Associated bundle
In mathematics, the theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F 1 {\displaystyle F_{1}} to F 2 {\displaystyle F_{2}} , which are both topological spaces with a group action of G {\displaystyle G} . For a fiber bundle F {\displaystyle F} with structure group G {\displaystyle G} , the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems U α {\displaystyle U_{\alpha }} and U β {\displaystyle U_{\beta }} are given as a G {\displaystyle G} -valued function g α β {\displaystyle g_{\alpha \beta }} on U α ∩ U β {\displaystyle U_{\alpha }\cap U_{\beta }} .