associated bundle construction

Let $G$ be a topological group, $\pi :P\to X$ a (right) principal $G$-bundle,$F$ a topological space and $\rho :G\to \text{Aut}(F)$ arepresentation of $G$ as homeomorphisms of $F$. Then the fiber bundleassociated to $P$ by $\rho$, is a fiber bundle ${\pi }_{\rho }:P{\times }_{\rho }F\to X$ with fiber $F$ and group $G$ that is defined as follows:

• The total space is defined as

  $$P{\times }_{\rho }F:=P\times F/G$$

  where the (left) action of $G$ on $P\times F$ is defined by

  $$g\cdot (p,f):=(p{g}^{-1},\rho (g)(f)),\forall g\in G,p\in P,F\in F.$$

• The projection ${\pi }_{\rho }$ is defined by

  $${\pi }_{\rho }[p,f]:=\pi (p),$$

  where $[p,f]$ denotes the $G$–orbit of $(p,f)\in P\times F$.

Theorem 1.

The above is well defined and defines a $G$–bundle over $X$ with fiber$F$. Furthermore $P{\mathrm{\times }}_{\rho }F$ has the same transition functions as $P$.

Sketch of proof.

To see that ${\pi }_{\rho }$ is well defined just notice that for $p\in P$ and$g\in G$, $\pi (pg)=\pi (p)$. To see that the fiber is $F$ notice that sincethe principal action is simply transitive, given $p\in P$ any orbit of the$G$–action on $P\times F$ contains a unique representative of theform $(p,f)$ for some $f\in F$. It is clear that an open cover thattrivializes $P$ trivializes $P{\times }_{\rho }F$ as well.To see that $P{\times }_{\rho }F$ has the sametransition functions as $P$ notice that transition functions of $P$ act on theleft and thus commute with the principal $G$–action on $P$.∎

Notice that if $G$ is a Lie group, $P$ a smooth principal bundle and $F$ is asmooth manifold and $\rho$ maps inside the diffeomorphism group of $F$, theabove construction produces a smooth bundle. Also quite often $F$ has extrastructure and $\rho$ maps into the homeomorphisms of $F$ that preserve thatstructure. In that case the above construction produces a “bundle of suchstructures.” For example when $F$ is a vector space and$\rho (G)\subset \mathrm{GL}(F)$, i.e. $\rho$ is a linearrepresentation of $G$ we geta vector bundle; if $\rho (G)\subset \mathrm{SL}(F)$ we get anoriented vector bundle, etc.