In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to, which are both topological spaces with a group action of . 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 Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ. One may then construct a fibre bundle F′ as a new fibre bundle having the same transition functions, but possibly a different fibre.
Read more about Associated Bundle: An Example, Construction, Reduction of The Structure Group
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—James Russell Lowell (1819–1891)