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.

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

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.

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