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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 function]]s. 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 space]]s and let ''G'' be a [[topological group]] with a [[continuous group action|continuous left action]] on ''F''. Given an [[open cover]] {''U''<sub>''i''</sub>} of ''X'' and a set of [[continuous function (topology)|continuous function]]s | |||
:<math>t_{ij} : U_i \cap U_j \to G\, </math> | |||
defined on each nonempty overlap, such that the ''cocycle condition'' | |||
:<math>t_{ik}(x) = t_{ij}(x)t_{jk}(x) \qquad \forall x \in U_i \cap U_j \cap U_k</math> | |||
holds, there exists a fiber bundle ''E'' → ''X'' with fiber ''F'' and structure group ''G'' that is trivializable over {''U''<sub>''i''</sub>} with transition functions ''t''<sub>''ij''</sub>. | |||
Let ''E''′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions ''t''′<sub>''ij''</sub>. If the action of ''G'' on ''F'' is [[Faithful_group_action#faithful|faithful]], then ''E''′ and ''E'' are isomorphic [[if and only if]] there exist functions | |||
:<math>t_i : U_i \to G\,</math> | |||
such that | |||
:<math>t'_{ij}(x) = t_i(x)^{-1}t_{ij}(x)t_j(x) \qquad \forall x \in U_i \cap U_j.</math> | |||
Taking ''t''<sub>''i''</sub> 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 manifold]]s, ''G'' is a [[Lie group]] with a smooth left action on ''Y'' and the maps ''t''<sub>''ij''</sub> are all smooth. | |||
==Construction== | |||
The proof of the theorem is [[constructive proof|constructive]]. That is, it actually constructs a fiber bundle with the given properties. One starts by taking the [[disjoint union (topology)|disjoint union]] of the [[product space (topology)|product space]]s ''U''<sub>''i''</sub> × ''F'' | |||
:<math>T = \coprod_{i\in I}U_i \times F = \{(i,x,y) : i\in I, x\in U_i, y\in F\}</math> | |||
and then forms the [[quotient topology|quotient]] by the [[equivalence relation]] | |||
:<math>(j,x,y) \sim (i,x,t_{ij}(x)\cdot y)\qquad \forall x\in U_i \cap U_j, y\in F.</math> | |||
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 | |||
:<math>\phi_i : \pi^{-1}(U_i) \to U_i \times F\,</math> | |||
are then defined by | |||
:<math>\phi_i^{-1}(x,y) = [(i,x,y)].</math> | |||
==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'' a with 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== | |||
*{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year = 1997 | isbn = 0-387-94732-9}} | |||
*{{cite book | last = Steenrod | first = Norman | title = The Topology of Fibre Bundles | publisher = Princeton University Press | location = Princeton | year = 1951 | isbn = 0-691-00548-6}} See Part I, §2.10 and §3. | |||
[[Category:Fiber bundles]] | |||
[[Category:Theorems in topology]] | |||
Revision as of 16:58, 20 April 2013
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 {Ui} 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 {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 faithful, then E′ and E are isomorphic if and only if there exist functions
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 smooth manifolds, G is a Lie group with a smooth left action on Y and the maps tij 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 Ui × 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 a with 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
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See Part I, §2.10 and §3.