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 $\{\phi_{i}:\pi^{-1}U_{i}\to F\}$ 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 $e\mapsto(\pi(e),\phi_{i}(e))$ is a homeomorphism for each $i$ ,
2.

for all indices $i, j$ and $e\in\pi^{-1}(U_{i}\cap U_{j})$ , $g_{ji}(\pi(e))\cdot\phi_{i}(e)=\phi_{j}(e)$ 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

$\xymatrix{E\ar[rr]^{\xi}\ar[dr]_{\pi}&&E^{\prime}\ar[dl]^{\pi^{\prime}}\\ &B&}.$

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	2013-03-22 13:07:06
Last modified on	2013-03-22 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