Let$F$ be atopological space and$G$ be atopological group which acts on$F$ on the left. Afiber bundle with fiber$F$ andstructure group$G$ consists of the following data:

• •

  a topological space$B$ called thebase space, a space$E$ called thetotal space and acontinuous surjective map$\pi :E\to B$ called the projection of the bundle,

• •

  anopen cover$\{U_i\}$ of$B$ along with acollection of continuous maps$\{{\varphi }_i:{\pi }^{-1}{U}_i\to F\}$ calledlocal trivializations and

• •

  a collection of continuous maps$\{g_{ij}:U_i\cap U_j\to G\}$ calledtransition functions

whichsatisfy the followingproperties

1. 1.

   the map${\pi }^{-1}{U}_i\to U_i\times F$ given by$e\mapsto (\pi (e),{\varphi }_i(e))$ is ahomeomorphism for each$i$,

2. 2.

   for all indices$i,j$ and$e\in {\pi }^{-1}(U_i\cap U_j)$,$g_{ji}(\pi (e))\cdot {\varphi }_i(e)={\varphi }_j(e)$ 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 thecocycle condition. Note, this imples that$g_{ii}(b)$ is theidentity in$G$ for each$b$, and$g_{ij}(b)=g_{ji}{(b)}^{-1}$.

If the total space$E$ is homeomorphic to theproduct$B\times F$ so that thebundle projection is essentially projection onto the first factor, then$\pi :E\to B$ is called atrivial bundle. Some examples of fiber bundles arevector bundles and covering spaces.

There is a notion ofmorphism 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

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