# Principal bundle

Principal bundles have important applications in topology and differential geometry and mathematical gauge theory. They have also found application in physics where they form part of the foundational framework of physical gauge theories.

One of the most important questions regarding any fiber bundle is whether or not it is trivial, *i.e.* isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:

*A principal bundle is trivial if and only if it admits a global section.*

The same is not true for other fiber bundles. For instance, vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.

The same fact applies to local trivializations of principal bundles. Let *π* : *P* → *X* be a principal *G*-bundle. An open set *U* in *X* admits a local trivialization if and only if there exists a local section on *U*. Given a local trivialization

where *e* is the identity in *G*. Conversely, given a section *s* one defines a trivialization Φ by

The simple transitivity of the *G* action on the fibers of *P* guarantees that this map is a bijection, it is also a homeomorphism. The local trivializations defined by local sections are *G*-equivariant in the following sense. If we write

Equivariant trivializations therefore preserve the *G*-torsor structure of the fibers. In terms of the associated local section *s* the map *φ* is given by

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

By gluing the local trivializations together using these transition functions, one may reconstruct the original principal bundle. This is an example of the fiber bundle construction theorem.
For any *x* ∈ *U*_{i} ∩ *U*_{j} we have

Any topological group *G* admits a **classifying space** *BG*: the quotient by the action of *G* of some weakly contractible space *EG*, *i.e.* a topological space with vanishing homotopy groups. The classifying space has the property that any *G* principal bundle over a paracompact manifold *B* is isomorphic to a pullback of the principal bundle *EG* → *BG*.^{[5]} In fact, more is true, as the set of isomorphism classes of principal *G* bundles over the base *B* identifies with the set of homotopy classes of maps *B* → *BG*.