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_i} 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 → X with fiber F and structure group G that is trivializable over {U_i} with transition functions t_ij.

Let E′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions t′_ij. 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.