Fiber bundle construction theorem

The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S¹. 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.

In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundles with a structure group from a given base space, fiber, group, and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic.

The theorem is used in the associated bundle construction, where one starts with a given bundle and changes just the fiber, while keeping all other data the same.

Formal statement

Existence

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

${\displaystyle t_{ij}:U_{i}\cap U_{j}\to G}$

defined on each nonempty overlap, such that the cocycle condition

${\displaystyle t_{ik}(x)=t_{ij}(x)t_{jk}(x)\qquad \forall x\in U_{i}\cap U_{j}\cap U_{k}}$

holds, there exists a fiber bundle E → X with fiber F and structure group G that is trivializable over {U_i} with transition functions t_ij.

Isomorphism

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

${\displaystyle t_{i}:U_{i}\to G}$

such that

${\displaystyle t'_{ij}(x)=t_{i}(x)^{-1}t_{ij}(x)t_{j}(x)\qquad \forall x\in U_{i}\cap U_{j}.}$

i.e. a gauge transformation on transition data.

In particular, given a base, fiber, structure group, group action on the fiber, trivializing neighborhoods, and a set of transition functions, if the action is faithful, then any two fiber bundles constructed are isomorphic. To see it, use the "if" direction of the isomorphism theorem with ${\displaystyle t_{i}(x)=1_{G}}$ , where ${\displaystyle 1_{G}\in G}$ is the identity element of ${\displaystyle G}$ . In other words, the construction is unique up to isomorphism.

Smooth category

The above pair of theorems hold in the topological category. A similar pair of theorems hold 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

Existence is proven constructively by the standard coequalizer construction in category theory.

Take the disjoint union of the product spaces ${\displaystyle U_{i}\times F}$

${\displaystyle T=\coprod _{i\in I}U_{i}\times F=\{(i,x,y):i\in I,x\in U_{i},y\in F\}.}$

Define the equivalence relation

${\displaystyle (j,x,y)\sim (i,x,t_{ij}(x)\cdot y)\qquad \forall x\in U_{i}\cap U_{j},y\in F.}$

Take the quotient ${\displaystyle E:=T/\sim }$ , with the projection map $${\displaystyle \pi :E\to X,\quad \pi ([(i,x,y)])=x}$$ The local trivializations are

${\displaystyle \phi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times F,\quad \phi _{i}^{-1}(x,y)=[(i,x,y)].}$

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

• 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.