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# Fiber bundle

In mathematics, in particular topology, a fiber bundle is a continuous surjective map, π from a topological space E to another topological space B, satisfying a further condition making it locally of a particularly simple form. Putting it in intuitive terms, by locally here is meant locally on B; that is, if we imagine a small creature living on B and describing E, as mapped to B, only within a limited horizon on B, π has the description of a projection map inside a cartesian product. Introducing a further topological space F, we use the projection map from to B as the model. For example in the case of a vector bundle, F is a vector space over the real numbers. To qualify as a vector bundle, the matching conditions between local trivializable neighborhoods would have to be linear as well.

Saying it more formally, for any x in B, there is a neighborhood such that is homeomorphic to , in such a way that π carries over to the projection onto the first factor.

B is called the base space of the fiber bundle and for any , the preimage of x, is called the fiber at x and the map π is called the projection map.

A standard example is a Möbius strip as E, in which B can be taken as a circle and F a line segment. The 'twisting' in the band is only apparent globally, while locally the ribbon structure defines the topology.

### Structure groups

Sometimes, there exists a topological group G of transformations of E, such that if ρ denotes the action,

for g in G and e in E. The condition states that every G-orbit lies within a single fiber.

In that case, G is called the structure group of the fiber bundle. To qualify as a G-bundle, the matching conditions between local trivializable neighborhoods would have to be intertwiners of G-actions as well.

If, in addition, G acts freely, transitively and continuously upon each fiber, then we call the fiber bundle a principal bundle. An example of a principal bundle that occurs naturally in geometry is the bundle of all bases for the tangent space to a manifold, with G the general linear group; restricting in Riemannian geometry to orthonormal bases, one would limit G to the orthogonal group. See vierbein for more details.

Making G explicit is essential for the operations of creating an associated bundle, and making precise the reduction of the structure group of a bundle.