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fiber bundle (Definition)

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:

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'$ over the same base $ B$ with the same structure group $ G$. Such a morphism is a $ G$-equivariant map $ \xi:E\to E'$, making the following diagram commute

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

Thus we have a category of fiber bundles over a fixed base with fixed structure group.



"fiber bundle" is owned by bwebste. [ full author list (2) | owner history (1) ]
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See Also: reduction of structure group, section of a fiber bundle, fibration

Other names:  fibre bundle
Also defines:  trivial bundle, local trivializations, structure group, cocycle condition, local trivialization

Attachments:
reduction of structure group (Definition) by antonio

Cross-references: fixed, category, morphism, covering spaces, vector bundles, factor, homeomorphic, identity, cohomology, homeomorphism, properties, satisfy, transition functions, collection, open cover, projection, map, surjective, continuous, base, fiber, topological group, topological space
There are 19 references to this object.

This is version 7 of fiber bundle, born on 2002-10-31, modified 2003-06-24.
Object id is 3551, canonical name is FiberBundle.
Accessed 7802 times total.

Classification:
AMS MSC55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

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