# 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. 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. 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. 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