fiber bundle
Let $F$ be a topological space^{} and $G$ be a topological group^{} which acts on $F$ on the left. A fiber bundle^{} with fiber $F$ and structure group $G$ consists of the following data:

•
a topological space $B$ called the base space, a space $E$ called the total space and a continuous^{} surjective map $\pi :E\to B$ called the projection of the bundle,

•
an open cover $\{{U}_{i}\}$ of $B$ along with a collection^{} of continuous maps $\{{\varphi}_{i}:{\pi}^{1}{U}_{i}\to F\}$ called local trivializations and

•
a collection of continuous maps $\{{g}_{ij}:{U}_{i}\cap {U}_{j}\to G\}$ called transition functions^{}
which satisfy the following properties

1.
the map ${\pi}^{1}{U}_{i}\to {U}_{i}\times F$ given by $e\mapsto (\pi (e),{\varphi}_{i}(e))$ is a homeomorphism for each $i$,

2.
for all indices $i,j$ and $e\in {\pi}^{1}({U}_{i}\cap {U}_{j})$, ${g}_{ji}(\pi (e))\cdot {\varphi}_{i}(e)={\varphi}_{j}(e)$ and

3.
for all indices $i,j,k$ and $b\in {U}_{i}\cap {U}_{j}\cap {U}_{k}$, ${g}_{ij}(b){g}_{jk}(b)={g}_{ik}(b)$.
Readers familiar with Čech cohomology may recognize condition 3), it is often called the cocycle condition. Note, this imples that ${g}_{ii}(b)$ is the identity in $G$ for each $b$, and ${g}_{ij}(b)={g}_{ji}{(b)}^{1}$.
If the total space $E$ is homeomorphic to the product^{} $B\times F$ so that the bundle projection is essentially projection onto the first factor, then $\pi :E\to B$ is called a trivial bundle. Some examples of fiber bundles are vector bundles^{} and covering spaces.
There is a notion of morphism of fiber bundles $E,{E}^{\prime}$ over the same base $B$ with the same structure group $G$. Such a morphism is a $G$equivariant map $\xi :E\to {E}^{\prime}$, making the following diagram commute
$$\text{xymatrix}E\text{ar}{[rr]}^{\xi}\text{ar}{[dr]}_{\pi}\mathrm{\&}\mathrm{\&}{E}^{\prime}\text{ar}{[dl]}^{{\pi}^{\prime}}\mathrm{\&}B\mathrm{\&}.$$ 
Thus we have a category of fiber bundles over a fixed base with fixed structure group.
Title  fiber bundle 
Canonical name  FiberBundle 
Date of creation  20130322 13:07:06 
Last modified on  20130322 13:07:06 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  10 
Author  bwebste (988) 
Entry type  Definition 
Classification  msc 55R10 
Synonym  fibre bundle 
Related topic  ReductionOfStructureGroup 
Related topic  SectionOfAFiberBundle 
Related topic  Fibration^{} 
Related topic  Fibration2 
Related topic  HomotopyLiftingProperty 
Related topic  SurfaceBundleOverTheCircle 
Defines  trivial bundle 
Defines  local trivializations 
Defines  structure group 
Defines  cocycle condition 
Defines  local trivialization 