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Associated Fiber Bundle

An associated fiber bundle on M is constructed from a continuous left group action G×F->F and a principal bundle pi:A->M with a right group action of G. It is

pi^~:(A×F)/G->M,
(1)

where pi^~([a,x])=pi(a). The total space is the quotient space (A×F)/G with (a,x)∼(ag,g^(-1)x). The projection is well-defined since pi(ag)=pi(a). Given a local bundle section s:U->A, write a=s(b)g uniquely for b in U. Then [a,x]|->(b,gx) gives a local trivialization with model fiber F. It is independent of the representative because replacing (a,x) by (ah,h^(-1)x) leaves gx unchanged (Mitchell 2011, pp. 4-6).

For example, the torus T={(e^(is),e^(it))} has an S^1 action given by

phi(e^(itheta))(e^(is),e^(it))=(e^(i(s+theta)),e^(i(t+theta)))
(2)

and the oriented orthonormal frame bundle on the sphere,

pi:SO(3)->S^2,
(3)

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.

See also

Bundle, Fiber Bundle, Group Action, Principal Bundle, Quotient Space

Portions of this entry contributed by Todd Rowland

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References

Mitchell, S. A. Notes on Principal Bundles and Classifying Spaces. Jun. 2011. https://sites.math.washington.edu/~mitchell/Quillen/prin.pdf.

Referenced on Wolfram|Alpha

Associated Fiber Bundle

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Associated Fiber Bundle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AssociatedFiberBundle.html

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