Associated bundle

In mathematics, the theory of fiber bundles with a structure group ${\displaystyle G}$ (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from ${\displaystyle F_{1}}$ to ${\displaystyle F_{2}}$, which are both topological spaces with a group action of ${\displaystyle G}$. 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 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.

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