Fiber Bundle Examples Facts Sections Structure Groups And

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.

1 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 mapTopology Algebraic topology Homotopy theory In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X with C and X being topological spaces, which has the following property: :to every x in X there exists an open is a fiber bundle; here the fiber space F is discreteTopology General topology In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense. Definitions.

2 Facts

Every fiber bundle π : E → B is an open map, since projections of cartesian products are open maps.

3 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 topologyTopology Algebraic topology Abstract algebra Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further ca.

4 Structure groups and principal bundles

Sometimes, there exists a topological groupIn mathematics, a topological group ''G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the produ G actingIn mathematics, groups are often used to describe symmetries of objects. This is formalized by the notion of a group action every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a transf 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.

5 Applications

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

6 See also

7 External link

Differential topology Algebraic topology Homotopy theory

Bundles (mathematics)

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