Fibre bundles

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 9 September 2012

Bourbaki, §6 Varietes Differentielles et Analytiques

A bundle or fibre bundle, is a morphism $p:E⟶B$ such that if $b\in B$ then there exists

1. an open neighbourhood $U$ of $b$,
2. a variety $F$
3. an isomorphism $\phi :p^{-1}\left(U\right)\stackrel{\sim }{⟶}U\times F$

such that

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

$\begin{array}{ccc}U\times F& \stackrel{\sim }{⟵}& p^{-1}\left(U\right)\\ p{r}_1\downarrow & & \downarrow p\\ U& \stackrel{}{\hookrightarrow }& B\end{array}$

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

$\begin{array}{ccc}E& \stackrel{g}{⟶}& E^{\prime }\\ p\downarrow & & \downarrow p^{\prime }\\ B& \underset{f}{⟶}& B^{\prime }\end{array}$

The trivial bundle with base $B$ and fibre $F$ is $p{r}_1:B\times F⟶B$ given by $p{r}_1(b,f)=b$.

A section of $p:E⟶B$ is $s:B⟶E$ such that $p\circ s={\text{id}}_B$.

A principal $G$–bundle is a morphism $p:E⟶B$ 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 $\psi :U\times G⟶p^{-1}\left(U\right)$

such that

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

$\begin{array}{ccc}U\times G& \stackrel{\psi }{⟶}& p^{-1}\left(U\right)\\ p{r}_1\downarrow & & \downarrow p\\ U& \stackrel{}{\hookrightarrow }& B\end{array}$

A morphism from a principal $G$–bundle $p:E⟶B$ to a principal $G^{\prime }$–bundle $p^{\prime }:E^{\prime }⟶B^{\prime }$ is a triple $(f,\phi ,h)$ with

$f:E⟶R^{\prime },h:B⟶B^{\prime },\phi :G⟶G^{\prime }$

such that

$h\circ \phi ={\phi }^{\prime }\circ f\text{and if}x\in E,g\in G\text{then}f\left(xg\right)=f\left(x\right)\phi \left(g\right)\text{.}$

$\begin{array}{ccc}E& \stackrel{f}{⟶}& E^{\prime }\\ p\downarrow & & \downarrow p^{\prime }\\ B& \underset{h}{⟶}& B^{\prime }\end{array}$

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 ${\left({\gamma }_{uv}\right)}_{u,v\in U}$,

${\gamma }_{uv}:U\cap V⟶G$

such that

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

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

$h_u:U⟶G$

such that

$\text{if}x\in U\cap V\text{then}{\gamma }_{uv}^{\prime }\left(x\right)=h_u{\left(x\right)}^{-1}{\gamma }_{uv}\left(x\right)h_v\left(x\right)\text{.}$

Let $p:E⟶B$ be a principal $G$–bundle.

A trivialization is an isomorphism $p:E⟶B\text{to}B\times G\stackrel{p{r}_1}{⟶}B$

The map

${\displaystyle \begin{array}{ccc}\left\{\genfrac{}{}{0ex}{}{\text{sections of}}{p:E⟶B}\right\}& ⟷& \left\{\genfrac{}{}{0ex}{}{\text{trivializations of}}{p:E⟶B}\right\}\\ s& ⟼& f_s\end{array}}$

given by

$f_s^{-1}(b,g)=s\left(b\right)g,\text{for}b\in B,g\in G,$

is a bijection.

Let $p:E⟶B$ be a principal $G$–bundle. Let ${\left(s_u\right)}_{u\in 𝒰}$ be a family if it sections over $U\in 𝒰$. Then

$s_v\left(b\right)=s_u\left(b\right)g_{uv}\left(b\right),\text{for}b\in U\cap V$

Let $f_u:p^{-1}\left(U\right)⟶U\times G$ be the trivialisation corresponding to $\left(s_u\right)$.

Then

$f_u\left(x\right)=g_{uv}\left(p\left(x\right)\right)f_v\left(x\right)\text{for}x\in p^{-1}(Y\cap V)\text{.}$

Let $\phi :E⟶E^{\prime }$ be an isomorphism of principal $G$–bundles $p:E⟶B$ to $p^{\prime }:E^{\prime }⟶B$. 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

$\phi \left(s_u^{\prime }\left(x\right)\right)=s_u\left(x\right)h_u\left(x\right),\text{for}u\in 𝒰\text{and}x\in U\text{.}$

§6.5

Let $F$ be a $G$–variety.

Then there is a map

${\displaystyle \begin{array}{ccc}\left\{\genfrac{}{}{0ex}{}{\text{principal}G\text{–}\text{bundles}}{p:E⟶B}\right\}& ⟷& \left\{\genfrac{}{}{0ex}{}{\text{bundles on}B}{\text{with fibre}F}\right\}\\ E& ⟼& E{\times }_{G}F& =& {\displaystyle \frac{E\times F}{⟨(xg,f)=(x,gf)⟩}}& (e,f)\\ p\downarrow & & \downarrow & & & \downarrow \\ B& & B& & & p\left(b\right)\end{array}}$

Vector Bundles

Let $B$ be a space, $E$ a set, and $p:E⟶B$ a function.

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

1. $U$ is an open set of $B$
2. $F$ is a vector space
3. $\phi :p\left(U\right)⟶U\times F$ is a bijection

such that

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

Two charts $\phi :p^{-1}\left(U\right)⟶U\times F$ and $\psi :p^{-1}\left(V\right)⟶V\times F^{\prime }$ are compatible is there exists

$\lambda :U\cap V⟶{\text{Hom}}_k(F,F^{\prime })\text{such that}{t}_b=t_b^{\prime }·\lambda \left(b\right),$

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

$\begin{array}{cc}\begin{array}{ccccc}{t}_b& :& F& ⟶& p^{-1}\left(b\right)\\ & & h& ⟼{\phi }^{-1}(b,h)\end{array}& \begin{array}{ccccc}(U\cap V)\times F& \stackrel{\phi }{⟵}& p^{-1}(U\cap V)& \stackrel{\psi }{⟶}& (U\cap V)\times F^{\prime }\\ & & \downarrow p\\ & & U\cap V\end{array}\end{array}$

A vector bundle is a collection of compatible charts for $p:E⟶B$.

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

$\begin{array}{c}\text{if}{b}_0\in B\text{then}\\ \text{there exists a chart}(U,\phi ,F)\text{of}p:E⟶B\\ \text{and a chart}(U^{\prime },\phi \prime ,F^{\prime })\text{of}{p}^{\prime }:E⟶B\\ \text{and a map}\lambda :U⟶{\text{Hom}}_k(F,F^{\prime })\text{such that}\\ f\left(U\right)\subseteq U^{\prime }\text{and}{g}_b\circ t_b=t_{f\left(b\right)}^{\prime }\circ \lambda \left(b\right)\end{array}$

where $g_b=\text{something}$

$\begin{array}{ccc}E& \stackrel{g}{⟶}& E^{\prime }\\ \downarrow & & \downarrow \\ B& \underset{f}{⟶}& B^{\prime }\end{array}$

Let $f:B^{\prime }⟶B$ be a morphism of spcaes and $p:E⟶B$ a vector bundle.

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

$\begin{array}{ccc}\begin{array}{ccc}{f}^{*}\left(E\right)& =& B^{\prime }{\times }_{B}E\\ & & p{r}_1\downarrow \\ & & B^{\prime }\end{array}& \text{where}& B^{\prime }{\times }_{B}E=\{(b^{\prime },e)\mid f\left(b^{\prime }\right)=p\left(e\right)\}\\ \begin{array}{ccc}{f}^{*}\left(B\right)& ⟶& B\\ \downarrow & & \downarrow \\ B^{\prime }& \underset{f}{⟶}& B\end{array}& \text{and}& {\displaystyle f^{*}:\left\{\genfrac{}{}{0ex}{}{\text{vector bundles}}{\text{on}B}\right\}⟶\left\{\genfrac{}{}{0ex}{}{\text{vector bundles}}{\text{on}{B}^{\prime }}\right\}}\end{array}$

is a functor.

Let $p:E⟶B$ 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

$\begin{array}{c}(s_1+s_2)\left(b\right)=s_1\left(b\right)+s_2\left(b\right)\text{and}\\ \left(\phi s\right)\left(b\right)=\phi \left(b\right)s\left(b\right)\end{array}$

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

${\displaystyle \begin{array}{ccc}\left\{\genfrac{}{}{0ex}{}{\text{vector bundles}}{\text{on}B}\right\}& ⟶& \left\{\genfrac{}{}{0ex}{}{\text{locally free sheaves}}{\text{of}{𝒪}_B–\text{modules}}\right\}\\ E& ⟼& {ℱ}_E\end{array}}$

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

$\text{if}b\in B\text{then}{M}_b\simeq F\text{.}$

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

Let

${\displaystyle P=\{(b,u)\mid b\in B\text{and}u:F⟶M_b\text{is a linear isomorphism}\}}$

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

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

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

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

${\displaystyle \begin{array}{ccc}\left\{\genfrac{}{}{0ex}{}{\text{rank}n\text{vector}}{\text{bundles}M\stackrel{p}{⟶}B}\right\}& ⟶& \left\{\genfrac{}{}{0ex}{}{\text{principal}}{G{L}_n\text{something}\text{bundles}}\right\}\\ M& ⟼& P\stackrel{\pi }{⟶}B\\ P{\times }_{G{L}_n}{ℂ}^n& \stackrel{}{↤}& P\end{array}}$

is an equivalence of categories.

The map

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

is a bijection.

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