---- > [!definition] Definition. ([[fiber bundle]]) > Let $B$ be a [[topological space]]. A **fiber bundle** over $B$ is a [[continuous]] [[surjection]] $E \xrightarrow{\pi}B$ (the **bundle projection**), together with a [[topological space]] (the **fiber**) $F$, such that the following *local triviality* condition is satisfied: for every $x \in B$, there exists a **trivializing [[neighborhood]]** $U \ni x$ in the sense that the following diagram commutes: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB120sA9YAWgCMAXwAUAVQCUIYaXSZc+QijKCqtRizbiZckBmx4CRQeXX1mrRCHEACTngC28WwDFbM9TCgBzeEVAAMwAnCEckMhAcCCRTDUs2Tm4QagY6ACMYBgAFBSNlEGCsHwALHF0g0PDEACZqaNjqCy1rJKwAfR1ZSrCI+pjaps0rDnZskqxPYSA > \begin{tikzcd} > \pi^{-1}(U) \arrow[d, "\pi"'] \arrow[r, "\Phi"] & U \times F \arrow[ld, "\pi_U"] \\ > U & > \end{tikzcd} > \end{document} > ``` > > where $\Phi:\pi ^{-1}(U) \to U \times F$ is a [[homeomorphism]] and $\pi_{U}$ is projection onto the $U$-factor. > > > $B$ is called the **base space**, while $E$ is called the **total space**. The set of all $\{ U_{\alpha}, \Phi_{\alpha} \}_{\alpha \in I}$ is called a **local trivialization** of the bundle. For any $p \in B$, one has $\pi ^{-1}(p) \cong \{ p \} \times F \cong F$ and this is called the **fiber over $p$**. We often write $F \to E \xrightarrow{\pi}B$ to denote the fiber bundle. > > A **smooth fiber bundle** is a fiber bundle in the [[category]] of [[smooth manifold|smooth manifolds]]. That is, $E,B,F$ are required to be [[smooth manifold|smooth manifolds]] and all relevant maps are required to be [[smooth manifold|smooth]]. > [!basicexample] > - A [[covering space]] is a [[fiber bundle]] such that $\pi$ is a [[local homeomorphism]]. > - A [[vector bundle]] is a [[fiber bundle]] whose fibers are [[vector space|vector spaces]] > - A [[principal bundle over a smooth manifold|principal bundle]] is a [[fiber bundle]] are bundles ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```