---- > [!definition] Definition. ([[associated vector bundle and principal bundle]]) > Let $V$ be a [[vector space]] of [[dimension]] $k$. Let $G$ be a [[Lie group]] with [[faithful group representation|faithful]] [[group representation|representation]] $G \hookrightarrow \text{GL}(V)$[[general linear group|.]] Let $B$ be a "base [[smooth manifold|manifold]]" with [[cover|open cover]] $B=\bigcup_{\alpha}U_{\alpha}$ and, for all $\alpha, \beta$, [[smooth maps between manifolds|smooth]] $\psi_{\beta \alpha}:U_{\alpha} \cap U_{\beta} \to G$ satisfying the [[cocycle conditions]]. > > From these premises, we may construct two objects: > 1. [[the Steenrod construction of a vector bundle over a smooth manifold|A]] rank-$k$ [[vector bundle]] $E \xrightarrow{ \pi}B$ with trivializing cover $\{ U_{\alpha} \}$, [[transition functions for a vector bundle over a smooth manifold|transition functions]] $\psi_{\beta \alpha}$, [[structure group of a real vector bundle over a smooth manifold|structure group]] $G \hookrightarrow \text{GL}(V)$, and typical fiber $V$; > 2. A [[principal bundle over a smooth manifold|principal]] $G$-[[principal bundle over a smooth manifold|bundle]] $P \xrightarrow{ \pi} B$ with trivializing cover $\{ U_{\alpha} \}$ and [[transition functions for a vector bundle over a smooth manifold|transition functions]] $\psi_{\beta \alpha}$. > > $E$ and $P$ have the same transition functions $\psi_{\beta \alpha}:U_{\alpha} \cap U_{\beta} \to G$. For $E$, we think of them as acting on the typical fiber $V \cong \mathbb{R}^{k}$ via matrix-vector multiplication. For $P$, we think of them as acting on $G$ via left [[topological group|translation]]. > Given a principal $G$-bundle $P$ and representation $G \curvearrowright V$, the associated vector bundle is sometimes denoted $E=P \times_{G} V$. (Not to be confused with the [[categorical pullback|fiber product]] notation.) Concretely, $P \times _{G} V= (P \times V) / {\sim}, \text{ where } (pg, v) \sim (p, g\cdot v)$ > where the bundle [[projection]] is [[quotient set|descended]] from $P\times V \to P \to B$ (well-defined because the fibers of $P$ are $G$-[[orbit|orbits]]). > > [[section|Sections]] of an associated vector bundle $P \times_{G} V$ are equivalent to [[group-equivariant map|equivariant functions]] $P \to V$, in the sense that[^1] $\Gamma(E) \cong \{ f :P \to V \text{ s.t. } f(pg)= \rho(g ^{-1})f(p) \}.$ [^1]: Indeed, given $s \in \Gamma(E)$, define $f:P\to V$ as $f(p)=v$, where $s\big(\pi(p)\big)=[p,v]$. Then $f(pg)=w$, where $s(\pi(pg))=[pg, w]=[p, g \cdot w]$. Since $\pi(p)=\pi(pg)$ (fibers are orbits), we conclude $v=g \cdot w$, i.e. $f(p)=g \cdot f(pg)$, and so $f$ is equivariant. (A bit more to check, but this is the basic map.) > [!basicexample] > - A **sign bundle** is the line bundle associated to a principal bundle whose representation $\rho:G \to \operatorname{GL_{1}}$ has image in $\{ \pm 1 \}$ (a [[sign representation]]). ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```