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 fiber bundle ${\displaystyle F}$ with structure group ${\displaystyle G}$, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems ${\displaystyle U_{\alpha }}$ and ${\displaystyle U_{\beta }}$ are given as a ${\displaystyle G}$-valued function ${\displaystyle g_{\alpha \beta }}$ on ${\displaystyle U_{\alpha }\cap U_{\beta }}$. One may then construct a fiber bundle ${\displaystyle F'}$ as a new fiber bundle having the same transition functions, but possibly a different fiber.