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  "body": "\n## Definition\n\nLet $G$ be a topological group and let $\\pi \\colon P \\to B$ be a [[principal $G$-bundle|8E37E74XVP]], with $G$ acting on $P$ from the right.\nLet $F$ be a topological space with a continuous left [[action|0WCY42EV9V]] of $G$.\nOn $P \\times F$, with the [[product topology|0GBG8B49AA]], let $G$ act by $g \\cdot (p, f) = (p g^{-1}, g f)$.\nThe *associated bundle* with fiber $F$ is the [[quotient space|0BR3FY5Q8M]]\n$$P \\times_G F = (P \\times F) / G,$$\ntogether with the map $\\pi_F \\colon P \\times_G F \\to B$, $\\pi_F([p, f]) = \\pi(p)$.\n\n## Remarks\n\nThe map $\\pi_F$ is well defined because $\\pi(p g^{-1}) = \\pi(p)$.\nIt is a fiber bundle with fiber $F$: a local trivialization $\\pi^{-1}(U) \\cong U \\times G$ of $P$ induces $\\pi_F^{-1}(U) \\cong U \\times F$.\nThe class $[p, f]$ is also written $p \\times_G f$; in it, $[p g, f] = [p, g f]$.\n\nExample: 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 $z \\mapsto -z$.\nLet the generator act on $F = \\R$ by $t \\mapsto -t$.\nThen $P \\times_G \\R$ 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.\n\nNon-example: the product $P \\times F$ itself, with the map $(p, f) \\mapsto \\pi(p)$, is not the associated bundle when $G$ is nontrivial.\nIts fibers are copies of $G \\times F$, not of $F$, because the quotient by $G$ was not taken.\n\nIf $F = \\R^n$ with $G = GL_n(\\R)$ acting linearly, then $P \\times_G \\R^n$ is a real vector bundle of rank $n$.\n",
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