in mathematics, the theory of fiber bundles with a structure group g (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from f_1 to f_2, which are both topological spaces with a group action of g. for a fibre bundle f with structure group g, the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems uÎ± and uÎ² are given as a g-valued function gÎ±Î² on uÎ±â©uÎ². one may then construct a fibre bundle fâ² as a new fibre bundle having the same transition functions, but possibly a different fibre. ==an example== a simple case comes with the mÃ¶bius strip, for which g is the cyclic group of order 2, \mathbb{z}_2. we can take as f any of: the real number line \mathbb{r}, the interval [-1,\ 1], the real number line less the point 0, or the two-point set \{-1,\ 1\}. the action of g on these (the non-identity element acting as x\ \rightarrow\ -x in each case) is comparable, in an intuitive sense. we could say that more formally in terms of gluing two rectangles [-1,\ 1] \times i and [-1,\ 1] \times j together: what we really need is the data to identify [-1,\ 1] to itself directly at one end, and with the twist over at the other end. this data can be written down as a patching function, with values in g. the associated bundle construction is just the observation that this data does just as well for \{-1,\ 1\} as for [-1,\ 1]. ==construction== in general it is enough to explain the transition from a bundle with fiber f, on which g acts, to the associated principal bundle (namely the bundle where the fiber is g, considered to act by translation on itself). for then we can go from f_1 to f_2, 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.all of these constructions are due to ehresmann (1941-3). attributed by steenrod (1951) page 36 ===associated bundles in general=== let &pi; : e &rarr; x be a fibre bundle over a topological space x with structure group g and typical fibre f. by definition, there is a left action of g (as a transformation group) on the fibre f. suppose furthermore that this action is [[group action#types of actions|effective]
Algebraic_topology Differential_geometry Differential_topology Fiber_bundles