## Fibre bundles

Last update: 9 September 2012

## Bourbaki, §6 Varietes Differentielles et Analytiques

A bundle or fibre bundle, is a morphism $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ such that if $b\in B$ then there exists

1. an open neighbourhood $U$ of $b$,
2. a variety $F$
3. an isomorphism $\phi :\phantom{\rule{0.2em}{0ex}}{p}^{-1}\left(U\right)\stackrel{\sim }{⟶}U×F$

such that

$\text{if}\phantom{\rule{1em}{0ex}}x\in U\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y\in F\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}p\left({\phi }^{-1}\left(x,y\right)\right)=x\text{.}$

$\begin{array}{ccc}U×F& \stackrel{\sim }{⟵}& {p}^{-1}\left(U\right)\\ p{r}_{1}↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}p\\ U& \stackrel{\phantom{\rule{2em}{0ex}}}{↪}& B\end{array}$

A morphism of bundles from $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ to ${p}^{\prime }:\phantom{\rule{0.2em}{0ex}}{E}^{\prime }⟶{B}^{\prime }$ is a pair of morphisms $f:B⟶{B}^{\prime }$ and $g:\phantom{\rule{0.2em}{0ex}}E⟶{E}^{\prime }$ such that ${p}^{\prime }\circ g=f\circ p$.

$\begin{array}{ccc}E& \stackrel{g}{⟶}& {E}^{\prime }\\ p↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}{p}^{\prime }\\ B& \underset{f}{⟶}& {B}^{\prime }\end{array}$

The trivial bundle with base $B$ and fibre $F$ is $p{r}_{1}:\phantom{\rule{0.2em}{0ex}}B×F⟶B$ given by $p{r}_{1}\left(b,f\right)=b$.

A section of $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ is $s:\phantom{\rule{0.2em}{0ex}}B⟶E$ such that $p\circ s={\text{id}}_{B}$.

A principal $G$–bundle is a morphism $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ where $P$ is a variety with a right $G$–action and if $b\in B$ then there exists

1. an open neighborhood $U$ of $b$,
2. an isomorphism $\psi :\phantom{\rule{0.2em}{0ex}}U×G⟶{p}^{-1}\left(U\right)$

such that

$\text{if}\phantom{\rule{1em}{0ex}}u\in U\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g,{g}^{\prime }\in G\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}p\left(\psi \left(u,g\right)\right)=u\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\psi \left(u,g{g}^{\prime }\right)=\psi \left(u,g\right)\text{something}$

$\begin{array}{ccc}U×G& \stackrel{\psi }{⟶}& {p}^{-1}\left(U\right)\\ p{r}_{1}↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}p\\ U& \stackrel{\phantom{\rule{2em}{0ex}}}{↪}& B\end{array}$

A morphism from a principal $G$–bundle $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ to a principal ${G}^{\prime }$–bundle ${p}^{\prime }:\phantom{\rule{0.2em}{0ex}}{E}^{\prime }⟶{B}^{\prime }$ is a triple $\left(f,\phi ,h\right)$ with

$f:\phantom{\rule{0.2em}{0ex}}E⟶{R}^{\prime },\phantom{\rule{2em}{0ex}}h:\phantom{\rule{0.2em}{0ex}}B⟶{B}^{\prime },\phantom{\rule{2em}{0ex}}\phi :\phantom{\rule{0.2em}{0ex}}G⟶{G}^{\prime }$

such that

$h\circ \phi ={\phi }^{\prime }\circ f\phantom{\rule{1em}{0ex}}\text{and if}\phantom{\rule{1em}{0ex}}x\in E,\phantom{\rule{0.2em}{0ex}}g\in G\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}f\left(xg\right)=f\left(x\right)\phi \left(g\right)\text{.}$

$\begin{array}{ccc}E& \stackrel{f}{⟶}& {E}^{\prime }\\ p↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}{p}^{\prime }\\ B& \underset{h}{⟶}& {B}^{\prime }\end{array}$

Let $G$ be a group.

Let $B$ be a space and $𝒰$ an open cover of $B$.

A cocycle on $B$ with values in $G$ subordinate to $U$ is a collection of morphisms ${\left({\gamma }_{uv}\right)}_{u,v\in U}$,

${\gamma }_{uv}:\phantom{\rule{0.2em}{0ex}}U\cap V⟶G$

such that

$\text{if}\phantom{\rule{1em}{0ex}}x\in U\cap V\cap W\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{\gamma }_{uw}={\gamma }_{uv}\left(x\right){\gamma }_{vw}\left(x\right)\text{.}$

Two cocycles are cohomologous $\left({\gamma }_{uv}\right)$ and $\left({\gamma }_{uv}^{\prime }\right)$ if there exists a collection of morphisms ${\left({h}_{u}\right)}_{u\in 𝒰}$

${h}_{u}:\phantom{\rule{0.2em}{0ex}}U⟶G$

such that

$\text{if}\phantom{\rule{1em}{0ex}}x\in U\cap V\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{\gamma }_{uv}^{\prime }\left(x\right)={h}_{u}{\left(x\right)}^{-1}{\gamma }_{uv}\left(x\right){h}_{v}\left(x\right)\text{.}$

Let $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ be a principal $G$–bundle.

A trivialization is an isomorphism $p:\phantom{\rule{0.2em}{0ex}}E⟶B\phantom{\rule{1em}{0ex}}\text{to}\phantom{\rule{1em}{0ex}}B×G\stackrel{p{r}_{1}}{⟶}B$

The map

$\begin{array}{ccc}\left\{\genfrac{}{}{0}{}{\text{sections of}}{p:\phantom{\rule{0.2em}{0ex}}E⟶B}\right\}& ⟷& \left\{\genfrac{}{}{0}{}{\text{trivializations of}}{p:\phantom{\rule{0.2em}{0ex}}E⟶B}\right\}\\ s& ⟼& {f}_{s}\end{array}$

given by

${f}_{s}^{-1}\left(b,g\right)=s\left(b\right)g,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}b\in B,\phantom{\rule{0.2em}{0ex}}g\in G,$

is a bijection.

Let $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ be a principal $G$–bundle. Let ${\left({s}_{u}\right)}_{u\in 𝒰}$ be a family if it sections over $U\in 𝒰$. Then

${s}_{v}\left(b\right)={s}_{u}\left(b\right){g}_{uv}\left(b\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}b\in U\cap V$

Let ${f}_{u}:\phantom{\rule{0.2em}{0ex}}{p}^{-1}\left(U\right)⟶U×G$ be the trivialisation corresponding to $\left({s}_{u}\right)$.

Then

${f}_{u}\left(x\right)={g}_{uv}\left(p\left(x\right)\right){f}_{v}\left(x\right)\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in {p}^{-1}\left(Y\cap V\right)\text{.}$

Let $\phi :\phantom{\rule{0.2em}{0ex}}E⟶{E}^{\prime }$ be an isomorphism of principal $G$–bundles $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ to ${p}^{\prime }:\phantom{\rule{0.2em}{0ex}}{E}^{\prime }⟶B$. Let ${\left({s}_{u}\right)}_{u\in 𝒰}$ be a family of sections of $p$, and ${\left({s}_{u}^{\prime }\right)}_{u\in 𝒰}$ a family of sections of ${p}^{\prime }$. Then

$\phi \left({s}_{u}^{\prime }\left(x\right)\right)={s}_{u}\left(x\right){h}_{u}\left(x\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}u\in 𝒰\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}x\in U\text{.}$

## §6.5

Let $F$ be a $G$–variety.

Then there is a map

$\begin{array}{ccc}\left\{\genfrac{}{}{0}{}{\text{principal}\phantom{\rule{0.2em}{0ex}}G\text{–}\phantom{\rule{0.2em}{0ex}}\text{bundles}}{p:\phantom{\rule{0.2em}{0ex}}E⟶B}\right\}& ⟷& \left\{\genfrac{}{}{0}{}{\text{bundles on}\phantom{\rule{0.2em}{0ex}}B}{\text{with fibre}\phantom{\rule{0.2em}{0ex}}F}\right\}\\ E& ⟼& E{×}_{G}F& =& \frac{E×F}{⟨\left(xg,f\right)=\left(x,gf\right)⟩}& \left(e,f\right)\\ p↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}\\ B& & B& & & p\left(b\right)\end{array}$

## Vector Bundles

Let $B$ be a space, $E$ a set, and $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ a function.

A chart of $p$ is a triple $\left(U,\phi ,F\right)$ where

1. $U$ is an open set of $B$
2. $F$ is a vector space
3. $\phi :\phantom{\rule{0.2em}{0ex}}p\left(U\right)⟶U×F$ is a bijection

such that

$p\left({\phi }^{-1}\left(b,h\right)\right)=b,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}b\in B\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}h\in F\text{.}$

Two charts $\phi :\phantom{\rule{0.2em}{0ex}}{p}^{-1}\left(U\right)⟶U×F$ and $\psi :\phantom{\rule{0.2em}{0ex}}{p}^{-1}\left(V\right)⟶V×{F}^{\prime }$ are compatible is there exists

$\lambda :U\cap V⟶{\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(F,{F}^{\prime }\right)\phantom{\rule{1em}{0ex}}\text{such that}\phantom{\rule{1em}{0ex}}{t}_{b}={t}_{b}^{\prime }·\lambda \left(b\right),$

for $b\in U\cap V$, where

$\begin{array}{cc}\begin{array}{ccccc}{t}_{b}& :& F& ⟶& {p}^{-1}\left(b\right)\\ & & h& ⟼{\phi }^{-1}\left(b,h\right)\end{array}& \begin{array}{ccccc}\left(U\cap V\right)×F& \stackrel{\phi }{⟵}& {p}^{-1}\left(U\cap V\right)& \stackrel{\psi }{⟶}& \left(U\cap V\right)×{F}^{\prime }\\ & & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}p\\ & & U\cap V\end{array}\end{array}$

A vector bundle is a collection of compatible charts for $p:\phantom{\rule{0.2em}{0ex}}E⟶B$.

A morphism of vector bundles is a morphism of bundles such that

$\begin{array}{c}\text{if}\phantom{\rule{1em}{0ex}}{b}_{0}\in B\phantom{\rule{1em}{0ex}}\text{then}\\ \text{there exists a chart}\phantom{\rule{1em}{0ex}}\left(U,\phi ,F\right)\phantom{\rule{1em}{0ex}}\text{of}\phantom{\rule{1em}{0ex}}p:E⟶B\\ \text{and a chart}\phantom{\rule{1em}{0ex}}\left({U}^{\prime },\phi \prime ,{F}^{\prime }\right)\phantom{\rule{1em}{0ex}}\text{of}\phantom{\rule{1em}{0ex}}{p}^{\prime }:\phantom{\rule{0.2em}{0ex}}E⟶B\\ \text{and a map}\phantom{\rule{1em}{0ex}}\lambda :\phantom{\rule{0.2em}{0ex}}U⟶{\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(F,{F}^{\prime }\right)\phantom{\rule{1em}{0ex}}\text{such that}\\ f\left(U\right)\subseteq {U}^{\prime }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{g}_{b}\circ {t}_{b}={t}_{f\left(b\right)}^{\prime }\circ \lambda \left(b\right)\end{array}$

where ${g}_{b}=\text{something}$

$\begin{array}{ccc}E& \stackrel{g}{⟶}& {E}^{\prime }\\ ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}\\ B& \underset{f}{⟶}& {B}^{\prime }\end{array}$

Let $f:\phantom{\rule{0.2em}{0ex}}{B}^{\prime }⟶B$ be a morphism of spcaes and $p:E⟶B$ a vector bundle.

The pullback $f*E$ of $E$ by $f$ is

$\begin{array}{ccc}\begin{array}{ccc}{f}^{*}\left(E\right)& =& {B}^{\prime }{×}_{B}E\\ & & p{r}_{1}↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}\\ & & {B}^{\prime }\end{array}& \text{where}& {B}^{\prime }{×}_{B}E=\left\{\left({b}^{\prime },e\right)\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}f\left({b}^{\prime }\right)=p\left(e\right)\right\}\\ \begin{array}{ccc}{f}^{*}\left(B\right)& ⟶& B\\ ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}\\ {B}^{\prime }& \underset{f}{⟶}& B\end{array}& \text{and}& {f}^{*}:\phantom{\rule{0.2em}{0ex}}\left\{\genfrac{}{}{0}{}{\text{vector bundles}}{\text{on}\phantom{\rule{0.2em}{0ex}}B}\right\}⟶\left\{\genfrac{}{}{0}{}{\text{vector bundles}}{\text{on}\phantom{\rule{0.2em}{0ex}}{B}^{\prime }}\right\}\end{array}$

is a functor.

Let $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ be a vector bundle on $B$.

Let $U$ be an open set of $B$.

The space of sections ${ℱ}_{E}\left(U\right)$ over $U$ is an ${𝒪}_{B}$ module with operations

$\begin{array}{c}\left({s}_{1}+{s}_{2}\right)\left(b\right)={s}_{1}\left(b\right)+{s}_{2}\left(b\right)\phantom{\rule{1em}{0ex}}\text{and}\\ \left(\phi s\right)\left(b\right)=\phi \left(b\right)s\left(b\right)\end{array}$

for $s,{s}_{1},{s}_{2}\in ℱ\left(U\right)$ and $\phi \in {𝒪}_{B}$. Then ${ℱ}_{E}$ is the sheaf of sections of $E$. The functor

$\begin{array}{ccc}\left\{\genfrac{}{}{0}{}{\text{vector bundles}}{\text{on}\phantom{\rule{0.2em}{0ex}}B}\right\}& ⟶& \left\{\genfrac{}{}{0}{}{\text{locally free sheaves}}{\text{of}\phantom{\rule{0.2em}{0ex}}{𝒪}_{B}–\text{modules}}\right\}\\ E& ⟼& {ℱ}_{E}\end{array}$

is an equivalence of categories.

## Bourbaki, Varietes Differentielles et Analytiques §7.10

A vector bundle $M\stackrel{p}{⟶}B$ is pure of type $F$ if $M\stackrel{p}{⟶}B$ satisfies

$\text{if}\phantom{\rule{1em}{0ex}}b\in B\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{M}_{b}\simeq F\text{.}$

Let $M\stackrel{p}{⟶}B$ be pure vector bundle of type $F$.

Let

$P=\left\{\left(b,u\right)\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}b\in B\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}u:\phantom{\rule{0.2em}{0ex}}F⟶{M}_{b}\phantom{\rule{1em}{0ex}}\text{is a linear isomorphism}\right\}$

with $GL\left(F\right)$ action given by

$\left(b,u\right)g=\left(b,u\circ g\right)$

and $P\stackrel{\pi }{⟶}B$ given by $\pi \left(b,u\right)=b$.

Then $P\stackrel{\pi }{⟶}B$ is a principal $GL\left(F\right)$ bundle and

$\begin{array}{ccc}\left\{\genfrac{}{}{0}{}{\text{rank}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{vector}}{\text{bundles}\phantom{\rule{0.2em}{0ex}}M\stackrel{p}{⟶}B}\right\}& ⟶& \left\{\genfrac{}{}{0}{}{\text{principal}}{G{L}_{n}\text{something}\phantom{\rule{0.2em}{0ex}}\text{bundles}}\right\}\\ M& ⟼& P\stackrel{\pi }{⟶}B\\ P{×}_{G{L}_{n}}{ℂ}^{n}& \stackrel{\phantom{\rule{2em}{0ex}}}{↤}& P\end{array}$

is an equivalence of categories.

The map

$\begin{array}{ccc}\left\{\genfrac{}{}{0}{}{\text{charts}\phantom{\rule{0.2em}{0ex}}t=\left(U,\phi ,F\right)}{\genfrac{}{}{0}{}{\phi :\phantom{\rule{0.2em}{0ex}}{p}^{-1}\left(U\right)\stackrel{r}{⟶}U×F}{\text{of}\phantom{\rule{0.2em}{0ex}}M\stackrel{p}{⟶}B}}\right\}& ⟶& \left\{\genfrac{}{}{0}{}{\text{sections}\phantom{\rule{0.2em}{0ex}}s:B\phantom{\rule{0.2em}{0ex}}⟶P}{\text{of}\phantom{\rule{0.2em}{0ex}}P\stackrel{\pi }{⟶}B}\right\}\\ t& ⟼& \genfrac{}{}{0}{}{s:\phantom{\rule{0.2em}{0ex}}B⟶P}{b⟼\left(b,{t}_{b}\right)}\end{array}$

is a bijection.