G-bundle - Knowlpedia

Let $G$ be a topological group with a continuous left action on a nonempty space $F$ . A $G$ -bundle with fiber $F$ is a fiber bundle $\pi:E\to B$ equipped with local trivializations $\Phi_i$ over an open cover $\{U_i\}$ and continuous maps $g_{ij}:U_i\cap U_j\to G$ such that

\Phi_i\Phi_j^{-1}(b,f)=(b,g_{ij}(b)\cdot f), \qquad g_{ii}=e,\qquad g_{ij}g_{jk}=g_{ik}

on the respective overlaps. The last identity is required on every triple overlap. Here $G$ is the structure group. The chosen data are considered up to refinement of the cover and changes of trivializations by continuous maps $h_i:U_i\to G$ ; replacing $\Phi_i$ by $h_i\Phi_i$ replaces $g_{ij}$ by $h_i g_{ij}h_j^{-1}$ .

Smooth version and examples

For a smooth fiber bundle, a Lie group acting smoothly on $F$ , and smooth maps $g_{ij}$ and $h_i$ , this defines a smooth $G$ -bundle. A rank- $n$ vector bundle has structure group $\mathrm{GL}(n,\mathbb K)$ acting on $\mathbb K^n$ .

Principal bundles and conventions

Taking $F=G$ with left multiplication gives a principal bundle: right multiplication on each local copy of $G$ commutes with the transition maps and therefore defines a global right action. Its smooth version is a principal G-bundle. For general $F$ , a $G$ -bundle need not carry a global $G$ -action on its total space.

Terminology. In gauge theory and algebraic geometry, “ $G$ -bundle” often abbreviates “principal $G$ -bundle.” This knowl uses the broader structure-group meaning; principal-bundle statements link to their explicit definition.

If the action on $F$ is not faithful, the maps $g_{ij}$ and their cocycle identities are extra data: the underlying changes of fiber coordinates do not determine them uniquely.

References

Fiber Bundles lecture notes, §1.2, pp. 6–7: fiber bundles, G-atlases, equivalent atlases, and the non-effective-action convention.