Editing Principal bundle (section)

==Formal definition==

A principal <math>G</math>-bundle, where <math>G</math> denotes any [[topological group]], is a [[fiber bundle]] <math>\pi:P \to X</math> together with a [[continuous (topology)|continuous]] [[Group action (mathematics)|right action]] <math>P \times G \to P</math> such that <math>G</math> preserves the fibers of <math>P</math> (i.e. if <math>y \in P_x</math> then <math>yg \in P_x</math> for all <math>g \in G</math>) and acts [[free action|freely]] and [[transitive action|transitively]] (meaning each fiber is a [[torsor|G-torsor]]) on them in such a way that for each <math>x \in X</math> and <math>y \in P_x</math>, the map <math>G \to P_x</math> sending <math>g</math> to <math>yg</math> is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group <math>G</math> itself. Frequently, one requires the base space <math>X</math> to be [[Hausdorff space|Hausdorff]] and possibly [[paracompact]].

Since the group action preserves the fibers of <math>\pi:P \to X</math> and acts transitively, it follows that the [[orbit (group theory)|orbits]] of the <math>G</math>-action are precisely these fibers and the orbit space <math>P/G</math> is [[homeomorphic]] to the base space <math>X</math>. Because the action is free and transitive, the fibers have the structure of G-torsors. A <math>G</math>-torsor is a space that is homeomorphic to <math>G</math> but lacks a group structure since there is no preferred choice of an [[identity element]].

An equivalent definition of a principal <math>G</math>-bundle is as a <math>G</math>-bundle <math>\pi:P \to X</math> with fiber <math>G</math> where the structure group acts on the fiber by left multiplication. Since right multiplication by <math>G</math> on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by <math>G</math> on <math>P</math>. The fibers of <math>\pi</math> then become right <math>G</math>-torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal <math>G</math>-bundles in the [[category (mathematics)|category]] of [[smooth manifold]]s. Here <math>\pi:P \to X</math> is required to be a [[smooth map]] between smooth manifolds, <math>G</math> is required to be a [[Lie group]], and the corresponding action on <math>P</math> should be smooth.