Associate Bundle

Associate Bundle

Let $\pi :P\to X$ be a principal $G$-bundle. Given a faithful left action $\rho :G\to \text{Homeo}(F)$ of $G$ on a fiber space $F$ (in the smooth category, we should have a smooth action on a smooth manifold), the goal is to construct a $G$-bundle ${\pi }_{\rho }:E\to X$ of the fiber space $F$ over the base space $X$ such that it is associated with $P$. Define a right action of $G$ on $P\times F$ via: $(p,f)\cdot g=(p\cdot g,g^{-1}\cdot f)$ Take the quotient of this action to obtain the space $E=P{\times }_{\rho }F=(P\times F)/G$. Denote the equivalence class of $(p,f)$ by $[p,f]$. Note that: $[p\cdot g,f]=[p,g\cdot f]\text{ for all }g\in G$ Define a projection map ${\pi }_{\rho }:E\to X$ by ${\pi }_{\rho }([p,f])=\pi (p)$. This is well-defined, since $\pi (p)=\pi (p\cdot g)$, i.e. the action of $G$ on $P$ preserves its fibers. Then ${\pi }_{\rho }:E\to X$ is a fiber bundle with fiber $F$ and structure group $G$, where the transition functions are given by $\rho \circ t_{ij}$, where $t_{ij}$ are the transition functions of the principal bundle $P$.

Remark

Intuitively, associate bundle is like “changing fiber from G to F ” following the way of how G act on F.