Principal bundle

In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G {\displaystyle X\times G} of a topological space X {\displaystyle X} with a group G {\displaystyle G} , but without requiring a product structure. In the same way as with the Cartesian product, a principal bundle P {\displaystyle P} is equipped with An action of G {\displaystyle G} on P {\displaystyle P} , analogous to ( x , g ) h = ( x , g h ) {\displaystyle (x,g)h=(x,gh)} for a product space, where ( x , g ) {\displaystyle (x,g)} is an element of P {\displaystyle P} and h {\displaystyle h} is the group element from G {\displaystyle G} (the group action is conventionally a right action). A projection onto X {\displaystyle X} . For a product space, this is just the projection onto the first factor, ( x , g ) ↦ x {\displaystyle (x,g)\mapsto x} . Unless it is the product space X × G {\displaystyle X\times G} , a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of x ↦ ( x , e ) {\displaystyle x\mapsto (x,e)} . Likewise, there is not generally a projection onto G {\displaystyle G} generalizing the projection onto the second factor, X × G → G {\displaystyle X\times G\to G} that exists for the Cartesian product. It may also have a complicated topology that prevents it from being realized as a product space. An example of a principal bundle is the bundle π : R → S 1 ⊆ C {\displaystyle \pi :\mathbb {R} \to S^{1}\subseteq \mathbb {C} } where π {\displaystyle \pi } is defined by π ( t ) = exp ⁡ ( 2 π...