Fiber bundle construction theorem explained

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

t _{ij} : U _{i} \cap U _{j} \to G defined on each nonempty overlap, such that the cocycle conditiont _{ik} (x) = t _{ij} (x)t _{jk} (x) \forall x \in U _{i} \cap U _{j} \cap U _{k} 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

t _{i} : U _{i} \to G such thatt' _{ij} (x) = t ^{-1} _{i(x)} t _{ij} (x)t _{j(x)} \forall x \in U _{i} \cap U _{j.} 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

T = \coprod _{i\in} U _{i} x F = \{(i,x,y) : i\in I, x\in U _{i,} y\in F\} and then forms the quotient by the equivalence relation(j,x,y) \sim (i,x,t _{ij} (x) ⋅ y) \forall x\in U _{i} \cap U _{j,} y\in F. 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\phi _{i} : \pi ^{-1} (U _{i)} \to U _{i} x F are then defined by\phi ^{-1} _{i} (x,y) = [(i,x,y)].

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

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