Principal G-bundle

Principal G-bundle

A principal $G$-bundle, where $G$ is any topological group, is a fiber bundle $\pi :P\to X$ with a continuous right action $P\times G\to P$ such that $G$ preserve fibers in $P$(i.e. if $y\in P_x$ then $yg\in P_x$ for all $g\in G$) and acts freely($g\cdot x=x$ for some $x\in X⟹g=e$) and transitively($\forall x,y\in X,\exists g\in Gs.t.g\cdot x=y$) (meaning each fiber is a $G$-torsor) on them in such a way that for each $x\in X$ and $y\in P_x$, the map $G\to P_x$ sending $g$ to $yg$ is a homeomorphism.

In particular, each fiber of the bundle is homeomorphic to the group $G$ itself.

Milnor's Construction (Classification Theorem for Principal G-bundle)

For any topological group $G$, there exists a universal principal $G$-bundle $p:EG\to BG$ with $BG:=EG/G$, $EG$ a contractible (or some cases weakly contractible) $G$-space. We call $BG$ the classifying space.