Fibre bundles

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 9 September 2012

Bourbaki, §6 Varietes Differentielles et Analytiques

A bundle or fibre bundle, is a morphism p:EB such that if bB then there exists

  1. an open neighbourhood U of b,
  2. a variety F
  3. an isomorphism φ:p-1(U) U×F

such that

ifxU andyF then p ( φ-1 (x,y) ) =x.

U×F p-1(U) pr1 p U B

A morphism of bundles from p:EB to p: EB is a pair of morphisms f:BB and g:EE such that pg= fp .

E g E p p B f B

The trivial bundle with base B and fibre F is pr1:B×FB given by pr1(b,f)=b.

A section of p:EB is s:BE such that ps=idB.

A principal G–bundle is a morphism p:EB where P is a variety with a right G–action and if bB then there exists

  1. an open neighborhood U of b,
  2. an isomorphism ψ:U×Gp-1(U)

such that

ifuUand g,gG then p(ψ(u,g))=u and ψ(u,gg)= ψ(u,g)something

U×G ψ p-1(U) pr1 p U B

A morphism from a principal G–bundle p:EB to a principal G–bundle p:E B is a triple (f,φ,h) with

f:ER, h:BB, φ:GG

such that

hφ=φf and if xE,gG then f(xg)=f(x)φ(g) .

E f E p p B h B

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 (γuv) u,vU ,

γuv: UVG

such that

ifxUVW then γuw=γuv(x) γvw(x).

Two cocycles are cohomologous (γuv) and (γuv) if there exists a collection of morphisms (hu)u𝒰


such that

ifxUVthen γuv(x)= hu(x)-1 γuv(x)hv(x) .

Let p:EB be a principal G–bundle.

A trivialization is an isomorphism p:EB to B×Gpr1B

s E f B×G p B idB B

The map

{ sections of p:EB } { trivializations of p:EB } s fs

given by

fs-1(b,g) =s(b)g,for bB,gG,

is a bijection.

Let p:EB be a principal G–bundle. Let (su) u𝒰 be a family if it sections over U𝒰. Then

sv(b)= su(b) guv(b), forbUV

Let fu:p-1 (U)U×G be the trivialisation corresponding to (su).


fu(x)= guv(p(x)) fv(x) for xp-1 (YV).

Let φ:EE be an isomorphism of principal G–bundles p:EB to p:EB. Let (su) u𝒰 be a family of sections of p, and (su) u𝒰 a family of sections of p. Then

φ(su(x))= su(x) hu(x), foru𝒰 andxU.


Let F be a G–variety.

Then there is a map

{ principalGbundles p:EB } { bundles onB with fibreF } E E×GF = E×F (xg,f)= (x,gf) (e,f) p B B p(b)

Vector Bundles

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

A chart of p is a triple (U,φ,F) where

  1. U is an open set of B
  2. F is a vector space
  3. φ:p(U) U×F is a bijection

such that

p ( φ-1 (b,h) ) =b,for bBand hF.

Two charts φ:p-1 (U) U×F and ψ:p-1 (V) V×F are compatible is there exists

λ:UV Homk(F,F) such that tb=tb· λ(b),

for bUV, where

tb : F p-1(b) h φ-1(b,h) (UV)×F φ p-1(UV) ψ (UV)×F p UV

A vector bundle is a collection of compatible charts for p:EB.

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

ifb0Bthen there exists a chart(U,φ,F) ofp:EB and a chart (U,φ,F) of p:EB and a map λ:U Homk(F,F) such that f(U)Uand gbtb= tf(b) λ(b)

where gb=something

E g E B f B

Let f:BB be a morphism of spcaes and p:EB a vector bundle.

The pullback f*E of E by f is

f*(E) = B×BE pr1 B where B×BE= { (b,e) f(b)=p (e) } f*(B) B B f B and f*: { vector bundles onB } { vector bundles onB }

is a functor.

Let p:EB be a vector bundle on B.

Let U be an open set of B.

The space of sections E(U) over U is an 𝒪B module with operations

(s1+s2)(b) =s1(b)+s2 (b)and (φs)(b)=φ (b)s(b)

for s,s1,s2(U) and φ𝒪B. Then E is the sheaf of sections of E. The functor

{ vector bundles onB } { locally free sheaves of𝒪Bmodules } E E

is an equivalence of categories.

Bourbaki, Varietes Differentielles et Analytiques §7.10

A vector bundle MpB is pure of type F if MpB satisfies

ifbBthen MbF.

Let MpB be pure vector bundle of type F.


P= { (b,u) bBand u:FMb is a linear isomorphism }

with GL(F) action given by

(b,u)g= (b,ug)

and PπB given by π(b,u)=b.

Then PπB is a principal GL(F) bundle and

{ ranknvector bundles MpB } { principal GLnsomethingbundles } M PπB P×GLn n P

is an equivalence of categories.

The map

{ chartst= (U,φ,F) φ:p-1 (U) r U×F of MpB } { sectionss:B P ofP πB } t s:BP b(b,tb)

is a bijection.

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