Fiber bundle

A fiber bundle (also known as a fibre bundle by the British), 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 $B\times F$ 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 $U_{x}$ such that ${\pi }^{-1}(U_{x})$ is homeomorphic to $U\times F$ , in such a way that &pi carries over to the projection onto the first factor.

B is called the base space of the fiber bundle and for any $x\in B$ , the preimage of x, $\pi ^{-1}(x)$ 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,

\pi (\rho (g)[e])=\pi (e)

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.

A section of a fiber bundle is a continuous map, $f:B\rightarrow E$ such that $\pi (f(x))=x$ , for x in B. Since bundles do not in general have sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology.

One of the primary applications of fiber bundles is in gauge theory.