Fiber bundle construction theorem In mathematics , the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space , fiber and a suitable set of transition functions . The theorem also gives conditions under which two such bundles are isomorphic . The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same.Formal statement 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 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 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 Taking t i to be constant functions to the identity in G , we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic. A similar theorem holds 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 ij are all smooth.Construction The proof of the theorem is constructive . That is, it actually constructs a fiber bundle with the given properties. One starts by taking the disjoint union of the product spaces U i × F and then forms the quotient by the equivalence relation The total space E of the bundle is T /~ and the projection π : E → X is the map which sends the equivalence class of to x . The local trivializations are then defined byAssociated bundle Let E → 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 ′ → 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 .
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