associated bundle construction

Let $G$ be a topological group  , $\pi\colon\thinspace P\to X$ a (right) principal $G$-bundle, $F$ a topological space  and $\rho\colon\thinspace G\to\text{Aut}(F)$ a representation of $G$ as homeomorphisms of $F$. Then the fiber bundle  associated to $P$ by $\rho$, is a fiber bundle $\pi_{\rho}\colon\thinspace P\times_{\rho}F\to X$ 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

 $g\cdot(p,f):=\left(pg^{-1},\rho(g)(f)\right),\quad\forall g\in G,p\in P,F\in F\,.$
• 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 $P\times_{\rho}F$ 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 since the principal action is simply transitive     , given $p\in P$ any orbit of the $G$–action on $P\times F$ contains a unique representative of the form $(p,f)$ for some $f\in F$. It is clear that an open cover that trivializes $P$ trivializes $P\times_{\rho}F$ as well. To see that $P\times_{\rho}F$ has the same transition functions as $P$ notice that transition functions of $P$ act on the left 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 a smooth manifold and $\rho$ maps inside the diffeomorphism group of $F$, the above construction produces a smooth bundle. Also quite often $F$ has extra structure  and $\rho$ maps into the homeomorphisms of $F$ that preserve that structure. In that case the above construction produces a “bundle of such structures.” For example when $F$ is a vector space and $\rho(G)\subset\operatorname{GL}(F)$, i.e. $\rho$ is a linear representation of $G$ we get a vector bundle  ; if $\rho(G)\subset\operatorname{SL}(F)$ we get an oriented vector bundle, etc.

Title associated bundle construction AssociatedBundleConstruction 2013-03-22 13:26:46 2013-03-22 13:26:46 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Definition msc 55R10 associated bundle