Fiber bundle construction theorem
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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 {Ui} of X and a set of continuous functions
t i j : U i ∩ U j → G {\displaystyle t_{ij}:U_{i}\cap U_{j}\to G}
defined on each nonempty overlap, such that the cocycle condition
t i k ( x ) = t i j ( x ) t j k ( x ) ∀ x ∈ U i ∩ U j ∩ U k {\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 {Ui} with transition functions tij.
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
t i : U i → G {\displaystyle t_{i}:U_{i}\to G}
such that
t i j ′ ( x ) = t i ( x ) − 1 t i j ( x ) t j ( x ) ∀ x ∈ U i ∩ U j . {\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 t i ( x ) = 1 G {\displaystyle t_{i}(x)=1_{G}}, where 1 G ∈ G {\displaystyle 1_{G}\in G} is the identity element of G {\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 tij are all smooth.
Construction
Existence is proven constructively by the standard coequalizer construction in category theory.
Take the disjoint union of the product spaces U i × F {\displaystyle U_{i}\times F}
T = ∐ i ∈ I U i × F = { ( i , x , y ) : i ∈ I , x ∈ U i , y ∈ 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
( j , x , y ) ∼ ( i , x , t i j ( x ) ⋅ y ) ∀ x ∈ U i ∩ U j , y ∈ F . {\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 E := T / ∼ {\displaystyle E:=T/\sim }, with the projection map π : E → X , π ( [ ( i , x , y ) ] ) = x {\displaystyle \pi :E\to X,\quad \pi ([(i,x,y)])=x}The local trivializations are
ϕ i : π − 1 ( U i ) → U i × F , ϕ i − 1 ( x , y ) = [ ( i , x , y ) ] . {\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.