Online Encyclopedia
Fiber bundle
In mathematics, in particular in 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: if we imagine a small creature living on B with a limited horizon, it will describe the map π as a projection map from a cartesian product onto one of its factors.
In the sequel, we will assume B to be connected. The formal definition then proceeds as follows: there exists a topological space F such that for any x in B, there is a neighborhood U_{x} of x such that π^{-1}(U_{x}) is homeomorphic to the product space U_{x} x F, in such a way that π carries over to the projection onto the first factor. (Completely formally: if p : U_{x} x F → U_{x} is the natural projection and h : π^{-1}(U_{x}) → U_{x} x F is the homeomorphism, we require that p o h = π restricted to π^{-1}(U_{x}).)
B is called the base space of the fiber bundle and E the total space. The map π is called the projection map. For any x in B, the preimage of x, π^{-1}(x) (which is homeomorphic to F) is called the fiber at x. The homeomorphism between π^{-1}(U_{x}) and U_{x} x F is a local trivialization.
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Examples
Every natural projection map p : B x F → B is a fiber bundle, but this example is somewhat besides the point. Bundles like these are called trivial bundles.
A standard example is the Möbius strip as E, B a circle and F a line segment. The 'twisting' in the band is only apparent globally, while locally the ribbon structure defines the topology.
Every vector bundle is a fiber bundle; here 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.
Every covering map is a fiber bundle; here the fiber space F is discrete.
Facts
Every fiber bundle π : E → B is an open map, since projections of cartesian products are open maps.
Sections
A section of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all 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.
Structure groups and principal bundles
Sometimes, there exists a topological group G acting on E, such that
for every g in G and every e in E. (The condition states that every G-orbit lies within a single fiber.) If furthermore the matching conditions between local trivializable neighborhoods are equivariant maps, we speak of a G-bundle.
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.
Furthermore, in addition to the above, if the total space E is contractible, then, we say that the above principal bundle is a universal principal bundle. Once it exists, it is unique up to homeomorphism.
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.
Applications
One of the primary applications of fiber bundles is in gauge theory.
See also
External links
- PlanetMath: Fiber Bundle http://planetmath.org/encyclopedia/FiberBundle.html
- MathWorld: Fiber Bundle http://mathworld.wolfram.com/FiberBundle.html