Given a group action G×F->F and a principal bundle π:A->M, the associated fiber bundle on M is π^~ :A×F/G->M. In particular, it is the quotient space A×F/G where (a, x)~(g a, g^(-1) x). For example, the torus T = {(e^(i s), e^(i t))} has a S^1 action given by ϕ(e^(i θ))(e^(i s), e^(i t)) = (e^(i(s + θ)), e^(i(t + θ))) and the frame bundle on the sphere, π:S O(3)->S^2, is a principal S^1 bundle. The associated fiber bundle is a fiber bundle on the sphere, with fiber the torus. It is an example of a four-dimensional manifold.

bundle | fiber bundle | group action | principal bundle | quotient space