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 {*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.

## 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 (*i*, *x*, *y*) to *x*. The local trivializations

are then defined by

## Associated 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.

## References

- Sharpe, R. W. (1997).
*Differential Geometry: Cartan's Generalization of Klein's Erlangen Program*. New York: Springer. ISBN 0-387-94732-9. - Steenrod, Norman (1951).
*The Topology of Fibre Bundles*. Princeton: Princeton University Press. ISBN 0-691-00548-6. See Part I, §2.10 and §3.