---- > [!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]]. ---- #### ---- #### 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 > ```