In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundles with a structure group from a given base space, fiber, group, and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic.
The theorem is used in the associated bundle construction, where one starts with a given bundle and changes just the fiber, while keeping all other data the same.
Let X and F be topological spaces and let G be a topological group with a continuous left action on F. Given an open cover {U<sub>i</sub>} of X and a set of continuous functions
defined on each nonempty overlap, such that the cocycle condition
holds, there exists a fiber bundle E â X with fiber F and structure group G that is trivializable over {U<sub>i</sub>} with transition functions t<sub>ij</sub>.
Let E′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions t′<sub>ij</sub>. If the action of G on F is faithful, then E′ and E are isomorphic if and only if there exist functions
such that
i.e. a gauge transformation on transition data.
In particular, given a base, fiber, structure group, group action on the fiber, trivializing neighborhoods, and a set of transition functions, if the action is faithful, then any two fiber bundles constructed are isomorphic. To see it, use the "if" direction of the isomorphism theorem with , where is the identity element of . In other words, the construction is unique up to isomorphism.
The above pair of theorems hold in the topological category. A similar pair of theorems hold in the smooth category, where X and Y are smooth manifolds, G is a Lie group with a smooth left action on Y and the maps t<sub>ij</sub> are all smooth.
Existence is proven constructively by the standard coequalizer construction in category theory.
Take the disjoint union of the product spaces
Define the equivalence relation
Take the quotient , with the projection map The local trivializations are
Let E â X a fiber bundle with fiber F and structure group G, and let F′ be another left G-space. One can form an associated bundle E′ â X with a fiber F′ and structure group G by taking any local trivialization of E and replacing F by F′ in the construction theorem. If one takes F′ to be G with the action of left multiplication then one obtains the associated principal bundle.