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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{topological vector bundle} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles} [[!include bundles - contents]] \hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra} [[!include homotopy - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{InTermsOfSliceCategories}{In terms of slice categories}\dotfill \pageref*{InTermsOfSliceCategories} \linebreak \noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak \noindent\hyperlink{TransitionFunctionsAndCechCohomology}{Transition functions and Cech cohomology}\dotfill \pageref*{TransitionFunctionsAndCechCohomology} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{BasicProperties}{Basic properties}\dotfill \pageref*{BasicProperties} \linebreak \noindent\hyperlink{DirectSummandBundles}{Direct summand bundles}\dotfill \pageref*{DirectSummandBundles} \linebreak \noindent\hyperlink{ConcordanceOfTopolgicslVectorBundles}{Concordance}\dotfill \pageref*{ConcordanceOfTopolgicslVectorBundles} \linebreak \noindent\hyperlink{OverClosedSubspaces}{Over closed subspaces}\dotfill \pageref*{OverClosedSubspaces} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{topological vector bundle} is a [[vector bundle]] in the context of \emph{[[topology]]}: a continuously varying collection of [[vector space]] over a given [[topological space]]. For more survey and motivation see at \emph{[[vector bundle]]}. Here we discuss the details of the general concept in [[topology]]. See also \emph{[[differentiable vector bundle]]} and \emph{[[algebraic vector bundle]]}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} We first give the more abstract definiton in terms of slice categories (def. \ref{TopologicalVectorBundleInTermsOfSliceCategories} below) and then unwind this to the traditional definition (def \ref{TopologicalVectorBundle} below). In the following \begin{itemize}% \item $k$ is either the [[topological field]] \begin{itemize}% \item $k = \mathbb{R}$ of [[real numbers]] \item or $k = \mathbb{C}$ of [[complex numbers]] \end{itemize} equipped with the [[Euclidean space|Euclidean]] [[metric topology]]. \item \emph{[[vector space]]} means \emph{[[finite dimensional vector space]]}. \end{itemize} \hypertarget{InTermsOfSliceCategories}{}\subsubsection*{{In terms of slice categories}}\label{InTermsOfSliceCategories} \begin{defn} \label{TopologicalVectorBundleInTermsOfSliceCategories}\hypertarget{TopologicalVectorBundleInTermsOfSliceCategories}{} \textbf{(topological vector bundles in terms of slice categories)} Write [[Top]] for the [[category]] of [[topological spaces]], and for $X \in Top$ a space, write $Top_{/X}$ for its [[slice category]] over $X$. The [[Cartesian product]] in $Top_{/X}$ is the [[fiber product]] over $X$ in $Top$, which we denote by $(-) \times_X (-)$. Observe $[X \times k \to X] \in Top_{/X}$ is canonically a [[field]] [[internalization|internal]] to $Top_{/X}$ A \emph{topological vector bundle} over $X \in Top$ is \begin{enumerate}% \item an [[object ]] $[E \overset{\pi}{\to} X]$ of $Top_{/X}$ \item with the [[structure]] of an $X \times k$-[[vector space]]-object [[internalization|interal]] to $Top_{/X}$, hence \begin{enumerate}% \item a [[morphism]] $(-)+(-) \;\colon\; E \times_X E \to E$ \item a morphism $(-)\cdot(-) \;\colon\; k \times E \to E$ \end{enumerate} which satisfy the vector space axioms \end{enumerate} such that \begin{itemize}% \item ([[local trivialization|local triviality]]) there exists \begin{enumerate}% \item an [[open cover]] $\{U_i \subset X\}_{i \in I}$, regarded via the [[disjoint union space]] $U \coloneqq \underset{i \in I}{\sqcup} U_i$ of the patches as the object $[U \to X] \in Top_{/X}$, \item an [[isomorphism]] of vector space objects in $Top_{/U}$ \begin{displaymath} U \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} U \times_X E \,, \end{displaymath} for some $n \in \mathbb{N}$, where $[U \times k^n \overset{pr_1}{\to} X]$ and $[U \times_X E \overset{pr_1}{\to} U]$ are regarded as a vector space objects in $Top_{U}$ in the canonical way. \end{enumerate} \end{itemize} It follows that $n \in \mathbb{N}$ is constant on [[connected components]]. Often this is required to be constant on all of $X$ and then called the \emph{[[rank]]} of the vector bundle. A \emph{[[homomorphism]]} of topological vector bundles is simple a homomorphism of vector space objects in $Top_{/X}$. Topological vector bundles over $X$ and homomorphisms between them constitutes a [[category]], usually denoted [[Vect(X)]]. \end{defn} Notice that viewed in [[Top]], the last condition means that there is a [[diagram]] of the form \begin{displaymath} \itexarray{ U \times k^n &\overset{\simeq}{\longrightarrow}& U \times_X E &\overset{}{\longrightarrow}& E \\ & \searrow & \downarrow &(pb)& \downarrow^{\mathrlap{\pi}} \\ && U &\longrightarrow& X } \end{displaymath} where the square is a [[pullback square]] and the [[homeomorphism]] in the top left is fiber-wise linear. If we say this yet more explicitly, it yields the definition as found in the traditional textbooks: \hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components} \begin{defn} \label{TopologicalVectorBundle}\hypertarget{TopologicalVectorBundle}{} \textbf{(topological vector bundle in components)} Let $X$ be a [[topological space]]. Then a \emph{topological vector bundle} over $X$ is \begin{enumerate}% \item a [[topological space]] $E$; \item a [[continuous function]] $E \overset{\pi}{\to} X$ \item for each $x \in X$ the stucture of a [[finite-dimensional vector space|finite-dimensional]] $k$-[[vector space]] on the [[pre-image]] \begin{displaymath} E_x \coloneqq \pi^{-1}(\{x\}) \subset E \end{displaymath} \end{enumerate} such that this is [[local trivialization|locally trivial]] in that there exists \begin{enumerate}% \item an [[open cover]] $\{U_i \subset X\}_{i \in I}$, \item for each $i \in I$ an $n_i \in \mathbb{N}$ and a [[homeomorphism]] \begin{displaymath} \phi_i \;\colon\; U_i \times k^{n_i} \overset{\simeq}{\longrightarrow} \pi^{-1}(U_i) \subset E \end{displaymath} from the [[product topological space]] of $U_i$ with the [[real numbers]] (equipped with their [[Euclidean space]] [[metric topology]]) to the restriction of $E$ over $U_i$, such that \begin{enumerate}% \item $\phi_i$ is a map over $U_i$ in that $\pi \circ \phi_i = pr_1$, hence in that $\phi_i(\{x\} \times k^{n_i}) \subset \pi^{-1}(\{x\})$ \item $\phi_i$ is a [[linear map]] in each fiber in that \begin{displaymath} \underset{x \in U_i}{\forall} \left( \phi_i(x) \;\colon\; k^{n_i} \overset{\text{linear}}{\longrightarrow} E_x = \pi^{-1}(\{x\}) \right) \,. \end{displaymath} \end{enumerate} \end{enumerate} Often, but not always, it is required that the numbers $n_i$ are all equal to some $n \in \mathbb{N}$, for all $i \in I$, hence that the vector space fibers all have the same [[dimension]]. In this case one says that the vector bundle has \emph{[[rank of a vector bundle|rank]]} $n$. (Over a [[connected topological space]] this is automatic, but the fiber dimension may be distinct over distinct [[connected components]].) For $[E_1 \overset{\pi_1}{\to} X]$ and $[E_2 \overset{\pi_2}{\to} X]$ two topological vector bundles over the same base space, then a \emph{[[homomorphism]]} between them is \begin{itemize}% \item a [[continuous function]] $f \colon E_1 \longrightarrow E_2$ \end{itemize} such that \begin{enumerate}% \item $f$ respects the [[projections]]: $\pi_2 \circ f = \pi_1$; \item for each $x \in X$ we have that $f|_x \colon (E_1)_x \to (E_2)_x$ is a [[linear map]]. \end{enumerate} \end{defn} \begin{remark} \label{TopologicalVectorBundlesCategory}\hypertarget{TopologicalVectorBundlesCategory}{} \textbf{([[Vect(X)|category of topological vector bundles]])} For $X$ a [[topological space]], there is the [[category]] whose \begin{itemize}% \item [[objects]] are the topological vector bundles over $X$, \item [[morphisms]] are the topological vector bundle homomorphisms \end{itemize} according to def. \ref{TopologicalVectorBundle}. This category usually denoted [[Vect(X)]]. We write $Vect(X)_{/\sim}$ for the [[set]] of [[isomorphism classes]] of this category. \end{remark} \begin{remark} \label{TerminologyVectorBundles}\hypertarget{TerminologyVectorBundles}{} \textbf{(some terminology)} Let $k$ and $n$ be as in def. \ref{TopologicalVectorBundle}. Then: For $k = \mathbb{R}$ one speaks of \emph{[[real vector bundles]]}. For $k = \mathbb{C}$ one speaks of \emph{[[complex vector bundles]]}. For $n = 1$ one speaks of \emph{[[line bundles]]}, in particular of \emph{[[real line bundles]]} and of \emph{[[complex line bundles]]}. \end{remark} \begin{remark} \label{CommonOpenCoverLocalTrivialization}\hypertarget{CommonOpenCoverLocalTrivialization}{} \textbf{(any two topologial vector bundles have [[local trivialization]] over a common [[open cover]])} Let $[E_1 \to X]$ and $[E_2 \to X]$ be two topological vector bundles (def. \ref{TopologicalVectorBundle}). Then there always exists an [[open cover]] $\{U_i \subset X\}_{i \in I}$ such that both bundles have a [[local trivialization]] over this cover. \end{remark} \begin{proof} By definition we may find two possibly different open covers $\{U^1_{i_1} \subset X\}_{{i_1} \in I_1}$ and $\{U^2_{i_2} \subset X\}_{i_2 \in I_2}$ with local tivializations $\{ U^1_{i_1} \underoverset{\simeq}{\phi^1_{i_1}}{\to} E_1\vert_{U^1_{i_1}} \}_{i_1 \in I_1}$ and $\{ U^2_{i_2} \underoverset{\simeq}{\phi^2_{i_2}}{\to} E_2\vert_{U^2_{i_2}} \}_{i_2 \in I_2}$. The \emph{joint [[refinement]]} of these two covers is the open cover \begin{displaymath} \left\{ U_{i_1, i_2} \coloneqq U^1_{i_1} \cap U^2_{i_2} \subset X \right\}_{(i_1, i_2) \in I_1 \times I_2} \,. \end{displaymath} The original local trivializations restrict to local trivializations on this finer cover \begin{displaymath} \left\{ U_{i_1, i_2} \underoverset{\simeq}{\phi^1_{i_1}\vert_{U^2_{i_2}}}{\longrightarrow} E_1\vert_{U_{i_1, i_2}} \right\}_{(i_1, i_2) \in I_1 \times I_2} \end{displaymath} and \begin{displaymath} \left\{ U_{i_1, i_2} \underoverset{\simeq}{\phi^2_{i_2}\vert_{U^1_{i_1}}}{\longrightarrow} E_2\vert_{U_{i_1, i_2}} \right\}_{(i_1, i_2) \in I_1 \times I_2} \,. \end{displaymath} \end{proof} \begin{example} \label{TrivialTopologicalVectorBundle}\hypertarget{TrivialTopologicalVectorBundle}{} \textbf{(trivial topological vector bundle and (local) trivialization)} For $X$ any [[topological space]], and $n \in \mathbb{N}$, we have that the [[product topological space]] \begin{displaymath} X \times k^n \overset{pr_1}{\to} X \end{displaymath} canonically becomes a topological vector bundle over $X$ (def. \ref{TopologicalVectorBundle}). This is called the \emph{[[trivial vector bundle]]} of [[rank]] $n$ over $X$. Given any topological vector bundle $E \to X$, then a choice of [[isomorphism]] to a trivial bundle (if it exists) \begin{displaymath} E \overset{\simeq}{\longrightarrow} X \times k^n \end{displaymath} is called a \emph{trivialization} of $E$. A vector bundle for which a trivialization exists is called \emph{trivializable}. Accordingly, the [[local trivialization|local triviality]] condition in the definition of topological vector bundles (def. \ref{TopologicalVectorBundle}) says that they are locally isomorphic to the trivial vector bundle. One also says that the data consisting of an open cover $\{U_i \subset X\}_{i \in I}$ and the [[homeomorphisms]] \begin{displaymath} \left\{ U_i \times k^n \overset{\simeq}{\to} E|_{U_i} \right\}_{i \in I} \end{displaymath} as in def. \ref{TopologicalVectorBundle} constitute a \emph{[[local trivialization]]} of $E$. \end{example} \begin{example} \label{VectorBundleSections}\hypertarget{VectorBundleSections}{} \textbf{([[section]] of a [[topological vector bundle]])} Let $E \overset{\pi}{\to} X$ be a topological vector bundle (def. \ref{TopologicalVectorBundle}). Then a [[homomorphism]] of vector bundles from the [[trivial vector bundle|trivial]] [[line bundle]] (example \ref{TrivialTopologicalVectorBundle}, remark \ref{TerminologyVectorBundles}) \begin{displaymath} f \;\colon\; X \times k \longrightarrow E \end{displaymath} is, by fiberwise linearity, equivalently a [[continuous function]] \begin{displaymath} \sigma \;\colon\; X \longrightarrow E \end{displaymath} such that $\pi \circ \sigma = id_X$; \begin{displaymath} f(x, c) = c \sigma(x) \end{displaymath} Such functions $\sigma \colon X \to E$ are called \emph{[[sections]]} (or \emph{cross-sections}) of the vector bundle $E$. \end{example} \begin{example} \label{TopologicalVetorSubbundle}\hypertarget{TopologicalVetorSubbundle}{} \textbf{(topological vector sub-bundle)} Given a topological vector bundel $E \to X$ (def. \ref{TopologicalVectorBundle}), then a \emph{sub-bundle} is a homomorphism of topological vector bundles over $X$ \begin{displaymath} i\;\colon\; E' \hookrightarrow E \end{displaymath} such that for each point $x \in X$ this is a linear embedding of fibers \begin{displaymath} i|_x \;\colon\; E'_x \hookrightarrow E_x \,. \end{displaymath} (This is a [[monomorphism]] in the [[category]] $Vect(X)$ of topological vector bundles over $X$.) \end{example} \hypertarget{TransitionFunctionsAndCechCohomology}{}\subsubsection*{{Transition functions and Cech cohomology}}\label{TransitionFunctionsAndCechCohomology} We discuss how topological vector bundles are equivalently given by [[cocycles]] in [[Cech cohomology]] constituted by their [[transition functions]]. \begin{defn} \label{ContinuousFunctionWithValuesInGLn}\hypertarget{ContinuousFunctionWithValuesInGLn}{} \textbf{([[continuous functions]] on [[open subsets]] with values in the [[general linear group]])} For $n \in \mathbb{N}$, regard the [[general linear group]] $GL(n,k)$ as a [[topological group]] with its standard [[topological space|topology]], given as the [[Euclidean space|Euclidean]] [[subspace topology]] via $GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$ or as the or as the subspace topology $GL(n,k) \subset Maps(k^n, k^n)$ of the [[compact-open topology]] on the [[mapping space]]. (That these topologies coincide is the statement of \href{general+linear+group#AsSubspaceOfTheMappingSpace}{this prop.}. For $X$ a [[topological space]], we write \begin{displaymath} \underline{GL(n,k)} \;\colon\; U \mapsto Hom_{Top}(U, GL(n,k) ) \end{displaymath} for the assignment that sends an [[open subset]] $U \subset X$ to the [[set]] of [[continuous functions]] $g \colon U \to GL(n,k)$ (for $U \subset X$ equipped with its [[subspace topology]]), regarded as a [[group]] via the pointwise group operation in $GL(n,k)$: \begin{displaymath} g_1 \cdot g_2 \;\colon\; x \mapsto g_1(x) \cdot g_2(x) \,. \end{displaymath} Moreover, for $U' \subset U \subset X$ an inclusion of open subsets, and for $g \in \underline{GL(n,k)}(U)$, we write \begin{displaymath} g|_{U'} \in \underline{GL(n,k)}(U') \end{displaymath} for the restriction of the continuous function from $U$ to $U'$. \end{defn} \begin{remark} \label{}\hypertarget{}{} \textbf{([[sheaf]] of [[groups]])} In the language of [[category theory]] the assignment $\underline{GL(n,k)}$ from def. \ref{ContinuousFunctionWithValuesInGLn} of continuous functions to open subsets and the restriction operations between these is called a \emph{[[sheaf]] of groups on the [[site of open subsets]]} of $X$. \end{remark} \begin{defn} \label{TransitionFunctions}\hypertarget{TransitionFunctions}{} \textbf{([[transition functions]])} Given a topological vector bundle $E \to X$ as in def. \ref{TopologicalVectorBundle} and a choice of [[local trivialization]] $\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_i}\}$ (example \ref{TrivialTopologicalVectorBundle}) there are for $i,j \in I$ induced [[continuous functions]] \begin{displaymath} \left\{ g_{i j} \;\colon\; (U_i \cap U_j) \longrightarrow GL(n, k) \right\}_{i,j \in I} \end{displaymath} to the [[general linear group]] (as in def. \ref{ContinuousFunctionWithValuesInGLn}) given by composing the local trivialization isomorphisms: \begin{displaymath} \itexarray{ (U_i \cap U_j) \times k^n &\overset{ \phi_i|_{U_i \cap U_j} }{\longrightarrow}& E|_{U_i \cap U_j} &\overset{ \phi_j^{-1}\vert_{U_i \cap U_j} }{\longrightarrow}& (U_i \cap U_j) \times k^n \\ (x,v) && \overset{\phantom{AAA}}{\mapsto} && \left( x, g_{i j}(x)(v) \right) } \,. \end{displaymath} These are called the \emph{[[transition functions]]} for the given local trivialization. \end{defn} These functions satisfy a special property: \begin{defn} \label{CocycleCech}\hypertarget{CocycleCech}{} \textbf{([[Cech cohomology|Cech]] [[cocycles]])} Let $X$ be a [[topological space]]. A \emph{normalized [[Cech cohomology|Cech cocycle]] of degree 1 with [[coefficients]]} in $\underline{GL(n,k)}$ (def. \ref{ContinuousFunctionWithValuesInGLn}) is \begin{enumerate}% \item an [[open cover]] $\{U_i \subset X\}_{i \in I}$ \item for all $i,j \in I$ a continuous function $g_{i j} \colon U_i \cap U_j \to GL(n,k)$ as in def. \ref{ContinuousFunctionWithValuesInGLn} \end{enumerate} such that \begin{enumerate}% \item (normalization) $\underset{i \in I}{\forall}\left( g_{i i} = const_1 \right)$ (the [[constant function]] on the [[neutral element]] in $GL(n,k)$), \item (cocycle condition) $\underset{i,j \in I}{\forall}\left( g_{j k} \cdot g_{i j} = g_{i k}\;\;\text{on}\, U_i \cap U_j \cap U_k\right)$. \end{enumerate} Write \begin{displaymath} C^1(X, \underline{GL(n,k)} ) \end{displaymath} for the set of all such cocycles for given $n \in \mathbb{N}$ and write \begin{displaymath} C^1( X, \underline{GL}(k) ) \;\coloneqq\; \underset{n \in \mathbb{N}}{\sqcup} C^1(X, \underline{GL(n,k)}) \end{displaymath} for the [[disjoint union]] of all these cocycles as $n$ varies. \end{defn} \begin{example} \label{CocycleCechTransitionFunction}\hypertarget{CocycleCechTransitionFunction}{} \textbf{([[transition functions]] are [[Cech cohomology|Cech]] [[cocycles]])} Let $E \to X$ be a topological vector bundle (def. \ref{TopologicalVectorBundle}) and let $\{U_i \subset X\}_{i \in I}$, $\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_{i}}\}_{i \in I}$ be a local trivialization (example \ref{TrivialTopologicalVectorBundle}). Then the set of induced [[transition functions]] $\{g_{i j} \colon U_i \cap U_j \to GL(n)\}$ according to def. \ref{TransitionFunctions} is a \emph{normalized Cech cocycle on $X$ with coefficients in $\underline{GL(k)}$}, according to def. \ref{CocycleCech}. \end{example} \begin{proof} This is immediate from the definition: \begin{displaymath} \begin{aligned} g_{i i }(x) & = \phi_i^{-1} \circ \phi_i(x,-) \\ & = id_{k^n} \end{aligned} \end{displaymath} and \begin{displaymath} \begin{aligned} g_{j k}(x) \cdot g_{i j}(x) & = \left(\phi_k^{-1} \circ \phi_j\right) \circ \left(\phi_j^{-1}\circ \phi_i\right)(x,-) \\ & = \phi_k^{-1} \circ \phi_i(x,-) \\ & = g_{i k}(x) \end{aligned} \,. \end{displaymath} \end{proof} Conversely: \begin{example} \label{TopologicalVectorBundleFromCechCocycle}\hypertarget{TopologicalVectorBundleFromCechCocycle}{} \textbf{(topological vector bundle constructed from a [[Cech cohomology|Cech]] [[cocycle]])} Let $X$ be a [[topological space]] and let $c \in C^1(X, \underline{GL(k)})$ a Cech cocycle on $X$ according to def. \ref{CocycleCech}, with open cover $\{U_i \subset X\}_{i \in I}$ and component functions $\{g_{i j}\}_{i,j \in I}$. This induces an [[equivalence relation]] on the [[product topological space]] \begin{displaymath} \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \end{displaymath} (of the [[disjoint union space]] of the patches $U_i \subset X$ regarded as [[topological subspaces]] with the [[product space]] $k^n = \underset{\{1,\cdots, n\}}{\prod} k$) given by \begin{displaymath} \big( ((x,i), v) \;\sim\; ((y,j), w) \big) \;\Leftrightarrow\; \left( (x = y) \;\text{and}\; (g_{i j}(x)(v) = w) \right) \,. \end{displaymath} Write \begin{displaymath} E(c) \;\coloneqq\; \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) \end{displaymath} for the resulting [[quotient topological space]]. This comes with the evident projection \begin{displaymath} \itexarray{ E(c) &\overset{\phantom{AA}\pi \phantom{AA}}{\longrightarrow}& X \\ [(x,i,),v] &\overset{\phantom{AAA}}{\mapsto}& x } \end{displaymath} which is a [[continuous function]] (by the [[universal property]] of the [[quotient topological space]] construction, since the corresponding continuous [[function]] on the un-quotientd disjoint union space respects the equivalence relation). Moreover, each [[fiber]] of this map is identified with $k^n$, and hence canonicaly carries the structure of a [[vector space]]. Finally, the quotient co-projections constitute a local trivialization of this vector bundle over the given open cover. Therefore $E(c) \to X$ is a topological vector bundle (def. \ref{TopologicalVectorBundle}). We say it is the topological vector bundle \emph{glued from the transition functions}. \end{example} \begin{remark} \label{}\hypertarget{}{} \textbf{(bundle glued from [[Cech cohomology|Cech]] [[cocycle]] is a [[coequalizer]])} Stated more [[category theory|category theoretically]], the constructure of a topological vector bundle from Cech cocycle data in example \ref{TopologicalVectorBundleFromCechCocycle} is a \href{Top#UniversalConstructions}{universal construction in topological spaces}, namely the [[coequalizer]] of the two morphisms \begin{displaymath} i, \mu: \underset{i j}{\sqcup} (U_i \cap U_j) \times V \overset{\to}{\to} \underset{i}{\sqcup} U_i \times V \end{displaymath} in the category of vector space objects in the slice category $Top/X$. Here the restriction of $i$ to the coproduct summands is induced by inclusion: \begin{displaymath} (U_i \cap U_j) \times V \hookrightarrow U_i \times V \hookrightarrow \underset{i}{\sqcup} U_i \times V \end{displaymath} and the restriction of $\mu$ to the coproduct summands is via the action of the transition functions: \begin{displaymath} (U_i \cap U_j) \times V \overset{(\langle incl, g_{i j} \rangle) \times V}{\to} U_j \times GL(V) \times V \overset{action}{\to} U_j \times V \hookrightarrow \underset{j}{\sqcup} U_j \times V \end{displaymath} \end{remark} In fact, extracting transition functions from a vector bundle by def. \ref{TransitionFunctions} and constructing a vector bundle from Cech coycle data as above are operations that are inverse to each other, up to [[isomorphism]]. \begin{prop} \label{FromTransitionFunctionsReconstructVectorBundle}\hypertarget{FromTransitionFunctionsReconstructVectorBundle}{} \textbf{(topological vector bundle reconstructed from its [[transition functions]])} Let $[E \overset{\pi}{\to} X]$ be a [[topological vector bundle]] (def. \ref{TopologicalVectorBundle}), let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of the base space, and let $\left\{ U_i \times k^n \underoverset{\simeq}{\phi_i}{\longrightarrow} E|_{U_i} \right\}_{i \in I}$ be a [[local trivialization]]. Write \begin{displaymath} \left\{ g_{i j} \coloneqq \phi_j^{-1}\circ \phi_i \colon U_i \cap U_j \to GL(n,k) \right\}_{i,j \in I} \end{displaymath} for the corresponding [[transition functions]] (def. \ref{TransitionFunctions}). Then there is an [[isomorphism]] of vector bundles over $X$ \begin{displaymath} \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) \;\underoverset{\simeq}{(\phi_i)_{i \in I}}{\longrightarrow}\; E \end{displaymath} from the vector bundle glued from the transition functions according to def. \ref{TransitionFunctions} to the original bundle $E$, whose components are the original local trivialization isomorphisms. \end{prop} \begin{proof} By the [[universal property]] of the [[disjoint union space]] ([[coproduct]] in [[Top]]), continuous functions out of them are equivalently sets of continuous functions out of every summand space. Hence the set of local trivializations $\{U_i \times k^n \underoverset{\simeq}{\phi_i}{\to} E|_{U_i} \subset E\}_{i \in I}$ may be collected into a single [[continuous function]] \begin{displaymath} \underset{i \in I}{\sqcup} U_i \times k^n \overset{(\phi_i)_{i \in I}}{\longrightarrow } E \,. \end{displaymath} By construction this function respects the [[equivalence relation]] on the disjoint union space given by the transition functions, in that for each $x \in U_i \cap U_j$ we have \begin{displaymath} \phi_i((x,i),v) = \phi_j \circ \phi_j^{-1} \circ \phi_i((x,i),v) = \phi_j \circ ((x,j),g_{i j}(x)(v)) \,. \end{displaymath} By the [[universal property]] of the [[quotient space]] coprojection this means that $(\phi_i)_{i \in I}$ uniquely [[extension|extends]] to a [[continuous function]] on the quotient space such that the following [[commuting diagram|diagram commutes]] \begin{displaymath} \itexarray{ \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& E \\ \downarrow & \nearrow_{\exists !} \\ \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) } \,. \end{displaymath} It is clear that this continuous function is a [[bijection]]. Hence to show that it is a [[homeomorphism]], it is now sufficient to show that this is an [[open map]] (by \href{Introduction+to+Topology+--+1#HomeoContinuousOpenBijection}{this prop.}). So let $O$ be an subset in the quotient space which is open. By definition of the [[quotient topology]] this means equivalently that its restriction $O_i$ to $U_i \times k^n$ is open for each $i \in I$. Since the $\phi_i$ are homeomorphsms, it follows that the images $\phi_i(O_i) \subset E\vert_{U_ i}$ are open. By the nature of the [[subspace topology]], this means that these images are open also in $E$. Therefore also the union $f(O) = \underset{i \in I}{\cup} \phi_i(O_i)$ is open. \end{proof} \begin{defn} \label{CoboundaryCech}\hypertarget{CoboundaryCech}{} \textbf{([[coboundary]] between [[Cech cohomology|Cech]] [[cocycles]] )} Let $X$ be a [[topological space]] and let $c_1, c_2 \in C^1(X, \underline{GL(k)})$ be two [[Cech cohomology|Cech]] [[cocycles]] (def. \ref{CocycleCech}), given by \begin{enumerate}% \item $\{U_i \subset X\}_{i \in I}$ and $\{U'_i \subset X\}_{i' \in I'}$ two [[open covers]], \item $\{g_{i j} \colon U_i \cap U_j \to GL(k,n_)\}_{i,j \in I}$ and $\{g_'_{i',j'} \colon U'_{i'} \cap U'_{j'} \to GL(n',k) \}_{i', j'}$ the corrsponding component functions. \end{enumerate} Then a \emph{[[coboundary]]} between these two cocycles is \begin{enumerate}% \item the condition that $n = n'$, \item an [[open cover]] $\{V_\alpha \subset X\}_{\alpha \in A}$, \item [[functions]] $\phi \colon A \to I$ and $\phi' \colon A \to J$ such that $\underset{\alpha \in A}{\forall}\left( \left( V_\alpha \subset U_{\phi(\alpha)} \right) \,\text{and}\, \left( V_\alpha \subset U'_{\phi'(\alpha)} \right) \right)$ \item a set $\{ \kappa_\alpha \colon V_\alpha \to GL(n,k) \}$ of continuous functions as in def. \ref{CocycleCech} \end{enumerate} such that \begin{itemize}% \item $\underset{ \alpha, \beta \in A }{\forall} \left( \kappa_{\beta} \cdot g_{\phi(\alpha) \phi(\beta)} = g'_{\phi'(\alpha) \phi'(\beta)} \cdot \kappa_{\alpha} \,\, \text{on}\,\, V_\alpha \cap V_\beta \right)$, hence such that the following [[diagrams]] of [[linear maps]] [[commuting diagram|commute]] for all $\alpha, \beta \in A$ and $x \in V_{\alpha} \cap V_\beta$: \begin{displaymath} \itexarray{ k^n &\overset{ g_{\phi(\alpha) \phi(\beta)}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_{\alpha}(x)} }\downarrow && \downarrow^{\mathrlap{ \kappa_{\beta}(x) }} \\ k^n &\underset{ g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n } \,. \end{displaymath} \end{itemize} Say that two Cech cocycles are \emph{cohomologous} if there exists a coboundary between them. \end{defn} \begin{example} \label{FinerCoverCech}\hypertarget{FinerCoverCech}{} \textbf{([[refinement]] of a [[Cech cohomology|Cech]] [[cocycle]] is a [[coboundary]])} Let $X$ be a [[topological space]] and let $c \in C^1(X, \underline{GL(k)})$ be a Cech cocycle as in def. \ref{CocycleCech}, with respect to some open cover $\{U_i \subset X\}_{i \in I}$ given by component functions $\{g_{i j}\}_{i,j \in I}$. Then for $\{V_\alpha \subset X\}_{\alpha \in A}$ a [[refinement]] of the given open cover, hence an open cover such that there exists a [[function]] $\phi \colon A \to I$ with $\underset{\alpha \in A}{\forall}\left( V\alpha \subset U_{\phi(\alpha)} \right)$, then \begin{displaymath} g'_{ \alpha \beta } \coloneqq g_{\phi(\alpha) \phi(\beta)} \colon V_\alpha \cap V_\beta \longrightarrow GL(n,k) \end{displaymath} are the components of a Cech cocycle $c'$ which is cohomologous to $c$. \end{example} \begin{prop} \label{CechCoboundaryFromIsomorphismBetweenVectoreBundles}\hypertarget{CechCoboundaryFromIsomorphismBetweenVectoreBundles}{} \textbf{([[isomorphism]] of topological vector bundles induces [[Cech cohomology|Cech]] [[coboundary]] between their [[transition functions]])} Let $X$ be a topological space, and let $c_1, c_2 \in C^1(X, \underline{GL(n,k)} )$ be two Cech cocycles as in def. \ref{CocycleCech}. Every [[isomorphism]] of topological vector bundles \begin{displaymath} f \;\colon\; E(c_1) \overset{\simeq}{\longrightarrow} E(c_2) \end{displaymath} between the vector bundles glued from these cocycles according to def. \ref{TopologicalVectorBundleFromCechCocycle} induces a coboundary between the two cocycles, \begin{displaymath} c_1 \sim c_2 \,, \end{displaymath} according to def. \ref{CoboundaryCech}. \end{prop} \begin{proof} By example \ref{FinerCoverCech} we may assume without restriction that the two Cech cocycles are defined with respect to the same open cover $\{U_i \subset X\}_{i \in I}$ (for if they are not, then both are cohomologous to cocycles on a joint refinement of the original covers and we may argue with these). Accordingly, by example \ref{TopologicalVectorBundleFromCechCocycle} the two bundles $E(c_1)$ and $E(c_2)$ both have local trivializations of the form \begin{displaymath} \{ U_i \times k^n \underoverset{\simeq}{\phi^1_i}{\longrightarrow} E(c_1)\vert_{U_i}\} \end{displaymath} and \begin{displaymath} \{ U_i \times k^n \underoverset{\simeq}{\phi^2_i}{\longrightarrow} E(c_2)\vert_{U_i}\} \end{displaymath} over this cover. Consider then for $i \in I$ the function \begin{displaymath} f_i \coloneqq (\phi_i^2)^{-1}\circ f\vert_{U_i} \circ \phi^1_i \,, \end{displaymath} hence the unique function making the following [[commuting diagram|diagram commute]]: \begin{displaymath} \itexarray{ U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i} \\ {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }} \\ U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_i} } \,. \end{displaymath} This induces for all $i,j \in I$ the following composite commuting diagram \begin{displaymath} \itexarray{ (U_i \cap U_j) \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i \cap U_j} & \underoverset{\simeq}{(\phi^1_j)^{-1}}{\longrightarrow} & (U_i \cap U_j) \times k^n \\ {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }} && \downarrow^{\mathrlap{ f_j }} \\ (U_i \cap U_j) \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_1 \cap U_2} &\underoverset{(\phi^2_j)^{-1}}{\simeq}{\longrightarrow}& (U_i \cap U_j) \times k^n } \,. \end{displaymath} By construction, the two horizonal composites of this diagram are pointwise given by the components $g^1_{i j}$ and $g^2_{i j}$of the cocycles $c_1$ and $c_2$, respectively. Hence the commutativity of this diagram is equivalently the commutativity of these diagrams: \begin{displaymath} \itexarray{ k^n &\overset{ g^1_{i j}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{ f_i(x) } }\downarrow && \downarrow^{\mathrlap{ f_j(x) }} \\ k^n &\underset{ g^2_{ i j }(x) }{\longrightarrow}& k^n } \,. \end{displaymath} for all $i,j \in I$ and $x \in U_i \cap U_j$. By def. \ref{CoboundaryCech} this exhibits the required coboundary. \end{proof} \begin{defn} \label{CohomologyCech}\hypertarget{CohomologyCech}{} \textbf{([[Cech cohomology]])} Let $X$ be a [[topological space]]. The relation $\sim$ on [[Cech cohomology|Cech cocycles]] of being cohomologous (def. \ref{CoboundaryCech}) is an [[equivalence relation]] on the set $C^1( X, \underline{GL(k)} )$ of [[Cech cohomology|Cech]] [[cocycles]] (def. \ref{CocycleCech}). Write \begin{displaymath} H^1(X, \underline{GL(k)} ) \;\coloneqq\; C^1(X, \underline{GL(k)} )/\sim \end{displaymath} for the resulting set of [[equivalence classes]]. This is called the \emph{[[Cech cohomology]] of $X$ in degree 1 with [[coefficients]] in $\underline{GL(k)}$.} \end{defn} \begin{prop} \label{}\hypertarget{}{} \textbf{(degree-1 [[Cech cohomology]] computes [[topological vector bundles]])} Let $X$ be a [[topological space]]. The construction of gluing a topological vector bundle from a Cech cocycle (example \ref{TopologicalVectorBundleFromCechCocycle}) constitutes a [[natural bijection|bijection]] between the degree-1 Cech cohomology of $X$ with coefficients in $GL(n,k)$ (def. \ref{CohomologyCech}) and the set of [[isomorphism classes]] of topological vector bundles on $X$ (def. \ref{TopologicalVectorBundle}, remark \ref{TopologicalVectorBundlesCategory}): \begin{displaymath} \itexarray{ H^1(X,\underline{GL(k)}) &\overset{\phantom{AA}\simeq \phantom{AA}}{\longrightarrow}& Vect(X)_{/\sim} \\ c &\overset{\phantom{AAA}}{\mapsto}& E(c) } \,. \end{displaymath} \end{prop} \begin{proof} First we need to see that the function is well defined, hence that if cocycles $c_1, c_2 \in C^1(X,\underline{GL(k)})$ are related by a coboundary, $c_1 \sim c_2$ (def. \ref{CoboundaryCech}), then the vector bundles $E(c_1)$ and $E(c_2)$ are related by an isomorphism. Let $\{V_\alpha \subset X\}_{\alpha \in A}$ be the open cover with respect to which the coboundary $\{\kappa_\alpha \colon V_\alpha \to GL(n,k)\}_{\alpha}$ is defined, with refining functions $\phi \colon A \to I$ and $\phi' \colon A \to I'$. Let $\left\{ \mathbb{R}^n \underoverset{\simeq}{\psi_{\phi(\alpha)}\vert_{V_\alpha} }{\to} E(c_1)\vert_{V_\alpha} \right\}_{\alpha \in A}$ and $\left\{ V_\alpha \times k^n \underoverset{\simeq}{\psi'_{\phi'(\alpha)}\vert_{V_\alpha} }{\to} E(c_2)\vert_{V_\alpha} \right\}_{\alpha \in A}$ be the corresponding restrictions of the canonical local trivilizations of the two glued bundles. For $\alpha \in A$ define \begin{displaymath} f_\alpha \coloneqq \psi'_{\phi'(\alpha)}\vert_{V_\alpha} \circ \kappa_\alpha \circ (\psi_{\phi(\alpha)}\vert_{V_\alpha} )^{-1} \phantom{AAAA} \text{hence:} \phantom{AAA} \itexarray{ V_\alpha \times k^n &\overset{ \psi_{\phi(\alpha)}\vert_{V_\alpha} }{\longrightarrow}& E(c_1)\vert_{V_\alpha} \\ {}^{\mathllap{\kappa_\alpha}}\downarrow && \downarrow^{\mathrlap{f_\alpha}} \\ V_\alpha \times k^n &\overset{ (\psi'_{\phi'(\alpha)}\vert_{V_\alpha})^{-1} }{\longleftarrow}& E(c_1)\vert_{V_\alpha} } \,. \end{displaymath} Observe that for $\alpha, \beta \in A$ and $x \in V_\alpha \cap V_\beta$ the coboundary condition implies that \begin{displaymath} f_\alpha\vert_{V_\alpha \cap V_\beta} \;=\; f_\beta\vert_{V_\alpha \cap V_\beta} \end{displaymath} because in the diagram \begin{displaymath} \itexarray{ k^n &\overset{ g_{\phi(\alpha) \phi(\beta) }(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_\alpha(x)}}\downarrow && \downarrow^{\mathrlap{\kappa_{\beta}(x)}} \\ k^n &\underset{g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n } \phantom{AAAAA} = \phantom{AAAAA} \itexarray{ k^n &\overset{ \psi_{\phi(\alpha)}(x) }{\longrightarrow}& E(c_1)_x &\overset{ (\psi_{\phi(\beta)})^{-1}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_\alpha(x)}}\downarrow && \downarrow^{\mathrlap{\exists !} } && \downarrow^{\mathrlap{\beta_\alpha(x)}} \\ k^n &\overset{ \psi'_{\phi'(\alpha)}(x) }{\longrightarrow}& E(c_2)_x &\overset{ (\psi'_{\phi'(\beta)})^{-1}(x) }{\longrightarrow}& k^n } \end{displaymath} the vertical morphism in the middle on the right is unique, by the fact that all other morphisms in the diagram on the right are invertible. Therefore there is a unique vector bundle homomorphism \begin{displaymath} f\;\colon\; E(c_1) \to E(c_2) \end{displaymath} given for all $\alpha \in A$ by $f\vert_{V_\alpha} = f_\alpha$. Similarly there is a unique vector bundle homomorphism \begin{displaymath} f^{-1}\;\colon\; E(c_2) \to E(c_1) \end{displaymath} given for all $\alpha \in A$ by $f^{-1}\vert_{V_\alpha} = f^{-1}_\alpha$. Hence this is the required vector bundle isomorphism. Finally to see that the function from Cech cohomology classes to isomorphism classes of vector bundles thus defined is a bijection: By prop. \ref{FromTransitionFunctionsReconstructVectorBundle} the function is [[surjective function|surjective]], and by prop. \ref{CechCoboundaryFromIsomorphismBetweenVectoreBundles} it is injective. \end{proof} $\,$ \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \begin{example} \label{TautologicalLineBundle}\hypertarget{TautologicalLineBundle}{} \textbf{([[tautological line bundle]])} For $n \in \mathbb{N}$ then the [[projective space]] $k P^n$ carries the \emph{[[tautological line bundle]]} whose fiber over the $k$-line $[v] \in k P^n$ is that $k$-line. For details see \href{tautological+line+bundle#AsAtopologicalLieBundle}{there} \end{example} \begin{example} \label{}\hypertarget{}{} \textbf{([[cylinder]])} Let \begin{displaymath} S^1 = \left\{ (x,y) \;\vert\; x^2 + y^2 = 1 \right\} \;\subset\, \mathbb{R}^2 \end{displaymath} be the [[circle]] with its [[Euclidean space|Euclidean]] [[subspace]] [[metric topology]]. Then the [[trivial vector bundle|trivial]] [[real vector bundle|real]] [[line bundle]] on the circle is the the [[cylinder]] \begin{displaymath} S^1 \times \mathbb{R} \end{displaymath} \end{example} \begin{example} \label{MoebiusStrip}\hypertarget{MoebiusStrip}{} \textbf{([[Moebius strip]])} Let \begin{displaymath} S^1 = \left\{ (x,y) \;\vert\; x^2 + y^2 = 1 \right\} \;\subset\, \mathbb{R}^2 \end{displaymath} be the [[circle]] with its [[Euclidean space|Euclidean]] [[subspace]] [[metric topology]]. Consider the [[open cover]] \begin{displaymath} \left\{ U_n \subset S^1 \right\}_{n \in \{0,1,2\}} \end{displaymath} with \begin{displaymath} U_n \coloneqq \left\{ (cos(\alpha), sin(\beta)) \;\vert\; n \frac{2 \pi }{3} - \epsilon \lt \alpha \lt (n+1) \frac{2\pi }{3} + \epsilon \right\} \end{displaymath} for any $\epsilon \in (0,2\pi/6)$. Define a [[Cech cohomology]] cocycle (remark \ref{CechCoycleCondition}) on this cover by \begin{displaymath} g_{n_1 n_2} = \left\{ \itexarray{ const_{-1} & \vert & (n_1,n_2) = (0,2) \\ const_{-1} &\vert& (n_1,n_2) = (2,0) \\ const_1 &\vert& \text{otherwise} } \right. \end{displaymath} Since there are no non-trivial triple intersections, all cocycle conditions are evidently satisfied. Accordingly by example \ref{TopologicalVectorBundleFromCechCocycle} these functions define a vector bundle. This is the \emph{[[Moebius strip]]} \end{example} \begin{example} \label{}\hypertarget{}{} \textbf{([[basic complex line bundle on the 2-sphere]])} Let \begin{displaymath} S^2 \coloneqq \left\{ (x,y,z) \;\vert\; x^2 + y^2 + z^2 = 1 \right\} \subset \mathbb{R}^3 \end{displaymath} be the [[2-sphere]] with its [[Euclidean space|Euclidean]] [[subspace]] [[metric topology]]. Let \begin{displaymath} \left\{ U_{i} \subset S^2 \right\}_{i \in \{+,-\}} \end{displaymath} be the two [[complements]] of antipodal points \begin{displaymath} U_\pm \coloneqq S^2 \setminus \{(0, 0, \pm 1)\} \,. \end{displaymath} Define continuous functions \begin{displaymath} \itexarray{ U_+ \cap U_- &\overset{g_{\pm \mp}}{\longrightarrow}& GL(1,\mathbb{C}) \\ ( \sqrt{1-z^2} \, cos(\alpha), \sqrt{1-z^2} \, sin(\alpha), z) &\mapsto& \exp(\pm 2\pi i \alpha) } \,. \end{displaymath} Since there are no non-trivial triple intersections, the only cocycle condition is \begin{displaymath} g_{\mp \pm} g_{\pm \mp} = g_{\pm \pm} = id \end{displaymath} which is clearly satisfied. The [[complex line bundle]] this defined is called the \emph{[[basic complex line bundle on the 2-sphere]]}. With the 2-sphere identified with the [[complex projective space]] $\mathbb{C} P^1$ (the [[Riemann sphere]]), the basic complex line bundle is the [[tautological line bundle]] (example \ref{TautologicalLineBundle}) on $\mathbb{C}P^1$. \end{example} \begin{example} \label{ClutchingConstruction}\hypertarget{ClutchingConstruction}{} \textbf{([[clutching construction]])} Generally, for $n \in \mathbb{N}$, $n \geq 1$ then the [[n-sphere]] $S^n$ may be covered by two open [[hemispheres]] intersecting in an [[equator]] of the form $S^{n-1} \times (-\epsilon, \epsilon)$. A vector bundle is then defined by specifying a single function \begin{displaymath} g_{+-} \;\colon\; S^{n-1} \longrightarrow GL(n,k) \,. \end{displaymath} This is called the \emph{[[clutching construction]]} of vector bundles over [[n-spheres]]. \end{example} \begin{example} \label{}\hypertarget{}{} \textbf{([[tangent bundle]])} For $X$ the [[topological space]] underlyithe ng a [[differentiable manifold]] then its [[tangent bundle]] $T X$ is a [[real vector bundle]] over $X$ whose [[rank of a vector bundle|rank]] is the [[dimension]] of $X$. \end{example} \begin{example} \label{}\hypertarget{}{} \textbf{([[normal bundle]])} For $i X \hookrightarrow Y$ an [[embedding of differentiable manifolds]], then the \emph{[[normal bundle]]} \begin{displaymath} N_i X \coloneqq T Y/T X \end{displaymath} is the real vector bundle over $Y$ whose [[fiber]] at $x \in X$ is the [[quotient vector space]] $(N_i X)_x \coloneqq T_{i(x)} Y / T_x X$. \end{example} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{BasicProperties}{}\subsubsection*{{Basic properties}}\label{BasicProperties} \begin{lemma} \label{FiberwiseIsoisIsomorphismOfVectorBundles}\hypertarget{FiberwiseIsoisIsomorphismOfVectorBundles}{} \textbf{(homomorphism of vector bundles is isomorphism as soon as it is a fiberwise isomorphism)} Let $[E_1 \to X]$ and $[E_2 \to X]$ be two topological vector bundles (def. \ref{TopologicalVectorBundle}). If a [[homomorphism]] of vector bundles $f \colon E_1 \longrightarrow E_2$ restricts on the [[fiber]] over each point to a linear isomorphism \begin{displaymath} f\vert_x \;\colon\; (E_1)_x \overset{\simeq}{\longrightarrow} (E_2)_x \end{displaymath} then $f$ is already an isomorphism of vector bundles. \end{lemma} \begin{proof} It is clear that $f$ has an [[inverse function]] of underlying sets $f^{-1} \colon E_2 \to _E_1$ which is a function over $X$: Over each $x \in X$ it it the linear inverse $(f\vert_x)^{-1} \colon (E_2)_x \to (E_1)_x$. What we need to show is that this is a continuous function. By remark \ref{CommonOpenCoverLocalTrivialization} we find an open cover $\{U_i \subset X\}_{i \in I}$ over which both bundles have a local trivialization. \begin{displaymath} \left\{ U_i \underoverset{\simeq}{\phi^1_i}{\to} (E_1)\vert_{U_i}\right\}_{i \in I} \phantom{AA} \text{and} \phantom{AA} \left\{ U_i \underoverset{\simeq}{\phi^2_i}{\to} (E_2)\vert_{U_i} \right\}_{i \in I} \,. \end{displaymath} Restricted to any patch $i \in I$ of this cover, the homomorphism $f|_{U_i}$ induces a homomorphism of [[trivial vector bundles]] \begin{displaymath} f_i \coloneqq \phi^2_j^{-1} \circ f \circ \phi^1_i \phantom{AAAAAA} \itexarray{ U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& (E_1)\vert|_{U_i} \\ {}^{f_i}\downarrow && \downarrow^{\mathrlap{f\vert_{U_i}}} \\ U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& (E_2)\vert_{U_j} } \,. \end{displaymath} Also the $f_i$ are fiberwise invertible, hence are continuous bijections. We claim that these are [[homeomorphisms]], hence that their inverse functions $(f_i)^{-1}$ are also continuous. To this end we re-write the $f_i$ a little. First observe that by the [[universal property]] of the [[product topological space]] and since they fix the base space $U_i$, the $f_i$ are equivalently given by a continuous function \begin{displaymath} h_i \;\colon\; U_i \times k^n \longrightarrow k^n \end{displaymath} as \begin{displaymath} f_i(x,v) = (x, h_i(x,v)) \,. \end{displaymath} Moreovern since $k^n$ is [[locally compact topological space|locally compact]] (like every [[finite dimensional vector space]], by the [[Heine-Borel theorem]]), the [[mapping space]] [[adjunction]] says (by \href{Introduction+to+Topology+--+1#UniversalPropertyOfMappingSpace}{this prop.}) that there is a continuous function \begin{displaymath} \tilde h_i \;\colon\; U_i \longrightarrow Maps(k^n, k^n) \end{displaymath} (with $Maps(k^n,k^n)$ the set of continuous functions $k^n \to k^n$ equipped with the [[compact-open topology]]) which factors $h_i$ via the [[evaluation]] map as \begin{displaymath} h_i \;\colon\; U_i \times k^n \overset{\tilde h_i \times id_{k^n}}{\longrightarrow} Maps(k^n, k^n) \times k^n \overset{ev}{\longrightarrow} k^n \,. \end{displaymath} By assumption of fiberwise linearity the functions $\tilde h_i$ in fact take values in the [[general linear group]] \begin{displaymath} GL(n,k) \subset Maps(k^n, k^n) \end{displaymath} and this inclusion is a [[homeomorphism]] onto its image (by \href{general+linear+group#AsSubspaceOfTheMappingSpace}{this prop.}). Since passing to [[inverse matrices]] \begin{displaymath} (-)^{-1} \;\colon\; GL(n,k) \longrightarrow GL(n,k) \end{displaymath} is a [[rational function]] on its domain $GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$ inside [[Euclidean space]] and since [[rational functions are continuous]] on their domain of definition, it follows that the inverse of $f_i$ \begin{displaymath} (f_i)^{-1} \;\colon\; U_i \times k^n \overset{(id , \tilde h_i ) }{\longrightarrow} U_i \times k^n \times GL(n,k) \overset{ id \times (-)^{-1} }{\longrightarrow} U_i \times k^n \times GL(n,k) \overset{id \times ev}{\longrightarrow} U_i \times k^n \end{displaymath} is a continuous function. To conclude that also $f^{-1}$ is a continuous function we make use prop. \ref{FromTransitionFunctionsReconstructVectorBundle} to find an isomorphism between $E_2$ and a [[quotient topological space]] of the form \begin{displaymath} E_2 \simeq \left(\underset{i \in I}{\sqcup} (U_i \times k^n) \right) / \left( \left\{ g_{i j}\right\}_{i,j\in I} \right) \,. \end{displaymath} Hence $f^{-1}$ is equivalently a function on this quotient space, and we need to show that as such it is continuous. By the [[universal property]] of the [[disjoint union space]] (the [[coproduct]] in [[Top]]) the set of continuous functions \begin{displaymath} \{ U_i \times k^n \overset{f_i^{-1}}{\to} U_i \times k^n \overset{\phi^1_i}{\to} E_1 \}_{i \in I} \end{displaymath} corresponds to a single continuous function of the form \begin{displaymath} (\phi^1_i \circ f_i^{-1})_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} U_i \times k^n \longrightarrow E_1 \,. \end{displaymath} These functions respect the equivalence relation, since for each $x \in U_i \cap U_j$ we have \begin{displaymath} (\phi^1_i \circ f_i^{-1})((x,i),v) = (\phi^1_j \circ f_j^{-1})( (x,j), g_{i j}(x)(v) ) \phantom{AAAA} \text{since:} \phantom{AAAA} \itexarray{ && E_1 \\ & {}^{\mathllap{\phi^1_i \circ f_i^{-1}}}\nearrow & \uparrow^{\mathrlap{f^{-1}}} & \nwarrow^{\mathrlap{ \phi^1_j \circ f_j^{-1} }} \\ U_i \times k^n &\underset{\phi^2_i}{\longrightarrow}& (E_2)\vert_{U_i \cap U_i} &\underset{(\phi^2_j)^{-1}}{\longrightarrow}& U_i \times k^n } \,. \end{displaymath} Therefore by the [[universal property]] of the [[quotient topological space]] $E_2$, these functions [[extension|extend]] to a unique continuous function $E_2 \to E_1$ such that the following [[commuting diagram|diagram commutes]]: \begin{displaymath} \itexarray{ \underset{i \in i}{\sqcup} U_i \times k^n &\overset{( \phi^1_i \circ f_i^{-1} )_{i \in I}}{\longrightarrow}& E_1 \\ \downarrow & \nearrow_{\mathrlap{\exists !}} \\ E_2 } \,. \end{displaymath} This unique function is clearly $f^{-1}$ (by pointwise inspection) and therefore $f^{-1}$ is continuous. \end{proof} \begin{example} \label{FiberwiseLinearlyIndependentSectionsTrivialize}\hypertarget{FiberwiseLinearlyIndependentSectionsTrivialize}{} \textbf{(fiberwise linearly independent sections trivialize a vector bundle)} If a topological vector bundle $E \to X$ of [[rank of a vector bundle|rank]] $n$ admits $n$ [[sections]] (example \ref{VectorBundleSections}) \begin{displaymath} \{\sigma_k \;\colon\; X \longrightarrow E\}_{k \in \{1, \cdots, n\}} \end{displaymath} that are linearly independent at each point $x \in X$, then $E$ is trivializable (example \ref{TrivialTopologicalVectorBundle}). In fact, with the sections regarded as vector bundle homomorphisms out of the trivial vector bundle of rank $n$ (according to example \ref{VectorBundleSections}), these sections \emph{are} the trivialization \begin{displaymath} (\sigma_1, \cdots, \sigma_n) \;\colon\; (X \times k^n) \overset{\simeq}{\longrightarrow} E \,. \end{displaymath} This is because their linear independence at each point means precisely that this morphism of vector bundles is a fiber-wise linear isomorphsm and therefore an isomorphism of vector bundles by lemma \ref{FiberwiseIsoisIsomorphismOfVectorBundles}. \end{example} \hypertarget{DirectSummandBundles}{}\subsubsection*{{Direct summand bundles}}\label{DirectSummandBundles} We discuss properties of the [[direct sum of vector bundles]] for topological vector bundles. \begin{prop} \label{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}\hypertarget{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}{} \textbf{(sub-bundles over [[paracompact spaces]] are [[direct sum of vector bundles|direct summands]])} Let \begin{enumerate}% \item $X$ be a [[paracompact Hausdorff space]], \item $E \to X$ a [[topological vector bundle]] (def. \ref{TopologicalVectorBundle}). \end{enumerate} Then every topological vector sub-bundle $E_1 \hookrightarrow E$ (example \ref{TopologicalVetorSubbundle}) is a direct vector bundle summand, in that there exists another vector sub-bundle $E_2 \hookrightarrow E$ (example \ref{TopologicalVetorSubbundle}) such that their [[direct sum of vector bundles]] is $E$: \begin{displaymath} E_1 \oplus E_2 \simeq E \,. \end{displaymath} \end{prop} (\hyperlink{Hatcher}{e.g. Hatcher, prop. 1.3}) \begin{proof} Since $X$ is assumed to be paracompact Hausdorff, there exists an [[inner product on vector bundles]] \begin{displaymath} \langle -,-\rangle \;\colon\; E \oplus_X E \longrightarrow X \times k \end{displaymath} (by \href{inner+product+of+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces}{this prop.}). This defines at each $x \in X$ the [[orthogonal complement]] $(E'_x)^\perp \subset E_x$ of $E'_x \hookrightarrow E$. The [[subspace]] of these orthogonal complements is readily checked to be a [[topological vector bundle]] $(E')^\perp \to X$. Hence by construction we have \begin{displaymath} E \;\simeq\; E' \oplus_X (E')^\perp \,. \end{displaymath} \end{proof} \begin{prop} \label{TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle}\hypertarget{TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle}{} \textbf{(vector bundles over a [[compact Hausdorff space]] are [[direct sum of vector bundles|direct summands]] of a [[trivial vector bundle]])} Let \begin{enumerate}% \item $X$ be a [[compact Hausdorff space]]; \item $E \to X$ a [[topological vector bundle]] (def. \ref{TopologicalVectorBundle}). \end{enumerate} Then there exists another topological vector bundle $\tilde E \to X$ such that the [[direct sum of vector bundles]] of the two is [[isomorphism|isomorphic]] to a [[trivial vector bundle]] $X \times k^n$: \begin{displaymath} E \oplus \tilde E \;\simeq\; X \times k^n \,. \end{displaymath} \end{prop} \begin{proof} Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which $E \to X$ has a [[local trivialization]] \begin{displaymath} \left\{ \phi_i \;\colon\; U_i \times k^n \overset{\simeq}{\longrightarrow} E\vert_{U_i} \right\}_{i \in I} \,. \end{displaymath} By compactness of $X$, there is a [[finite cover|finite sub-cover]], hence a [[finite set]] $J \subset I$ such tat \begin{displaymath} \{U_i \subset X\}_{i \in J \subset I} \end{displaymath} is still an open cover over which $E$ trivializes. Since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]] there exists a [[partition of unity]] \begin{displaymath} \left\{ f_i \;\colon\; X \to [0,1] \right\}_{i \in J} \end{displaymath} with [[support]] $supp(f_i) \subset U_i$. Hence the functions \begin{displaymath} \itexarray{ E\vert_{U_i} &\overset{\phantom{AAAA}}{\longrightarrow}& U_i \times k^n \\ v &\overset{\phantom{AAA}}{\mapsto}& f_i(x) \cdot \phi_i^{-1}(v) } \end{displaymath} extend by 0 to vector bundle homomorphism of the form \begin{displaymath} f_i \cdot \phi^{-1}_i \;\colon\; E \longrightarrow X \times k^n \,. \end{displaymath} The finite pointwise [[direct sum]] of these yields a vector bundle homomorphism of the form \begin{displaymath} \underset{i \in J}{\oplus} f_i \cdot \phi_i \;\colon\; E \longrightarrow X \times \left( \underset{i \in J}{\oplus} k^n \right) \simeq X \times k^{n \dot {\vert J\vert}} \,. \end{displaymath} Observe that, as opposed to the single $f_i \cdot \phi^{-1}_i$, this is a fiber-wise injective, because at each point at least one of the $f_i$ is non-vanishing. Hence this is an injection of $E$ into a trivial vector bundle. With this the statement follows by prop. \ref{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}. \end{proof} \begin{remark} \label{}\hypertarget{}{} Prop. \ref{TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle} is key in the analysis of [[topological K-theory]] groups on [[compact Hausdorff spaces]]. See \href{topological+K-theory#DirectSumHasInverseUpToTrivialBundle}{there} for more. \end{remark} \hypertarget{ConcordanceOfTopolgicslVectorBundles}{}\subsubsection*{{Concordance}}\label{ConcordanceOfTopolgicslVectorBundles} We discuss that every [[concordance]] of topological vector bundles over a [[paracompact topological space]] makes the restriction of the vector bundle over the endpoints of the interval isomorphic (prop. \ref{ConcondanceOfTopologicalVectorBundles} below). In particular this implies tht the [[pullbacks of vector bundles]] along two [[homotopy|homotopic]] [[continuous functions]] are [[isomorphism|isomorphic]] (corollary \ref{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic} below). The proof below follows \hyperlink{Hatcher}{Hatcher, theorem 1.6}. For $X$ a [[topological space]] write $X \times I$ for the [[product topological space]] with the [[closed interval]] $[0,1]$ equipped with its [[Euclidean space|Euclidean]] [[metric topology]]. Write \begin{displaymath} X \overset{p_X}{\longleftarrow} X \times [0,1] \overset{p_{[0,1]}}{\longrightarrow} [0,1] \end{displaymath} for the two continuous [[projections]] out of the product space. \begin{lemma} \label{TrivilizationOfVectorBundleOverProductSpaceWithInterval}\hypertarget{TrivilizationOfVectorBundleOverProductSpaceWithInterval}{} For $X$ a [[topological space]], then a vector bundle $E \to X \times [0,1]$ is trivializable (example \ref{TrivialTopologicalVectorBundle}) if its restrictions to $X \times [0,1/2]$ and to $X \times [1/2,1]$ are trivializable. \end{lemma} \begin{lemma} \label{CoverForProductSpaceWithIntrval}\hypertarget{CoverForProductSpaceWithIntrval}{} For $X$ a [[topological space]], then for every topological vector bundle $E \to X \times I$ there exists an [[open cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ such that the vector bundle trivializes over $U_i \times [0,1] \subset X \times [0,1]$, for each $i \in I$. \end{lemma} \begin{proof} By [[local trivialization|local trvializability]] of the vector bundle, there exists an open cover $\{V_j \subset X \times I\}_{j \in J}$ over which the bundle trivializes. For each point $x \in X$ this induces a cover of $\{x\} \times [0,1]$. This is a [[compact topological space]] (for instance by the [[Heine-Borel theorem]]) and hence there exists a [[finite set|finite]] [[subset]] $J_x \subset I$ such that $\{V_i \subset X \times I\}_{i \in J_x}$ still covers $\{x\} \times [0,1]$. By finiteness of $J_x$, the [[intersection]] \begin{displaymath} U_x \coloneqq \underset{i \in J_x}{\cap} p_X(V_i) \end{displaymath} is an open neighbourhood of $x$ in $X$. Moreover \begin{displaymath} \{ p_{[0,1]}(V_i) \subset I \}_{i \in J_x} \end{displaymath} is an open cover of $[0,1]$ such that the given vector bundle trivializes over each element of $\{U_x \times p_{[0,1]}(V_i)\}_{i \in J_x}$. By the nature of the Euclidean [[metric topology]] each [[open subset]] of $[0,1]$ is a union of intervals. So we may pass to a [[refinement]] of this cover of $[0,1]$ such that each element is a single interval. Again by compactness of $[0,1]$, this refinement has a finite subcover \begin{displaymath} \{W_{x,k} \subset [0,1]\}_{k \in K_x} \end{displaymath} each element of which is an [[interval]]. Since this is a finite cover, we may find numbers $\{0 = t_0 \lt t_1 \lt t_2 \lt \cdots \lt t_{n_x} = 1\}$ such that \begin{displaymath} \{ [t_k, t_{k+1}] \subset [0,1] \}_{0 \leq k \lt n_x} \end{displaymath} is a cover of $[0,1]$, and such that the given vector bundle still trivializes over $V_x \times [t_k, t_{k+1}]$ for all $0 \leq k \lt n_x$. By lemma \ref{TrivilizationOfVectorBundleOverProductSpaceWithInterval} this implies that the vector bundle in fact trivializes over $U_x \times [0,1]$. Applying this procedure for all points $x \in X$ yields a cover \begin{displaymath} \{ U_x \subset X \}_{x \in X} \end{displaymath} with the required property. \end{proof} \begin{prop} \label{ConcondanceOfTopologicalVectorBundles}\hypertarget{ConcondanceOfTopologicalVectorBundles}{} \textbf{([[concordance]] of [[topological vector bundles]])} Let $X$ be a [[paracompact Hausdorff space]]. If $E \to X \times [0,1]$ is a [[topological vector bundle]] over the [[product space]] of $X$ with the [[closed interval]] (hence a \emph{[[concordance]]} of topological vector bundles on $X$), then the two endpoint-restrictions \begin{displaymath} E|_{X \times \{0\}} \phantom{AA} \text{and} \phantom{AA} E|_{X \times \{1\}} \end{displaymath} are [[isomorphism|isomorphic]] vector bundles over $X$. \end{prop} \begin{proof} By lemma \ref{CoverForProductSpaceWithIntrval} there exists an open cover $\{U_i \subset X\}_{i \in I}$ of $X$ such that the vector bundle $E$ trivializes over $U_i \times [0,1]$ for each $i \in I$. By \href{paracompact+topological+space#CountableCoverOfUnionsofOpenSubsetsInsideGivenCover}{this lemma} there exists a [[countable cover]] \begin{displaymath} \{V_n \subset X\}_{n \in \mathbb{N}} \end{displaymath} such that each element is a [[disjoint union]] of open subsets that each are contained in one of the $U_i$. This means that the vector bundle $E$ still trivializes over $V_n \times [0,1]$, for each $n \in \mathbb{N}$. Moreover, since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]], there exists a [[partition of unity]] $\left\{f_n \colon X \to [0,1] \right\}_{n \in \mathbb{N}}$ subordinate to this countable cover. For $n \in \mathbb{N}$ define \begin{displaymath} \psi_n \coloneqq \underoverset{k = 0}{n}{\sum} f_n \end{displaymath} (so $\psi_0 = 0$ and by local finiteness there is for each $x \in X$ an $n_x$ such that $\psi_{n \gt n_x} = 1$.) Now write \begin{displaymath} X_n \coloneqq graph( \psi_n ) \subset X \times [0,1] \end{displaymath} for the [[graph]] of the function $\psi_n$ equipped with its [[subspace topology]], and write \begin{displaymath} E_n \coloneqq \psi_n^\ast E \end{displaymath} for the restriction of $E$ to that subspace \begin{displaymath} \itexarray{ E_n &\longrightarrow& E \\ \downarrow && \downarrow \\ X_n = graph(\psi_n) &\hookrightarrow& X } \end{displaymath} Observe that the projection functions \begin{displaymath} \itexarray{ p_{n+1,n} \colon & X_{n+1} &\overset{}{\longrightarrow}& X_n \\ & (x,\psi_{n+1}(x)) &\overset{\phantom{AA}}{\mapsto}& (x, \psi_n(x)) = (x, \psi_{n+1}(x) - f_{n+1}(x)) } \end{displaymath} are [[continuous functions]]: By the nature of the [[product topology]] and the [[subspace topology]] it is sufficient to check for $U \subset X$ and $V \subset \mathbb{R}$ open subsets, that every point $(x,c)$ in the preimage $p_n^{-1}( U \times V ) \subset X \times [0,1]$ is contained in an open subset of the form $U_x \times V_x \subset X \times [0,1]$ such that every point of $X_{n+1}$ that is also in $U_x \times V_x$ is still mapped to $U \times V$. Such an open subset is $\left( U \cap \psi_n^{-1}(V) \right) \times [0,1]$. Also observe that the composites \begin{displaymath} E_n \longrightarrow X_n \overset{p_{n,0}}{\longrightarrow} X_0 = 0 \end{displaymath} make each $E_n$ a vector bundle over $X$: To see local trivializability over $X$ choose a local trivialization of $E$ over some open cover $\{U_i \subset X\}_{i \in I}$ and observe that then $E_n$ is trivial over the [[fiber product]]$X_n \times_X U_n$ and hence over $U_n$. Now by the pullback definition of the $E_n$, the [[pasting law]] says that for each $n \in \mathbb{N}$ we have a pullback square of vector bundles of the form \begin{displaymath} \itexarray{ E_{n+1} && \overset{h_n}{\longrightarrow} && E_n \\ \downarrow && (pb) && \downarrow \\ X_{n+1} && \longrightarrow && X_n \\ & \searrow && \swarrow \\ && X } \,. \end{displaymath} By the nature of pullbacks, the top horizontal function $h_n$ in this diagram is on each fiber a linear isomorphism. Therefore prop. \ref{FiberwiseIsoisIsomorphismOfVectorBundles} implies that each $h_n$ is in fact an isomorphism of vector bundles over $X$ By local finiteness, each point $x \in X$ has a neighbourhood $U_x$ such that only a finite number $n_x$ of these $h_n$ are non-trivial, and so it makes sense to consider the infinite composition \begin{displaymath} h \coloneqq h_1 \circ h_2 \circ h_3 \circ \cdots \end{displaymath} understood to be on each $U_x$ the finite composite \begin{displaymath} h(x) \coloneqq h_1 \circ \cdots \circ h_{n_x} \,. \end{displaymath} Since all the $h_k$ are vector bundle isomorphisms, so are all their composites. Thus $h$ is an isomorphism of the required form \begin{displaymath} h \;\colon\; E|_{X \times \{0\}} \overset{\simeq}{\longrightarrow} E|_{X \times \{1\}} \,. \end{displaymath} \end{proof} \begin{cor} \label{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic}\hypertarget{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic}{} Let $X$ be a [[paracompact Hausdorff space]], let $E \to Y$ be a topological [[vector bundle]], let $f,g \colon X \to Y$ be two [[continuous functions]], and let $\eta \colon f \to g$ be a [[left homotopy]] between them. Then there is an [[isomorphism]] of vector bundles over $X$ between the [[pullback of vector bundles]] of $E$ along $f$ and along $g$, respectively: \begin{displaymath} f^\ast E \simeq g^\ast E \,. \end{displaymath} \end{cor} \begin{proof} By definition, the [[left homotopy]] $\eta$ is a [[continuous function]] of the form \begin{displaymath} \eta \;\colon\; X \times [0,1] \longrightarrow Y \,. \end{displaymath} For $t \in [0,1]$ write $i_t$ for the continuous function \begin{displaymath} \itexarray{ X &\overset{\phantom{AA}i_t\phantom{AA}}{\longrightarrow}& X \times [0,1] \\ x &\overset{\phantom{AAAA}}{\mapsto}& (x,t) } \,. \end{displaymath} By the [[pasting law]] for pullbacks we have that \begin{displaymath} f^\ast E = (\eta \circ i_0)^\ast E \simeq i_0^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{0\}} \end{displaymath} and \begin{displaymath} g^\ast E = (\eta \circ i_1)^\ast E \simeq i_1^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{1\}} \end{displaymath} With this the statement follows by prop. \ref{ConcondanceOfTopologicalVectorBundles}. \end{proof} \begin{example} \label{HomotopyInvarianceOfIsomorphismClassesOfVectorBundles}\hypertarget{HomotopyInvarianceOfIsomorphismClassesOfVectorBundles}{} \textbf{([[homotopy invariance]] of isomorphism classes of vector bundles)} Let $X$ and $Y$ be [[paracompact Hausdorff spaces]] and let \begin{displaymath} f \;\colon\; X \longrightarrow Y \end{displaymath} be a [[continuous function]] which is a [[homotopy equivalence]]. Then pullback along $f$ constitutes a [[bijection]] on sets of isomorphism classes of topological vector bundles: \begin{displaymath} f^\ast \;\colon\; Vect(Y)/_\sim \overset{\simeq}{\longrightarrow} Vect(X)/_\sim \,. \end{displaymath} \end{example} \begin{proof} By definition of homotopy equivalence, there is a continuous function $g \colon Y \longrightarrow X$ and [[left homotopies]] \begin{displaymath} g \circ f \Rightarrow id \phantom{AAAA} f \circ g \Rightarrow id \,. \end{displaymath} Hence corollary \ref{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic} implies that \begin{displaymath} f^\ast \circ g^\ast = (g \circ f)^\ast = id \phantom{AAAAA} g^\ast \circ f^\ast = (g \circ g)^\ast = id \,. \end{displaymath} This mean that $g^\ast$ is the [[inverse function]] to $f^\ast$, and hence both are bijections. \end{proof} \begin{example} \label{TopologicalVectorBundleOverContractibleSpaceIsTrivializable}\hypertarget{TopologicalVectorBundleOverContractibleSpaceIsTrivializable}{} \textbf{(topological vector bundle on contractible topological space is trivializable)} If $X$ is a [[contractible topological space]], then every topological vector bundle over $X$ is isomorphic to a [[trivial vector bundle]]. \end{example} \begin{proof} That $X$ is contractible means by definition that there is a [[left homotopy]] of the form \begin{displaymath} \itexarray{ X &\longrightarrow& \ast \\ \mathllap{i_0}\downarrow & & \downarrow \\ X \times [0,1] &\overset{\eta}{\longrightarrow}& X \\ \mathllap{i_1}\uparrow & \nearrow_{\mathrlap{id}} \\ X & } \,. \end{displaymath} By cor \ref{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic} it follows that for $E \to X$ any topological vector bundle that there is an isomorphism between $id^\ast E = E$ and the result of first restricting the bundle to the point, and then forming the [[pullback bundle]] along $X \to \ast$. But the latter operation precisely produces the [[trivial vector bundles]] over $X$. \end{proof} \hypertarget{OverClosedSubspaces}{}\subsubsection*{{Over closed subspaces}}\label{OverClosedSubspaces} We discuss the behavour of vector bundles with respect to [[closed subspaces]] $A \subset X$ of [[compact Hausdorff spaces]]. \begin{lemma} \label{IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods}\hypertarget{IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods}{} \textbf{([[isomorphism]] of vector bundles on [[closed subset]] of [[compact Hausdorff spaces]] [[extension|extends]] to [[open neighbourhood]])} Let $k \in \{\mathbb{R}, \mathbb{C}\}$, let $X$ be a [[compact Hausdorff space]] and let $A \subset X$ a [[closed subset|closed]] [[subspace]]. Let $E_i \overset{p_i}{\to} X$ be two topological vector bundles over $X$, $i \in \{1,2\}$. If there exists an [[isomorphism]] \begin{displaymath} E_1\vert_A \overset{\simeq}{\longrightarrow} E_2\vert_A \end{displaymath} of the restricted vector bundles over $A$, then there also exists an [[open subset]] $U \subset X$ with $A \subset U$ such that there is also an isomorphism \begin{displaymath} E_1\vert_U \overset{\simeq}{\longrightarrow} E_2\vert_U \end{displaymath} of the vector bundles restricted to $U$. \end{lemma} \begin{proof} A bundle isomorphism $E_1\vert_A \simeq E_2\vert_A$ is equivalently a trivializing section (example \ref{FiberwiseLinearlyIndependentSectionsTrivialize}) of the [[tensor product of vector bundles]] $(E_1\vert_A)^\ast \otimes_A E_2\vert_A$ of $E_2\vert_A$ with the [[dual vector bundle]] $(E_2\vert_A)^\ast$. (by \href{tensor+product+of+vector+bundles#FinitrRankBundleHomomorphismIsSectionOfTensorProductWithDual}{this prop.}). Let $\{V_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which this tensor product bundle trivializes with trivializations \begin{displaymath} \left\{ V_i \times \mathbb{R}^{(n^2)} \underoverset{\simeq}{\phi_i}{\longrightarrow} (E_1^\ast \otimes_X E_2)\vert_{U_i} \right\} \,. \end{displaymath} Since [[compact Hausdorff spaces are normal]], the [[shrinking lemma]] applies and gives a refinement of this by a cover $\{U_i \subset X\}_{i \in I}$ by \emph{closed} subsets $U_i \subset X$. Then a trivializing section $\sigma \in \Gamma_A\left( (E_1\vert_A)^\ast \otimes_A E_2 \vert_A \right)$ as above is on each $U_i \cap A$ a [[continuous function]] \begin{displaymath} \sigma_i \;\colon\; U_i \cap A \longrightarrow GL(n,k) \end{displaymath} to the [[general linear group]] $GL(n,k) \subset Mat_{n \times n}(k)$, such that \begin{displaymath} \sigma\vert_{U_i \cap A} = \phi_i \circ \sigma_i \,. \end{displaymath} Regarded as a function to the $n \times n$ [[matrices]], this is a set of $n^2$ [[continuous function]] $((\sigma_i)_{a b})$ Now since $U_i \subset X$ is closed by construction, and $A \subset X$ is closed by assumption, also the intersections $U_i \cap X$ are closed. Since [[compact Hausdorff spaces are normal]] the [[Tietze extension theorem]] therefore applies to these component functions and yields [[extensions]] of each $\sigma_i$ to a [[continuous function]] of the form \begin{displaymath} \hat \sigma_i \;\colon\; U_i \longrightarrow Mat_{n \times n}(k) \,. \end{displaymath} Moreover, since compact Hausdorff spaces are evidently [[paracompact Hausdorff spaces]], and since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]], it follows that we find a [[partition of unity]] $\{f_i \colon U_i \to \mathbb{R} \}_{i \in I}$. Consider then the functions $f_i \cdot \hat \sigma_i$ given by pointwise multiplication and regarded, via extension by zero, as continuous functions on all of $X$ \begin{displaymath} f_i \cdot \hat \sigma_i \;\colon\; X \longrightarrow \mathbb{R} \,. \end{displaymath} Summing these up yields a single section $\hat \sigma$ of $E_1^\ast \otimes_X E_2$ \begin{displaymath} \hat \sigma \coloneqq \sum_{i \in I} \phi_i(f_i \cdot \hat \sigma_i) \in \Gamma_X(E_1^\ast \otimes_X E_2) \,, \end{displaymath} which by construction is an [[extension]] of the original section, in that \begin{displaymath} \hat \sigma\vert_A = \sigma \,. \end{displaymath} This is because for each $a \in A \subset X$ we have, using the above definitions, \begin{displaymath} \begin{aligned} \left(\underset{i \in I}{\sum} \phi_i(f_i \cdot \hat \sigma_i)\right)(a) & = \underset{i \in I}{\sum} (\phi_i (\hat \sigma_i(a))) \\ & = \underset{i \in I}{\sum} \phi_i( f_i(a) \sigma_i(a) ) \\ & = \underset{i \in I}{\sum} f_i(a) \cdot (\phi_i \circ \sigma_i)(a) \\ & = \underset{i \in I}{\sum} f_i(a) \cdot \sigma(a) \\ & = \left( \underset{i \in I}{\sum} f_i(a)\right) \cdot \sigma(a) \\ & = \sigma(a) \end{aligned} \end{displaymath} Here the last step uses the nature of the partition of unity. Now while $\hat \sigma$ is an extension of the section $\sigma$ to $X$, it will in general not be a trivializing section on $X$. But since the [[general linear group]] $GL(n,k) = det^{-1}(k \setminus \{0\}) \subset Mat_{n \times n}(k)$ is an [[open subset]] of the [[Euclidean space]] $Mat_{n \times n}(k) \simeq k^{(n^2)}$, it follows that each point $x \in A$ has an open neighbourhood $U_x \subset X$ such that $\hat \sigma\vert_{U_x}$ is still a trivializing section, namely choosing $i_x \in I$ such that $x \in U_{i_x}$ set \begin{displaymath} U_x \coloneqq (\hat \sigma_{i_x})^{-1}( GL(n,k) ) \,. \end{displaymath} The union of these \begin{displaymath} U \coloneqq \underset{x \in A}{\cup} U_x \end{displaymath} is hence an open subset containing $A$ such that $(E_1^\ast \otimes_X E_2)\vert_U$ has a trivializing section, extending $\sigma$, hence such that there is an isomorphism $E_1\vert_U \simeq E_2 \vert_U$ extending the original isomorphism on $A$. \end{proof} As a consequence: \begin{prop} \label{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}\hypertarget{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}{} \textbf{([[vector bundle]] [[trivial vector bundle|trivial]] over [[closed subspace]] of [[compact Hausdorff space]] is [[pullback bundle|pullback]] of bundle on [[quotient space]])} Let $X$ be a [[compact Hausdorff space]] and let $A \subset X$ be a [[closed subspace]]. If a topological vector bundle $E \overset{p}{\to} X$ is such that its restriction $E\vert_A$ is [[trivializable vector bundle|trivializable]], then $E$ is [[isomorphism|isomorphic]] to the [[pullback bundle]] $q^\ast E'$ of a topological vector bundle $E' \to X/A$ over the [[quotient space]]. \end{prop} \begin{proof} Let \begin{displaymath} A \times k^n \underoverset{\simeq}{\phi}{\longrightarrow} E\vert_A \end{displaymath} be an isomorphism of vector bundles over $A$, which exists by assumption. Consider then on the total space $E\vert_A$ the [[equivalence relation]] given by \begin{displaymath} \phi^{-1}(x,v) \sim \phi^{-1}(x',v) \end{displaymath} for all $x,x' \in A$ and $v \in k^n$. Let \begin{displaymath} E' \coloneqq E/\sim \end{displaymath} be the corresponding [[quotient topological space]]. Observe that for $x \in X$ we have $E'_x = E_x$ while for $x \in A$ we have a canonical identification $E'_{x/A} \simeq k^n$, and over these points quotient coprojection is identified with $\phi^{-1}$: \begin{displaymath} \itexarray{ E &\overset{}{\longrightarrow}& E' \\ (x,v) &\mapsto& \left\{ \itexarray{ (x,v) &\vert& x \in X \setminus A \\ \phi^{-1}_x(v) &\vert& x\in A } \right. } \,. \end{displaymath} Since the composite continuous function \begin{displaymath} E \overset{p}{\longrightarrow} X \overset{q}{\longrightarrow} X/A \end{displaymath} respects the equivalence relation (in that it sends any two equivalent points to the same image point) the [[universal property]] of the quotient space yields a continuous function \begin{displaymath} p' \;\colon\; E' \to X/A \end{displaymath} such that the following [[commuting diagram|diagram commutes]] \begin{displaymath} \itexarray{ E &\longrightarrow& E' \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{p'}} \\ X &\overset{q}{\longrightarrow}& X/A } \,. \end{displaymath} We claim that this is a [[pullback]] diagram in [[Top]]: By the above description of the top horizontal function, it is a pullback diagram of underlying sets. Hence we need to see that the topology on $E$ has a [[base for a topology|base]] given by the pre-images of the open subsets in $X$ and in $E'$. Now by definition of the quotient space topology on $E'$, its open subsets are those of $E$ that either do not contain a point $(x,v)$ with $x \in A$ or if they do, then they also contain all the points of the form $(x', \phi_{x'}^{-1}(\phi_x(v)))$ for $x' \in A$. Moreover, if $(x,v)$ is in the open subset for $x \in A$, then also $(x,v')$ for all $v'$ in some open ball in $k^n$ containing $v$. Hence intersecing these pre-images with pre-images of open subsets of $X$ under $p$ yields a basis for the topology. Hence it only remains to see that $E' \overset{p'}{\longrightarrow} X/A$ is a vector bundle. The fiberwise linearity is clear, we need to show that it is locally trivializable. To that end, let $\{U_i \subset X\}_{i \in I}$ be an open cover over which $E \overset{p}{\to} X$ has a local trivialization. Since $A \subset X$ is assumed to be closed, it follows that \begin{displaymath} \left\{ U_i \setminus A \subset X \setminus A\right\}_{i \in I} \end{displaymath} is an open cover of the complement of $A$ in $X$. By the nature of the [[quotient space topology]], this induces an open cover of $X\setminus A$. If we adjoin the quotient $U/A$ of an open neighbourhood $U \subset X$ of $A$ in $X$, then \begin{displaymath} \{ U_i \setminus A \subset X/A \} \sqcup \{ U/A \subset X/A \} \end{displaymath} is an open cover of $X/A$. Moreover, by the construction of $E' \overset{p'}{\to} X/A$ it is clear that this bundle has a local trivialization over $U_i$, since $E \overset{p}{\to} X$ does, and similarly $E'$ trivializes over $U/A$ if $E$ trivializes over $U$. But such a $U$ does indeed exist by lemma \ref{IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods}. \end{proof} \begin{remark} \label{}\hypertarget{}{} Prop \ref{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace} is the reason why [[reduced K-theory|reduced]] [[topological K-theory]] satisfies the [[long exact sequences in cohomology]] that make it a [[generalized (Eilenberg-Steenrod) cohomology theory]]. See \end{remark} \begin{prop} \label{VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace}\hypertarget{VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace}{} Let $X$ be a [[compact Hausdorff space]] and $A \subset X$ a [[closed subset|closed]] [[subspace]] and write $X/A$ for the corresponding [[quotient topological space]] (\href{quotient+space#QuotientBySubspace}{this example}) with quotient coprojection denoted $q \colon X \longrightarrow X/A$. If $A$ is a [[contractible topological space]] then the [[pullback bundle]] construction \begin{displaymath} q^\ast \;\colon\; Vect(X/A)_{/\sim} \longrightarrow Vect(X)_{/\sim} \end{displaymath} is an [[isomorphism]]. \end{prop} \begin{proof} By example \ref{TopologicalVectorBundleOverContractibleSpaceIsTrivializable} every vector bundle $E \overset{p}{\to} X$ is trivializable over the contractible subspace $A$. Therefore prop. \ref{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace} implies that it is in the image of the pullback bundle map $q^\ast$. This says that $q^\ast$ is surjective. Finally, it is clear that it is injective. Therefore it is bijective. \end{proof} \begin{example} \label{}\hypertarget{}{} Let $(X,x)$ be a [[pointed topological space|pointed]] [[compact topological space]]. For $[0,1] \subset \mathbb{R}$ the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]]. There is \begin{enumerate}% \item the ordinary [[cylinder]], being the [[product space]] $X \times I$ \item the [[reduced cylinder]] $X \wedge I_+ = (X \times I)/( \{x\} \times I )$ which is the [[smash product]] with the interval that has a base point freely adjoined \end{enumerate} and \begin{enumerate}% \item the ordinary [[suspension]] $S X \coloneqq (X \times I)/( X \times \{0,1\} )$; \item the [[reduced suspension]] $\Sigma X \coloneqq (S X)/( \{x\} \times I )$. \end{enumerate} In both cases the reduced space is obtained from the unreduced space by quotienting out the contractible closed subspace $I \simeq \{x\} \times I$ and hence topological vector bundles do not see the difference between the reduced and the unreduced spaces, by prop. \ref{VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace}. \end{example} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[algebraic vector bundle]] \item [[differentiable vector bundle]] \item [[topological K-theory]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Textbook accounts include \begin{itemize}% \item Glenys Luke, Alexander S. Mishchenko, \emph{Vector bundles and their applications}, Math. and its Appl. \textbf{447}, Kluwer 1998. viii+254 pp. \href{http://www.ams.org/mathscinet-getitem?mr=99m:55019}{MR99m:55019} \item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf}) \end{itemize} Lecture notes with an eye towards [[topological K-theory]] is in \begin{itemize}% \item [[Klaus Wirthmüller]], \emph{Vector bundles and K-theory}, 2012 (\href{ftp://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf}{pdf}) \item [[Allen Hatcher]], chapter 1 of \emph{Vector bundles and K-Theory}, (partly finished book) \href{http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web} \end{itemize} [[!redirects topological vector bundles]] \end{document}