Fiber bundle construction theorem

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 {Ui} 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 {Ui} with transition functions tij.

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 ti 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.

One starts by taking the disjoint union of the product spaces Ui × 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 (i, x, y) to x.

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.

The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S 1 . When glued trivially (with g UV =1 ) one obtains the trivial bundle, but with the non-trivial gluing of g UV =1 on one overlap and g UV =-1 on the second overlap, one obtains the non-trivial bundle E , the Möbius strip. This can be visualised as a "twisting" of one of the local charts.