associated bundle construction

Let G be a topological groupMathworldPlanetmath, π:PX a (right) principal G-bundle, F a topological spaceMathworldPlanetmath and ρ:GAut(F) a representation of G as homeomorphisms of F. Then the fiber bundleMathworldPlanetmath associated to P by ρ, is a fiber bundle πρ:P×ρFX with fiber F and group G that is defined as follows:

  • The total space is defined as


    where the (left) action of G on P×F is defined by

  • The projection πρ is defined by


    where [p,f] denotes the G–orbit of (p,f)P×F.

Theorem 1.

The above is well defined and defines a G–bundle over X with fiber F. Furthermore P×ρF has the same transition functionsMathworldPlanetmathPlanetmath as P.

Sketch of proof.

To see that πρ is well defined just notice that for pP and gG, π(pg)=π(p). To see that the fiber is F notice that since the principal action is simply transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath, given pP any orbit of the G–action on P×F contains a unique representative of the form (p,f) for some fF. It is clear that an open cover that trivializes P trivializes P×ρF as well. To see that P×ρF has the same transition functions as P notice that transition functions of P act on the left and thus commute with the principal G–action on P. ∎

Notice that if G is a Lie groupMathworldPlanetmath, P a smooth principal bundleMathworldPlanetmath and F is a smooth manifold and ρ maps inside the diffeomorphism group of F, the above construction produces a smooth bundle. Also quite often F has extra structureMathworldPlanetmath and ρ maps into the homeomorphisms of F that preserve that structure. In that case the above construction produces a “bundle of such structures.” For example when F is a vector space and ρ(G)GL(F), i.e. ρ is a linear representation of G we get a vector bundleMathworldPlanetmath; if ρ(G)SL(F) we get an oriented vector bundle, etc.

Title associated bundle construction
Canonical name AssociatedBundleConstruction
Date of creation 2013-03-22 13:26:46
Last modified on 2013-03-22 13:26:46
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Definition
Classification msc 55R10
Defines associated bundle