Fiber bundle construction theorem
In
Formal statement
Let X and F be
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.
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
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.
A similar theorem holds in the smooth category, where X and Y are
Construction
The proof of the theorem is
and then forms the
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.