associated bundle について

. 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 ''U''_α and ''U''_β are given as a ''G''-valued function ''g''_αβ on ''U''_α∩''U''_β. One may then construct a fibre bundle ''F''′ as a new fibre bundle having the same transition functions, but possibly a different fibre.
==An example==

A simple case comes with the Möbius strip, for which

G

is the cyclic group of order 2,

\mathbb_2

. We can take as

F

any of: the real number line

\mathbb

, the interval

(1)

, the real number line less the point 0, or the two-point set

\

. 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

) \times I

and

) \times J

together: what we really need is the data to identify

(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

\

as for

(1)

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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