c9c30cf95b14f86817fb3c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.857ex; height:3.343ex;" alt="{\displaystyle {\tilde {h}}_{0}={\tilde {h}}|_{X\times {0}}.}" loading="lazy">

Das folgende <a href="Kommutatives_Diagramm" title="Kommutatives Diagramm">kommutative Diagramm</a> zeigt die Situation:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[4]S.66}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle ^{[4]S.66}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/981af585d5313d167d22231f0c02cac3662ff425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.13ex; height:2.676ex;" alt="{\displaystyle ^{[4]S.66}}" loading="lazy">

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

Eine Faserung (oder auch Hurewicz-Faserung) ist eine Abbildung <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"> welche die Homotopie-Hochhebungseigenschaft für alle Räume <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"> erfüllt. Der 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"> wird Basisraum und der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<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"> wird Totalraum genannt. Als Faser über <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"> bezeichnet man den Unterraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(b)=F_{b}\subseteq E.}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee790d271801c50ff10ceb190ab87c59f4bfd138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.45ex; height:3.176ex;" alt="{\displaystyle p^{-1}(b)=F_{b}\subseteq E.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[4]S.66}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[4]S.66}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/981af585d5313d167d22231f0c02cac3662ff425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.13ex; height:2.676ex;" alt="{\displaystyle ^{[4]S.66}}" loading="lazy">

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

Eine Serre-Faserung (auch schwache Faserung genannt) ist eine Abbildung <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"> welche die Homotopie-Hochhebungseigenschaft für alle <a href="Zellkomplex" title="Zellkomplex">CW-Komplexe</a> erfüllt.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.375-376}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.375-376}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04801e88ad180f1c8bbd6521ebcdf0b184f175c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:9.697ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.375-376}}" loading="lazy">

Jede Hurewicz-Faserung ist eine Serre-Faserung.

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

Eine Abbildung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> wird Quasifaserung genannt, falls für jedes <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/b0c119df3e85bce0baaa3cc0df34f9d5f6eb15c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.249ex; height:2.509ex;" alt="{\displaystyle b\in B,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in p^{-1}(b)}">

<msup>

</mrow>

</msup>

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/549c8b2ab67f5ad102a0cddbc9712e08b5a509d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.233ex; height:3.176ex;" alt="{\displaystyle e\in p^{-1}(b)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\geq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/405e1424cb9c4fc171c433a8e8f04b3e5938e366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.063ex; height:2.343ex;" alt="{\displaystyle i\geq 0}" loading="lazy"> gilt, dass die induzierte Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{*}\colon \pi _{i}(E,p^{-1}(b),e)\to \pi _{i}(B,b)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{*}\colon \pi _{i}(E,p^{-1}(b),e)\to \pi _{i}(B,b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55a28216761969ba3ce4edddca533485b8760e2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:29.861ex; height:3.176ex;" alt="{\displaystyle p_{*}\colon \pi _{i}(E,p^{-1}(b),e)\to \pi _{i}(B,b)}" loading="lazy"> ein <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a> ist.

Jede Serre-Faserung ist eine Quasifaserung.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[5]S.241-242}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[5]S.241-242}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf3c4073291d6c69387f15bda6027ac9562fa138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:9.697ex; height:2.676ex;" alt="{\displaystyle ^{[5]S.241-242}}" loading="lazy">

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

<ul><li>Die <a href="Projektion_(Mengenlehre)" title="Projektion (Mengenlehre)">Projektion</a> auf den ersten Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon B\times F\to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b82c07bf682791ea2d5a52fdf6669c580cc4f6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.016ex; height:2.509ex;" alt="{\displaystyle p\colon B\times F\to B}" loading="lazy"> ist eine Faserung.</li>

<li>Jede <a href="%C3%9Cberlagerung_(Topologie)" title="Überlagerung (Topologie)">Überlagerung</a> <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> erfüllt die Homotopie-Hochhebungseigenschaft 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_/eb734a37dd21ce173a46342d1cc64c92/5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> Speziell gibt es für jede Homotopie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon X\times [0,1]\to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd9099735a17a7378fbcf1112b509cef6e8fd60e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.224ex; height:2.843ex;" alt="{\displaystyle h\colon X\times [0,1]\to B}" loading="lazy"> und jeden Lift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {h}}_{0}\colon X\to E}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}_{0}\colon X\to E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dec0e5f0d90b7a35936cc75bcdef5e85953fb4ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.797ex; height:3.009ex;" alt="{\displaystyle {\tilde {h}}_{0}\colon X\to E}" loading="lazy"> einen eindeutig definierten Lift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {h}}\colon X\to B}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}\colon X\to B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed975ef3448b877ec868d52a95b352b28cf6693d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.731ex; height:2.676ex;" alt="{\displaystyle {\tilde {h}}\colon X\to B}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ {\tilde {h}}=h.}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c6615944dd840f0de3b9033b15724ec56e65535.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.877ex; height:3.009ex;" alt="{\displaystyle p\circ {\tilde {h}}=h.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[2]S.159}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[2]S.159}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ef1269fca6589adc19cf28823804bc85672478a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[2]S.159}}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[3]S.50}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[3]S.50}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9cc5fa92ed9dc891a57bbfe03b66ce022b96c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.13ex; height:2.676ex;" alt="{\displaystyle ^{[3]S.50}}" loading="lazy"></li>

<li><a href="Faserb%C3%BCndel" title="Faserbündel">Faserbündel</a> <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> erfüllen die Homotopie-Hochhebungseigenschaft für alle CW-Komplexe.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.379}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.379}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75880ab5cbbcd6af0226007cd936703ac5ebc2c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.379}}" loading="lazy"></li>

<li>Ein Faserbündel mit <a href="Parakompakter_Raum" title="Parakompakter Raum">parakompaktem</a> Hausdorff Basisraum erfüllt die Homotopie-Hochhebungseigenschaft für alle Räume.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.379}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.379}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75880ab5cbbcd6af0226007cd936703ac5ebc2c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.379}}" loading="lazy"></li>

<li>Eine Faserung, welche kein Faserbündel ist, ist die von der Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon \partial I^{k}\to I^{k}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\colon \partial I^{k}\to I^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23c0b0d0dac53b1a9662fa9452842db602dfc420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.379ex; height:2.676ex;" alt="{\displaystyle i\colon \partial I^{k}\to I^{k}}" loading="lazy"> induzierte Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{*}\colon X^{I^{k}}\to X^{\partial I^{k}},}">

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{*}\colon X^{I^{k}}\to X^{\partial I^{k}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adab9a8167fd203f32d8089c7ca07aa2004b5382.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.981ex; height:3.343ex;" alt="{\displaystyle i^{*}\colon X^{I^{k}}\to X^{\partial I^{k}},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41f2d6e44ab029b7cee559d32df7149e9e221baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.377ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {N} ,}" loading="lazy"> <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"> ein topologischer Raum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{A}=\{f\colon A\to X\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{A}=\{f\colon A\to X\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73dddd3e7fad1513c850f1953bee0e140c5a7fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.535ex; height:3.176ex;" alt="{\displaystyle X^{A}=\{f\colon A\to X\}}" loading="lazy"> der Raum aller stetigen Abbildungen mit der <a href="Kompakt-Offen-Topologie" title="Kompakt-Offen-Topologie">Kompakt-Offen-Topologie</a> ist.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[2]S.198}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[2]S.198}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3372c43881d90c751050e1253bf5f416da3516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[2]S.198}}" loading="lazy"></li>

<li>Die <a href="Hopf-Faserung" title="Hopf-Faserung">Hopf-Faserung</a> <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"> ist ein nicht triviales Faserbündel und speziell eine Serre-Faserung.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Grundlegende_Konzepte">Grundlegende Konzepte</h2></div>

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

Eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon E_{1}\to E_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd57b96b5c68ea29b8d1413a1f9e7d4fdd1defce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.466ex; height:2.509ex;" alt="{\displaystyle f\colon E_{1}\to E_{2}}" loading="lazy"> zwischen Totalräumen von zwei Faserungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}\colon E_{1}\to B}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96b41d06dda27fdd10db49ac19528127da87a133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.495ex; height:2.509ex;" alt="{\displaystyle p_{1}\colon E_{1}\to B}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}\colon E_{2}\to B}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49dd517e7a74217fdaf5e6b3f6bb68ef7d1d37dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.495ex; height:2.509ex;" alt="{\displaystyle p_{2}\colon E_{2}\to B}" loading="lazy"> mit gleichem Basisraum ist ein Faserungs-Homomorphismus, falls das Diagramm

<a href="Kommutatives_Diagramm" title="Kommutatives Diagramm">kommutiert</a>. Die Abbildung <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_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ist eine Faser-Homotopieäquivalenz, falls zusätzlich ein Faserungs-Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon E_{2}\to E_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94979428e5cbed946389cc7f419f13d5bc5eb38a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.303ex; height:2.509ex;" alt="{\displaystyle g\colon E_{2}\to E_{1}}" loading="lazy"> existiert, sodass die Verknüpfungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}" loading="lazy"> homotop, durch Faserungs-Homomorphismen, zu den Identitäten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Id_{E_{2}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb6c40393df24357cbae6765ae410568690b9503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.657ex; height:2.843ex;" alt="{\displaystyle Id_{E_{2}}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Id_{E_{1}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/871938e8a362437a0c5df8444fbe18ceb1221cb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.657ex; height:2.843ex;" alt="{\displaystyle Id_{E_{1}}}" loading="lazy">sind.<a href="#cite_note-:0-1">[1]</a>

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

Gegeben seien eine Faserung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> und eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dec1893560fabff9fa9c17b83b71f7f97996119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.434ex; height:2.509ex;" alt="{\displaystyle f\colon A\to B}" loading="lazy">. Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{f}\colon f^{*}(E)\to A}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{f}\colon f^{*}(E)\to A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3fe2b32caaa31b14bc305b05ab00c60252e0a30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:14.746ex; height:3.009ex;" alt="{\displaystyle p_{f}\colon f^{*}(E)\to A}" loading="lazy"> ist eine Faserung, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(E)=\{(a,e)\in A\times E|f(a)=p(e)\}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}(E)=\{(a,e)\in A\times E|f(a)=p(e)\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a65a231b1195f9ba2f1611b2c0a057f20cafe338.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.865ex; height:2.843ex;" alt="{\displaystyle f^{*}(E)=\{(a,e)\in A\times E|f(a)=p(e)\}}" loading="lazy"> der <a href="Pullbackb%C3%BCndel" class="mw-redirect" title="Pullbackbündel">Pullback</a> ist und die Projektionen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(E)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89618fa2b7ae703593dffb7aa170aa5e2fa825a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.96ex; height:2.843ex;" alt="{\displaystyle f^{*}(E)}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> und <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_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.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"> das kommutative Diagramm liefern:

Die Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3281617d6e3bba7242afb5c1d727dd2d0a9c5aba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.395ex; height:2.343ex;" alt="{\displaystyle p_{f}}" loading="lazy"> wird Pullback-Faserung oder auch induzierte Faserung genannt.<a href="#cite_note-:0-1">[1]</a>

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

Mit der Wegeraumkonstruktion kann jede stetige Abbildung zu einer Faserung erweitert werden, indem man den <a href="Definitionsmenge" title="Definitionsmenge">Definitionsbereich</a> der Abbildung zu einem homotopieäquivalenten Raum vergrößert. Diese Faserung wird dann Wegeraum-Faserung genannt.

Der Totalraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{f}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12ce471791c1904f7401133fb39c86f5a088f205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle E_{f}}" loading="lazy"> der Wegeraum-Faserung für eine stetige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dec1893560fabff9fa9c17b83b71f7f97996119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.434ex; height:2.509ex;" alt="{\displaystyle f\colon A\to B}" loading="lazy"> zwischen topologischen Räumen besteht aus Paaren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,\gamma )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c39d9ed06fc64af403f53351221ab190c81bb811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.335ex; height:2.843ex;" alt="{\displaystyle (a,\gamma )}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a97387981adb5d65f74518e20b6785a284d7abd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}" loading="lazy"> und <a href="Weg_(Mathematik)" title="Weg (Mathematik)">Wegen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon I\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd98227c991415928a881c4e2f1e32849fec6ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.676ex;" alt="{\displaystyle \gamma \colon I\to B}" loading="lazy"> mit Startpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)=f(a)~}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma (0)=f(a)~}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddc9ac6477051521d819e198807b1087e6e62e57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.231ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=f(a)~}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87ec65159c44769434523e46928bc1b82681f842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}" loading="lazy"> das <a href="Einheitsintervall" class="mw-redirect" title="Einheitsintervall">Einheitsintervall</a> ist. Der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{f}=\{(a,\gamma )\in A\times B^{I}|\gamma (0)=f(a)\}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{f}=\{(a,\gamma )\in A\times B^{I}|\gamma (0)=f(a)\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d0e7d5bfdc3347869ff0115fb10ccc57caf4b7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.156ex; height:3.343ex;" alt="{\displaystyle E_{f}=\{(a,\gamma )\in A\times B^{I}|\gamma (0)=f(a)\}}" loading="lazy"> trägt die <a href="Teilraumtopologie" title="Teilraumtopologie">Teilraumtopologie</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B^{I},}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c49b798566d985e5d034fe4a11758a898a1bf6d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.055ex; height:3.009ex;" alt="{\displaystyle A\times B^{I},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{I}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c615f6c6d649f468aa611867b86f4eeed49c7ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.825ex; height:2.676ex;" alt="{\displaystyle B^{I}}" loading="lazy"> den Raum aller Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f04b7f5f095ad6938fe73a04055d0ffd81d60784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.55ex; height:2.176ex;" alt="{\displaystyle I\to B}" loading="lazy"> beschreibt und die <a href="Kompakt-Offen-Topologie" title="Kompakt-Offen-Topologie">Kompakt-Offen-Topologie</a> trägt.

Die Wegeraum-Faserung ist durch die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E_{f}\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef0b2165f9067abea13d30655f022165b0df025d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:10.522ex; height:2.843ex;" alt="{\displaystyle p\colon E_{f}\to B}" loading="lazy"> mit der Abbildungsvorschrift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(a,\gamma )=\gamma (1)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cdacb05749dc14e6466e579a4f1abd77cd6bffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.927ex; height:2.843ex;" alt="{\displaystyle p(a,\gamma )=\gamma (1)}" loading="lazy"> gegeben. Die Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{f}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89bf80a3fe301ea858f26f6c2b3ff9e3867e4bbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.631ex; height:2.843ex;" alt="{\displaystyle F_{f}}" loading="lazy"> wird auch <a href="Homotopie-Faser" title="Homotopie-Faser">Homotopie-Faser</a> von <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_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> genannt und besteht aus den Paaren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,\gamma )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c39d9ed06fc64af403f53351221ab190c81bb811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.335ex; height:2.843ex;" alt="{\displaystyle (a,\gamma )}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a97387981adb5d65f74518e20b6785a284d7abd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}" loading="lazy"> und Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon [0,1]\to B,}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb4d31b2eb7aa7194e451818b636fd24e6af8251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.974ex; height:2.843ex;" alt="{\displaystyle \gamma \colon [0,1]\to B,}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)=f(a)}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (0)=f(a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28c0cfbbe8ab42112731f76678d41577962d44dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.65ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=f(a)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (1)=b_{0}\in B}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma (1)=b_{0}\in B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26b2977a70b879d30ef643b687bd94235681f22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.989ex; height:2.843ex;" alt="{\displaystyle \gamma (1)=b_{0}\in B}" loading="lazy"> gilt.

Für den Spezialfall der Inklusion des Basispunktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon b_{0}\to B,}">

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<annotation encoding="application/x-tex">{\displaystyle i\colon b_{0}\to B,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d6fbc3ea488099b6d73e1f2514007ff27c7b635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle i\colon b_{0}\to B,}" loading="lazy"> ergibt sich ein wichtiges Beispiel der Wegeraum-Faserung. Der Totalraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" loading="lazy"> besteht aus allen Wegen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,}">

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</mrow>

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

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</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"> die am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy"> starten. Dieser Raum wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PB}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9152209e2f334627253ea50f939b320ec60df4ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle PB}" loading="lazy"> gekennzeichnet und Wegeraum genannt. Die Wege-Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon PB\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e15121145fdf1a8bc3f00eff21b39f9177e5c9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.18ex; height:2.509ex;" alt="{\displaystyle p\colon PB\to B}" loading="lazy"> ordnet jedem Weg seinen Endpunkt zu, weshalb die Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(b_{0})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e4b240ffb30c5c4875c6b0b838fa22479325751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.453ex; height:3.176ex;" alt="{\displaystyle p^{-1}(b_{0})}" loading="lazy"> aus allen geschlossenen Wegen besteht. Die Faser wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/158a20b94fba6a6ca346660cfac3334034fe09dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.442ex; height:2.176ex;" alt="{\displaystyle \Omega B}" loading="lazy"> gekennzeichnet und <a href="Schleifenraum" title="Schleifenraum">Schleifenraum</a> genannt.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.407-408}}">

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<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.407-408}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43d06b1dc10097dbe8111a066d2e3e2f0ea2a765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:9.697ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.407-408}}" loading="lazy">

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

<ul><li>Die Fasern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(b)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24cf889bd6790c53de93cb41e064a6bd8f08f94a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.398ex; height:3.176ex;" alt="{\displaystyle p^{-1}(b)}" loading="lazy"> über <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"> sind für die einzelnen <a href="Wegzusammenhangskomponente" class="mw-redirect" title="Wegzusammenhangskomponente">Wegzusammenhangskomponenten</a> von <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"> <a href="Homotopie%C3%A4quivalenz" title="Homotopieäquivalenz">homotopieäquivalent</a>.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.405}}">

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<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.405}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87ac01148a17913b491ec1d05e1c6fa017a0af87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.405}}" loading="lazy"></li>

<li>Für eine Homotopie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon [0,1]\times A\to B}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fbf15d17d22dc1c28d4b04ee3caced4be809842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.927ex; height:2.843ex;" alt="{\displaystyle f\colon [0,1]\times A\to B}" loading="lazy"> sind die Pullback Faserungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}^{*}(E)\to A}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{0}^{*}(E)\to A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bb24822644ce02a0e628adf9161de6d4fc1cd38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.317ex; height:3.009ex;" alt="{\displaystyle f_{0}^{*}(E)\to A}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}^{*}(E)\to A}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}^{*}(E)\to A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a5155538136bc39aa0f9c7f9df53feb939843b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.317ex; height:3.009ex;" alt="{\displaystyle f_{1}^{*}(E)\to A}" loading="lazy"> Faser homotopieäquivalent.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.406}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.406}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f243654f3765f72610397934735e7645b8ecc1fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.406}}" loading="lazy"></li>

<li>Ist der 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_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.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"> <a href="Zusammenziehbarer_Raum" title="Zusammenziehbarer Raum">zusammenziehbar</a>, dann ist <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> Faser homotopieäquivalent zu einer Produkt Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\times F\to B.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91d7c6764ce8722a9b6c53984c642b48514d45a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.37ex; height:2.176ex;" alt="{\displaystyle B\times F\to B.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.406}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.406}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f243654f3765f72610397934735e7645b8ecc1fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.406}}" loading="lazy"></li>

<li>Die Wegeraum-Faserung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> ist sich selbst sehr ähnlich. Genauer gilt: Die Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\hookrightarrow E_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2623869791385b87effd96d83f600c15853804b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.457ex; height:2.843ex;" alt="{\displaystyle E\hookrightarrow E_{p}}" loading="lazy"> ist eine Faser-Homotopieäquivalenz.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.408}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.408}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eae5055d79a8521bc7ad383db5aeeff0b7f6ba67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.408}}" loading="lazy"></li>

<li>Ist der Totalraum <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_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.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"> zusammenziehbar, dann gibt es eine <a href="Schwache_Homotopie%C3%A4quivalenz" class="mw-redirect" title="Schwache Homotopieäquivalenz">schwache Homotopieäquivalenz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\to \Omega B.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4051d241394978c3d5b5ebdc13db7dc519b344e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.444ex; height:2.176ex;" alt="{\displaystyle F\to \Omega B.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.408}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.408}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eae5055d79a8521bc7ad383db5aeeff0b7f6ba67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.408}}" loading="lazy"></li></ul>

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

Für eine Faserung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\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 Basispunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}\in B}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}\in B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/576f1554c09ba87c0bfcd2e0904d9b8228552808.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_{0}\in B}" loading="lazy"> ist die Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\hookrightarrow F_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12b149e482dcc1d7ef21518339b766a371ab8c5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.201ex; height:2.843ex;" alt="{\displaystyle F\hookrightarrow F_{p}}" loading="lazy"> der Faser in die Homotopie-Faser eine <a href="Homotopie%C3%A4quivalenz" title="Homotopieäquivalenz">Homotopieäquivalenz</a>. Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon F_{p}\to E}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ec870f90c592c4cbca7c3df099a925f27ca4db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.78ex; height:2.843ex;" alt="{\displaystyle i\colon F_{p}\to E}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(e,\gamma )=e,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50ccb573e0bcf9b0023fa802b8887bd7719ced89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.82ex; height:2.843ex;" alt="{\displaystyle i(e,\gamma )=e,}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34778736d9c6d607a4da3d25594b38dd3e8c82ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon I\to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd98227c991415928a881c4e2f1e32849fec6ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.676ex;" alt="{\displaystyle \gamma \colon I\to B}" loading="lazy"> ein Weg von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(e)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5260672870cc53cf3dea69bf1bd9f958e34ad39b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.152ex; height:2.843ex;" alt="{\displaystyle p(e)}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy"> im Basisraum sind, ist eine Faserung. Sie ist die Pullback-Faserung der Wege-Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PB\to B.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fdbad41a17e31b98fcb1796a0bf9e7d0ac8e358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.534ex; height:2.176ex;" alt="{\displaystyle PB\to B.}" loading="lazy"> Dieses Vorgehen kann nun wieder auf die Faserung <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/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> angewandt und iteriert werden. Dies führt zu einer langen Sequenz:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mpadded>

</mover>

<msub>

</mrow>

</msub>

<mover>

</mpadded>

</mover>

<mover>

</mpadded>

</mover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdots \to F_{j}\to F_{i}\xrightarrow {j} F_{p}\xrightarrow {i} E\xrightarrow {p} B.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c103983fc8a111453c5f8fdce15cd37b9a2d2c46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.401ex; width:32.232ex; height:4.509ex;" alt="{\displaystyle \cdots \to F_{j}\to F_{i}\xrightarrow {j} F_{p}\xrightarrow {i} E\xrightarrow {p} B.}" loading="lazy"></dd></dl>

Die Faser von <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/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> über einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{0}\in p^{-1}(b_{0})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{0}\in p^{-1}(b_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/289c7f1a971aa69ca62ca0c41470d4dfe3c4fe0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.342ex; height:3.176ex;" alt="{\displaystyle e_{0}\in p^{-1}(b_{0})}" loading="lazy"> besteht aus genau den Paaren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{0},\gamma )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc8bfcbef08eac035785ee28e2351efba4670523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.243ex; height:2.843ex;" alt="{\displaystyle (e_{0},\gamma )}" loading="lazy"> mit geschlossenen Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> und Startpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy">, also dem Schleifenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega B.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d28836557a5227ce69b689cdc394abdb53466c72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.089ex; height:2.176ex;" alt="{\displaystyle \Omega B.}" loading="lazy"> Die Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega B\to F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1e1806c5271146c48916f2f1e9eb048b2f11538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.797ex; height:2.176ex;" alt="{\displaystyle \Omega B\to F}" loading="lazy"> ist eine Homotopieäquivalenz und durch <a href="Iteration" title="Iteration">Iteration</a> ergibt sich die Sequenz:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdots \Omega ^{2}B\to \Omega F\to \Omega E\to \Omega B\to F\to E\to B.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e454ede34d971e1fff16a71a8227049b82e7fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:45.533ex; height:2.676ex;" alt="{\displaystyle \cdots \Omega ^{2}B\to \Omega F\to \Omega E\to \Omega B\to F\to E\to B.}" loading="lazy"></dd></dl>

Durch die Dualität von Faserung und <a href="Kofaserung" title="Kofaserung">Kofaserung</a> existiert auch eine Sequenz von Kofaserungen. Diese beiden Sequenzen sind unter dem Namen <a href="Puppe-Sequenz" class="mw-redirect" title="Puppe-Sequenz">Puppe-Sequenzen</a> oder auch Sequenz von Faserungen bzw. Kofaserungen bekannt.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.407-409}}">

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<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.407-409}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc5de4473e37a9557185aab4dbd3c09d1d6052d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:9.697ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.407-409}}" loading="lazy">

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

Eine Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> mit Faser <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 Hauptfaserung genannt, falls ein kommutatives Diagramm existiert:

Die untere Zeile ist eine Sequenz von Faserungen und die vertikalen Abbildungen sind schwache Homotopieäquivalenzen. Hauptfaserungen spielen eine wichtige Rolle bei <a href="Postnikow-Turm" title="Postnikow-Turm">Postnikow-Türmen</a>.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]p.412}}">

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<annotation encoding="application/x-tex">{\displaystyle ^{[1]p.412}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d45e9e3c2adcbda5a1cab117fec6d935dbff4dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.719ex; height:2.676ex;" alt="{\displaystyle ^{[1]p.412}}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Lange_exakte_Homotopiesequenz">Lange exakte Homotopiesequenz</h2></div>

Für eine Serre-Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\colon E\to B}" loading="lazy"> existiert eine lange <a href="Exakte_Sequenz" title="Exakte Sequenz">exakte Sequenz</a> von <a href="Homotopiegruppe" title="Homotopiegruppe">Homotopiegruppen</a>. Für Basispunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}\in B}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0}\in B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/576f1554c09ba87c0bfcd2e0904d9b8228552808.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_{0}\in B}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in F=p^{-1}(b_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in F=p^{-1}(b_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2686634f2018b6f275e6575d289aec576da16294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.427ex; height:3.176ex;" alt="{\displaystyle x_{0}\in F=p^{-1}(b_{0})}" loading="lazy"> ist diese gegeben durch:

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<annotation encoding="application/x-tex">{\displaystyle \cdots \rightarrow \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})\rightarrow \pi _{n-1}(F,x_{0})\rightarrow \cdots \rightarrow \pi _{0}(F,x_{0})\rightarrow \pi _{0}(E,x_{0}).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54641c8cad8b957aeba69179a68565f62d6a1954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:89.993ex; height:2.843ex;" alt="{\displaystyle \cdots \rightarrow \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})\rightarrow \pi _{n-1}(F,x_{0})\rightarrow \cdots \rightarrow \pi _{0}(F,x_{0})\rightarrow \pi _{0}(E,x_{0}).}" loading="lazy"></dd></dl>

Die <a href="Homomorphismus" title="Homomorphismus">Homomorphismen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06ca466327ccb21e747d13379a6ca89dd55b3e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.672ex; height:2.843ex;" alt="{\displaystyle \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb9561159c5504daeba146c66628bf0c3aa75525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.363ex; height:2.843ex;" alt="{\displaystyle \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})}" loading="lazy"> sind die induzierten Homomorphismen der Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon F\hookrightarrow E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a32c76750d3952f963eed27bf8f305a642ecfe38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.26ex; height:2.176ex;" alt="{\displaystyle i\colon F\hookrightarrow E}" loading="lazy"> und der Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\rightarrow B.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a9d37de8fdd86110074d7296fc5c05de2fb42e0.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\rightarrow B.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]S.376}}">

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<annotation encoding="application/x-tex">{\displaystyle ^{[1]S.376}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8599706d9eee698f4f4962853a773c2f039b4bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[1]S.376}}" loading="lazy">

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

Unter den <a href="Hopf-Faserung" title="Hopf-Faserung">Hopf-Faserungen</a> versteht man eine Familie von <a href="Faserb%C3%BCndel" title="Faserbündel">Faserbündeln</a>, deren Faser, Totalraum und Basisraum <a href="Sph%C3%A4re_(Mathematik)" title="Sphäre (Mathematik)">Sphären</a> sind:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ffc646b2a7de909eb0097ed97511191fa8699e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.895ex; height:3.009ex;" alt="{\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1},}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11f623124809dd7b93e78520af01d18b339dac4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.895ex; height:3.009ex;" alt="{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2},}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56e7906c37cd2c77bff8f92d1f99e95b3a8765f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.895ex; height:3.009ex;" alt="{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4},}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39b2465457cf6ca928064d98607f4ec9d237db08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.717ex; height:2.676ex;" alt="{\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}.}" loading="lazy"></dd></dl>

Die lange exakte Homotopiesequenz der Hopf-Faserung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dbd8bb37e502b6bd3cd888c3660a89f783e17fd.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}\rightarrow S^{2}}" loading="lazy"> liefert:

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<annotation encoding="application/x-tex">{\displaystyle \cdots \rightarrow \pi _{n}(S^{1},x_{0})\rightarrow \pi _{n}(S^{3},x_{0})\rightarrow \pi _{n}(S^{2},b_{0})\rightarrow \pi _{n-1}(S^{1},x_{0})\rightarrow \cdots \rightarrow \pi _{1}(S^{1},x_{0})\rightarrow \pi _{1}(S^{3},x_{0})\rightarrow \pi _{1}(S^{2},b_{0}).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62c33eb08879125a375c02352cc33125cd3c5f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:108.375ex; height:3.176ex;" alt="{\displaystyle \cdots \rightarrow \pi _{n}(S^{1},x_{0})\rightarrow \pi _{n}(S^{3},x_{0})\rightarrow \pi _{n}(S^{2},b_{0})\rightarrow \pi _{n-1}(S^{1},x_{0})\rightarrow \cdots \rightarrow \pi _{1}(S^{1},x_{0})\rightarrow \pi _{1}(S^{3},x_{0})\rightarrow \pi _{1}(S^{2},b_{0}).}" loading="lazy"></dd></dl>

Die Sequenz zerfällt in kurze exakte Sequenzen, da die Faser <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"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01e57c690f890937838c10ba57853ff21bf30ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}" loading="lazy"> zu einem Punkt zusammengezogen werden kann:

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<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08f48bf778963ee20a22021af8abd69cb6615fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.058ex; height:3.176ex;" alt="{\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.}" loading="lazy"></dd></dl>

Diese kurze exakte Sequenz <a href="Zerfallende_kurze_exakte_Sequenz" class="mw-redirect" title="Zerfallende kurze exakte Sequenz">zerfällt</a> wegen des <a href="Einh%C3%A4ngung" title="Einhängung">Einhängungshomomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon \pi _{i-1}(S^{1})\to \pi _{i}(S^{2})~}">

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<annotation encoding="application/x-tex">{\displaystyle \phi \colon \pi _{i-1}(S^{1})\to \pi _{i}(S^{2})~}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0ad3e8ec29728b2f748ec15b9567b7cd2d4129f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.734ex; height:3.176ex;" alt="{\displaystyle \phi \colon \pi _{i-1}(S^{1})\to \pi _{i}(S^{2})~}" loading="lazy">und es gibt Isomorphismen:

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<annotation encoding="application/x-tex">{\displaystyle \pi _{i}(S^{2})\cong \pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea83132db73c2cdeed75da34109b30750debbfe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.216ex; height:3.176ex;" alt="{\displaystyle \pi _{i}(S^{2})\cong \pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).}" loading="lazy"></dd></dl>

Die Homotopiegruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i-1}(S^{1})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f9ed4ae5e367650b6233230e0ec7959ceae4108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.61ex; height:3.176ex;" alt="{\displaystyle \pi _{i-1}(S^{1})}" loading="lazy"> sind für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geq 3}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e58e610383919acf4b940f9126b78b231c52137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.063ex; height:2.343ex;" alt="{\displaystyle i\geq 3}" loading="lazy"> trivial, weshalb es Isomorphismen zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}(S^{2})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{i}(S^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41b6695db5b207af218c7f97a46aa9ad546c0905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:3.176ex;" alt="{\displaystyle \pi _{i}(S^{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}(S^{3})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{i}(S^{3})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/970de47268a1c9099a1280d90f8ac2937d8f95b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:3.176ex;" alt="{\displaystyle \pi _{i}(S^{3})}" loading="lazy"> ab <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=3}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9aea65b32ee831db79412d978e090135c02e54c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=3}" loading="lazy"> gibt. Analog kann die Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01e57c690f890937838c10ba57853ff21bf30ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{7}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/893537d33492bb0eb25cfef3b631c456d7fbe5cf.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^{7}}" loading="lazy"> und die Faser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{7}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/893537d33492bb0eb25cfef3b631c456d7fbe5cf.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^{7}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{15}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7ffd3cec3f479db1972e3843813eb259c20b42c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.398ex; height:2.676ex;" alt="{\displaystyle S^{15}}" loading="lazy"> zu einem Punkt zusammengezogen werden. Die kurzen exakten Sequenzen zerfallen weiter, wodurch es Familien von Isomorphismen gibt:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{i}(S^{4})\cong \pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d49b707374e3f351212cf90606d6228634744477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.569ex; height:3.176ex;" alt="{\displaystyle \pi _{i}(S^{4})\cong \pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}(S^{8})\cong \pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{i}(S^{8})\cong \pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/478864621c3d37b23f9376b08107e1a30f656a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.038ex; height:3.176ex;" alt="{\displaystyle \pi _{i}(S^{8})\cong \pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[6]S.111}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[6]S.111}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d229c9e157c1545408782d80bc6b7d81ed6e21b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[6]S.111}}" loading="lazy"></dd></dl>

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

<a href="Spektralsequenz" title="Spektralsequenz">Spektralsequenzen</a> sind wichtige Hilfsmittel in der algebraischen Topologie zur Berechnung von (Ko-)Homologiegruppen.

Die <a href="Spektralsequenz#Leray-Serre-Spektralsequenz" title="Spektralsequenz">Leray-Serre-Spektralsequenz</a> stellt einen Zusammenhang zwischen der (Ko-)Homologie von Totalraum und Faser mit der (Ko-)Homologie des Basisraums einer Faserung her. Für eine Faserung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\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">, wobei der Basisraum ein wegzusammenhängender CW-Komplex ist, und einer additiven <a href="Homologietheorie" title="Homologietheorie">Homologietheorie</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{*}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b5054fe94e137bd0257646b0bc789a2efd8f702.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{*}}" loading="lazy"> existiert eine Spektralsequenz:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{k}(B;G_{q}(F))\cong E_{k,q}^{2}\implies G_{k+q}(E).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd33de0199e01cdc95b0c5f0a44cba1f27456356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:36.677ex; height:3.509ex;" alt="{\displaystyle H_{k}(B;G_{q}(F))\cong E_{k,q}^{2}\implies G_{k+q}(E).}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[7]S.242}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[7]S.242}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86155e155ea0ed44d88a145a74cdb905ed26d260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[7]S.242}}" loading="lazy"></dd></dl>

Faserungen liefern in der Homologie keine langen exakten Sequenzen, wie in der Homotopie. Aber unter bestimmten Bedingungen, liefern Faserungen exakte Sequenzen in der Homologie. Für eine Faserung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\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">, wobei Basisraum und Faser <a href="Wegzusammenh%C3%A4ngend" class="mw-redirect" title="Wegzusammenhängend">wegzusammenhängend</a> sind, die <a href="Fundamentalgruppe" title="Fundamentalgruppe">Fundamentalgruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(B)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ecc0042f43d56fe64dd6c3940c30d14caa2d435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.953ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(B)}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{*}(F)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e3e933a700d3f0dfb4f3df3b42c65d18bfa23be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.536ex; height:2.843ex;" alt="{\displaystyle H_{*}(F)}" loading="lazy"> trivial operiert und zusätzlich die Bedingungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{p}(B)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{p}(B)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fb37896d83cee7cd58142752bbfc5fdabd6f5b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.825ex; height:3.009ex;" alt="{\displaystyle H_{p}(B)=0}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<p<m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4454effb4ea3fce58a5aae4360c9d4195267af4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.569ex; height:2.509ex;" alt="{\displaystyle 0<p<m}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{q}(F)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{q}(F)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e69ad39b07bd6f139bd5040c18d350292ff37f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.731ex; height:3.009ex;" alt="{\displaystyle H_{q}(F)=0}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<q<n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<q<n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec454e631e6ba02a74dd2fb395c4eace0b92486d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.824ex; height:2.509ex;" alt="{\displaystyle 0<q<n}" loading="lazy"> gelten, existiert eine exakte Sequenz:

<msub>

</mrow>

</msub>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

<msub>

</mrow>

</msub>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

<msub>

</mrow>

</msub>

<mover>

</mpadded>

</mover>

<msub>

</mrow>

</msub>

<mover>

<msup>

</mrow>

</msup>

</mpadded>

</mover>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{m+n-1}(F)\xrightarrow {i_{*}} H_{m+n-1}(E)\xrightarrow {f_{*}} H_{m+n-1}(B)\xrightarrow {\tau } H_{m+n-2}(F)\xrightarrow {i^{*}} \cdots \xrightarrow {f_{*}} H_{1}(B)\to 0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d596c5d4d7b5ab345e575c77e236799973860ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.306ex; width:80.992ex; height:4.343ex;" alt="{\displaystyle H_{m+n-1}(F)\xrightarrow {i_{*}} H_{m+n-1}(E)\xrightarrow {f_{*}} H_{m+n-1}(B)\xrightarrow {\tau } H_{m+n-2}(F)\xrightarrow {i^{*}} \cdots \xrightarrow {f_{*}} H_{1}(B)\to 0.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[7]S.250}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[7]S.250}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b51d3606622927eaee717ac9182ebb8227cbcc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[7]S.250}}" loading="lazy"></dd></dl>

Diese Sequenz kann z. B. benutzt werden, um den <a href="Hurewicz-Theorem" title="Hurewicz-Theorem">Satz von Hurewicz</a> zu beweisen oder um die Homologiegruppen von Schleifenräumen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega S^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b99627d155e2fc268cd597bc295d31a686ab1bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.418ex; height:2.343ex;" alt="{\displaystyle \Omega S^{n}}" loading="lazy"> zu berechnen:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{k}(\Omega S^{n})={\begin{cases}\mathbb {Z} &\exists q\in \mathbb {Z} \colon k=q(n-1)\\0&sonst\end{cases}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5e87e4e84dec50ff4a774245006b3e48d5b9bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.733ex; height:6.176ex;" alt="{\displaystyle H_{k}(\Omega S^{n})={\begin{cases}\mathbb {Z} &\exists q\in \mathbb {Z} \colon k=q(n-1)\\0&sonst\end{cases}}.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[8]S.162}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[8]S.162}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/599746a595967e03a35de339e155f7a5e061f132.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[8]S.162}}" loading="lazy"></dd></dl>

Für den Spezialfall einer Faserung <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_/eb734a37dd21ce173a46342d1cc64c92/64a1efef3122f5a2920078b2dd223909708a934a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.069ex; height:2.676ex;" alt="{\displaystyle p\colon E\to S^{n},}" loading="lazy"> bei welcher der Basisraum eine <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 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"> ist, existieren exakte Sequenzen (auch <a href="Wang-Sequenz" title="Wang-Sequenz">Wang Sequenzen</a> genannt) für Homologie und Kohomologie:

<msub>

</mrow>

</msub>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdots \to H_{q}(F)\xrightarrow {i_{*}} H_{q}(E)\to H_{q-n}(F)\to H_{q-1}(F)\to \cdots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/582bdfb5da64be0f34564f31525f1e4b99243007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.365ex; width:54.195ex; height:4.343ex;" alt="{\displaystyle \cdots \to H_{q}(F)\xrightarrow {i_{*}} H_{q}(E)\to H_{q-n}(F)\to H_{q-1}(F)\to \cdots }" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mover>

<msup>

</mrow>

</msup>

</mpadded>

</mover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdots \to H^{q}(E)\xrightarrow {i^{*}} H^{q}(F)\to H^{q-n+1}(F)\to H^{q+1}(E)\to \cdots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4e66ae120313988234e4342c5b859043d70263e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.401ex; width:57.019ex; height:4.343ex;" alt="{\displaystyle \cdots \to H^{q}(E)\xrightarrow {i^{*}} H^{q}(F)\to H^{q-n+1}(F)\to H^{q+1}(E)\to \cdots }" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[4]S.456}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[4]S.456}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/360b78e76097309f486a116d1d11cf8662081f51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[4]S.456}}" loading="lazy"></dd></dl>

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

Für eine Faserung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\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 einem festen kommutativen Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> mit Eins existiert ein <a href="Kontravarianter_Funktor" class="mw-redirect" title="Kontravarianter Funktor">kontravarianter Funktor</a> von dem <a href="Fundamentalgruppoid" title="Fundamentalgruppoid">Fundamentalgruppoid</a> 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"> zur Kategorie von <a href="Graduierter_Modul" class="mw-redirect" title="Graduierter Modul">graduierten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy">-Moduln, welcher jedem <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"> den Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{*}(F_{b},R)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{*}(F_{b},R)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0877878cd6ea76f5e764c5a36c6017a6f976e66e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.025ex; height:2.843ex;" alt="{\displaystyle H_{*}(F_{b},R)}" loading="lazy"> und der Wegeklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\omega ]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e9322de5b680a27fc52afa084255303b3cba69a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle [\omega ]}" loading="lazy"> den Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[\omega ]_{*}\colon H_{*}(F_{\omega (0)},R)\to H_{*}(F_{\omega (1)},R)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h[\omega ]_{*}\colon H_{*}(F_{\omega (0)},R)\to H_{*}(F_{\omega (1)},R)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59ae575fc2732731f853f7ef19bdc47566fb9aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.667ex; height:3.176ex;" alt="{\displaystyle h[\omega ]_{*}\colon H_{*}(F_{\omega (0)},R)\to H_{*}(F_{\omega (1)},R)}" loading="lazy"> zuordnet, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[\omega ]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e5238254b8a6b4e2c3876a8428b1a893ebf9d47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.078ex; height:2.843ex;" alt="{\displaystyle h[\omega ]}" loading="lazy"> eine Homotopieklasse in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [F_{\omega (0)},F_{\omega (1)}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f77114bd0fc37b3a54077611ee022d50f9064f09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.029ex; height:3.176ex;" alt="{\displaystyle [F_{\omega (0)},F_{\omega (1)}]}" loading="lazy"> ist.

Eine Faserung wird orientierbar über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> genannt, falls für jeden geschlossenen Weg <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> in <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"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[\omega ]_{*}=1.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h[\omega ]_{*}=1.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c63d461f1303f65a7715736dd61c3fd4116a7d0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.04ex; height:2.843ex;" alt="{\displaystyle h[\omega ]_{*}=1.}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[4]S.476}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[4]S.476}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb0d284a7a1608fe92659e3b71cf425ed8b0a140.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[4]S.476}}" loading="lazy">

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

Für eine 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"> orientierbare Faserung <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/f7dbc7beb7b90e93453328d72d909a576816b033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.446ex; height:2.509ex;" alt="{\displaystyle p\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 wegzusammenhängendem Basisraum ist die <a href="Euler-Charakteristik" title="Euler-Charakteristik">Euler-Charakteristik</a> des Totalraums definiert durch:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/162d2168d6fc212ea6832cab7b192afd42fdd3e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.819ex; height:2.843ex;" alt="{\displaystyle \chi (E)=\chi (B)\chi (F).}" loading="lazy"></dd></dl>

Die Euler-Charakteristiken des Basisraums und der Faser sind dabei über dem 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"> definiert.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[4]S.481}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{[4]S.481}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da47863f5254abed8864eff9e3fadb9c811a617a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:5.952ex; height:2.676ex;" alt="{\displaystyle ^{[4]S.481}}" loading="lazy">

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

<ul><li>[1] Allen Hatcher: <cite style="font-style:italic">Algebraic Topology</cite>. Cambridge University Press, NY 2001, ISBN 0-521-79160-X.<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:Faserung&rft.au=Allen+Hatcher&rft.btitle=Algebraic+Topology&rft.date=2001&rft.genre=book&rft.isbn=052179160X&rft.place=NY&rft.pub=Cambridge+University+Press" style="display:none"> </li>

<li>[2] 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, <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:Faserung&rft.au=Gerd+Laures%2C+Markus+Szymik&rft.btitle=Grundkurs+Topologie&rft.date=2014-10&rft.doi=10.1007%2F978-3-662-45953-9&rft.edition=2&rft.genre=book&rft.isbn=9783662459522&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer+Spektrum" style="display:none"> </li>

<li>[3] J.P. May: <cite style="font-style:italic">A Concise Course in Algebraic Topology</cite>.<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:Faserung&rft.au=J.P.+May&rft.btitle=A+Concise+Course+in+Algebraic+Topology&rft.genre=book" style="display:none"> </li>

<li>[4] Edwin H. Spanier: <cite style="font-style:italic">Algebraic Topology</cite>. Springer Science & Business Media, ISBN 978-0-387-94426-5, <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:Faserung&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.pub=Springer+Science+%26+Business+Media" style="display:none"> </li>

<li>[5] Albrecht Dold, René Thom: <cite style="font-style:italic">Quasifaserungen und Unendlich Symmetrische Produkte</cite>. Annals of Mathematics, 1958, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307/1970005">10.2307/1970005</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:Faserung&rft.au=Albrecht+Dold%2C+Ren%C3%A9+Thom&rft.btitle=Quasifaserungen+und+Unendlich+Symmetrische+Produkte&rft.date=1958&rft.doi=10.2307%2F1970005&rft.genre=book&rft.pub=Annals+of+Mathematics" style="display:none"> </li>

<li>[6] 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:Faserung&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"> </li>

<li>[7] James F. Davis, Paul Kirk: <cite style="font-style:italic">Lecture Notes in Algebraic Topology</cite>. 1991.<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:Faserung&rft.au=James+F.+Davis%2C+Paul+Kirk&rft.btitle=Lecture+Notes+in+Algebraic+Topology&rft.date=1991&rft.genre=book" style="display:none"> </li>

<li>[8] Ralph L. Cohen: <cite style="font-style:italic">The Topology of Fiber Bundles Lecture Notes</cite>. August 1998.<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:Faserung&rft.au=Ralph+L.+Cohen&rft.btitle=The+Topology+of+Fiber+Bundles+Lecture+Notes&rft.date=1998-08&rft.genre=book" style="display:none"> </li></ul>

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

<li id="cite_note-:0-1">↑ <a href="#cite_ref-:0_1-0">a</a> <a href="#cite_ref-:0_1-1">b</a> Allen Hatcher: <cite style="font-style:italic">Algebraic topology</cite>. 14th printing 2015 Auflage. Cambridge University Press, Cambridge 2015, ISBN 978-0-521-79160-1, S. 405–406.<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:Faserung&rft.au=Allen+Hatcher&rft.btitle=Algebraic+topology&rft.date=2015&rft.edition=14th+printing+2015&rft.genre=book&rft.isbn=9780521791601&rft.pages=405-406&rft.place=Cambridge&rft.pub=Cambridge+University+Press" style="display:none">

</li>

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