Construction
In general it is enough to explain the transition from a bundle with fiber, on which acts, to the associated principal bundle (namely the bundle where the fiber is, considered to act by translation on itself). For then we can go from to, via the principal bundle. Details in terms of data for an open covering are given as a case of descent.
This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.
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