Associated bundle

Remarks

The map $\pi_F$ is well defined because $\pi(p g^{-1}) = \pi(p)$. It is a fiber bundle with fiber $F$: a local trivialization $\pi^{-1}(U) \cong U \times G$ of $P$ induces $\pi_F^{-1}(U) \cong U \times F$. The class $[p, f]$ is also written $p \times_G f$; in it, $[p g, f] = [p, g f]$.

Example: let $G = \Z/2\Z$ and $P = S^1 \to S^1$, $z \mapsto z^2$, a principal $G$-bundle where the generator acts by $z \mapsto -z$. Let the generator act on $F = \R$ by $t \mapsto -t$. Then $P \times_G \R$ is the quotient of $S^1 \times \R$ by $(z, t) \sim (-z, -t)$, which is the open Möbius band, a nontrivial line bundle over the circle.

Non-example: the product $P \times F$ itself, with the map $(p, f) \mapsto \pi(p)$, is not the associated bundle when $G$ is nontrivial. Its fibers are copies of $G \times F$, not of $F$, because the quotient by $G$ was not taken.

If $F = \R^n$ with $G = GL_n(\R)$ acting linearly, then $P \times_G \R^n$ is a real vector bundle of rank $n$.