\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\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*{vector bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{vector_bundles}{}\section*{{Vector bundles}}\label{vector_bundles}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{standard}{Standard}\dotfill \pageref*{standard} \linebreak
\noindent\hyperlink{sheaftheoretic_version}{Sheaf-theoretic version}\dotfill \pageref*{sheaftheoretic_version} \linebreak
\noindent\hyperlink{virtual_vector_bundles}{Virtual vector bundles}\dotfill \pageref*{virtual_vector_bundles} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given some context of [[geometry]], then a \emph{vector bundle} is a collection of [[vector spaces]] that varies in a geometric way over a given base [[space]] $X$: over each [[generalized element|element]] $x \in X$ there is a [[vector space]] $V_x$, called the \emph{[[fiber]]} over $x$, and as $x$ varies in $X$, the fibers vary along in a geometric way. One also says that vector bundles are \emph{[[fiber bundles]]} whose fiber carries [[vector space]]-structure. Hence the theory of vector bundle is \emph{parameterized} [[linear algebra]].

For example

\begin{itemize}%
\item in [[topology]] a \emph{[[topological vector bundle]]} is a collection of [[vector spaces]] which ``vary continuously'' over a [[topological space]] $X$,


\item in [[differential geometry]] a \emph{[[differentiable vector bundle]]} is a collection of vector space which ``varies differentiably'' over a [[differentiable manifold]],


\item in [[algebraic geometry]] an \emph{[[algebraic vector bundle]]} is a collection of vector spaces which vary algebraically over a [[scheme]].



\end{itemize}
and so on.

One requires that ``locally'', on small enough patches of the base space $X$, the variation of the fibers is constant up to isomorphism (one says the vector bundle is ``locally trivial''), but the key point of vector bundles is that there may be non-trivial structure in how the collection of vector spaces ``globally glues together''.

For example if $X = S^1$ is the [[circle]] regarded as a [[topological space]] in the standard way, and if we consider [[real vector spaces]], then there are up to [[isomorphism]] two different $\mathbb{R}$-vector bundles over $S^1$ whose [[fibers]] look like the 1-dimensional real vector space $\mathbb{R}$ itself, namely

\begin{enumerate}%
\item the [[cylinder]]




\item the [[Möbius strip]]:





\end{enumerate}
(In these pictures each vertical interval is to be thought of as a stand-in for a copy of the [[real line]] $\mathbb{R}$.)

Clearly for the cylinder nothing special happens to the fibers as one moves around the circle (one says this is a \emph{trivial vector bundle}) while the M\"o{}bius strip is ``locally trivial'' but globally has a twist: as one moves once around the circle the original fiber comes back identified with its reflection at the origin.

An important class of examples of vector bundles are [[tangent bundles]] of [[differentiable manifolds]] $X$. Here the [[vector space]] at each point of $X$ is the [[tangent space]] of that point, the space of all [[tangent vectors]] based at that point. The graphics on the right shows one of the tangent space of the [[2-sphere]].

Dually, given an [[embedding of differentiable manifolds]] into a [[Euclidean space]], then the [[normal vectors]] to the tangent bundle span a vector bundle called the \emph{[[normal bundle]]} of the embedding.

All the usual operations on [[finite dimensional vector spaces]] in [[linear algebra]] generalize to vector bundles by applying them [[fiber]]-wise. For instance there is [[direct sum of vector bundles]] and the [[tensor product of vector bundles]] over the same base space.

To the extent that the base [[space]] $X$ is encoded in its [[algebra of functions]] (tautologically in [[algebraic geometry]] or via [[Gelfand duality]] in [[topology]]), the [[Serre-Swan theorem]] asserts that vector bundles over $X$ are equivalently encoded in the [[projective modules]] over these algebras constitutes by their [[sections]].

Vector bundles have various applications and uses:

\begin{enumerate}%
\item their [[Grothendieck group]] under [[direct sum of vector bundles]] yields [[topological K-theory]], an interesting [[generalized (Eilenberg-Steenrod) cohomology theory]];


\item a [[reduction of the structure group]] of vector bundles encodes actual [[geometry]] on the base space; when applied to [[tangent bundles]] such \emph{[[G-structures]]} on vector bundles encode for instance [[orthogonal structure]], [[Riemannian geometry]], [[complex geometry]], [[symplectic geometry]], [[conformal geometry]] etc. (in general: [[Cartan geometry]]); when applied to [[normal bundles]] these [[G-structures]] give rise, via [[Thom's theorem]], to [[Thom spectra]] and [[cobordism theory]];


\item equipping differentiable vector bundles with [[connection on a vector bundle]] is the basis for [[Chern-Weil theory]] and for the application of vector bundles in [[physics]], where they model [[gauge fields]] and [[instanton sectors]]; see also at \emph{[[fiber bundles in physics]]}.



\end{enumerate}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{standard}{}\subsubsection*{{Standard}}\label{standard}

See at \emph{[[topological vector bundle]]}

\hypertarget{sheaftheoretic_version}{}\subsubsection*{{Sheaf-theoretic version}}\label{sheaftheoretic_version}

Vector bundles can also be defined via [[sheaf and topos theory|sheaf theory]], which permits easy transport to general [[Grothendieck toposes]]. Let $Sh(X)$ be the [[category]] of ([[set]]-valued) [[sheaf|sheaves]] on $X$. The sheaf of continuous local sections of the product projection

\begin{displaymath}
X \times \mathbb{R} \to X
\end{displaymath}
forms a [[local ring]] object $R$; when interpreted in the [[internal logic]] of $Sh(X)$, it is the Dedekind [[real numbers object]]. Then, according to a [[Serre-Swan theorem|theorem of Richard Swan]], in its sheaf-theoretic incarnation a vector bundle is the same thing as a [[projective module|projective R-module]].

\begin{itemize}%
\item A theorem of Kaplansky states ``every [[projective module]] over a [[local ring]] is [[free module|free]]''. When interpreted in [[sheaf semantics]] ([[Kripke-Joyal semantics]]), the [[existential quantifier]] implicit in ``free'' is interpreted \emph{locally}, so we can consider a vector bundle as a locally free module over the Dedekind reals.

\end{itemize}
\hypertarget{virtual_vector_bundles}{}\subsubsection*{{Virtual vector bundles}}\label{virtual_vector_bundles}

In one class of models for [[K-theory]] -- [[generalized (Eilenberg-Steenrod) cohomology]] theory -- cocycles are represented by $\mathbb{Z}_2$-graded vector bundles (pairs of vector bundles, essentially) modulo a certain equivalence relation. In that context it is sometimes useful to consider a certain variant of infinite-dimensional $\mathbb{Z}_2$-graded vector bundles called [[vectorial bundle]]s.

Much else to be discussed\ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[principal bundle]], [[associated bundle]]


\item \textbf{vector bundle}, [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]

\begin{itemize}%
\item [[universal vector bundle]]


\item [[dual vector bundle]]


\item [[module bundle]]


\item [[direct sum of vector bundles]]


\item [[connection on a vector bundle]]


\item [[flat vector bundle]]


\item [[real vector bundle]], [[complex vector bundle]]


\item [[super vector bundle]]



\end{itemize}

\item [[2-vector bundle]]


\item [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]



\end{itemize}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\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 . . , \emph{    } (Russian; A. S. Mishchenko, Vector bundles and their applications) Nauka, Moscow, 1984. 208 pp.


\item Howard Osborn, \emph{Vector bundles. Vol. 1. Foundations and Stiefel-Whitney classes}, Pure and Appl. Math. \textbf{101}, Academic Press 1982. xii+371 pp. \href{http://www.ams.org/mathscinet-getitem?mr=85e:55001}{MR85e:55001}


\item [[Dale Husemöller]], \emph{Fibre bundles}, McGraw-Hill 1966 (300 p.); Springer GTM 1975 (327 p.), 1994 (353 p.).


\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}
An exposition with an eye towards [[gauge theory]] is in section 16.1 of

\begin{itemize}%
\item [[Theodore Frankel]], \emph{[[The Geometry of Physics - An Introduction]]}


\item [[Raoul Bott]], [[Loring Tu]], \emph{Differential forms in algebraic topology}, Graduate Texts in Mathematics \textbf{82}, Springer 1982. xiv+331 pp.



\end{itemize}
Discussion with an eye towards [[K-theory]] is in

\begin{itemize}%
\item [[Max Karoubi]], \emph{K-theory. An introduction}, Grundlehren der Mathematischen Wissenschaften \textbf{226}, Springer 1978. xviii+308 pp.


\item [[Allen Hatcher]], \emph{Vector bundles and K-Theory}, (partly finished book) \href{http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web}



\end{itemize}
[[!redirects vector bundles]]



\end{document}