Proposition 9.2.7 (Principal bundles and vector bundles). Every principal \(\GL _n(\C )\)-bundle \(P \to B\) defines a complex vector bundle of rank \(n\) over \(B\) by taking \[ E := P \times _{\GL _n(\C )} \C ^n = (P \times \C ^n)_{/(p,Av) \sim (pA,v)}, \] where the equivalence relation identifies the pair \((p,Av)\) with \((pA,v)\) for all \(p \in P\), \(v \in \C ^n\) and \(A \in \GL _n(\C )\). This construction defines a bijection between the set of isomorphism classes of principal \(\GL _n(\C )\)-bundles and the set of isomorphism classes of rank \(n\) vector bundles.
Proof. Local trivializations of \(P\) induce local trivializations of the associated bundle. Conversely, the frame bundle \(\mathrm {Fr}(E)\to B\) of a rank \(n\) vector bundle is a principal \(\GL _n(\C )\)-bundle. The natural map \(\mathrm {Fr}(E)\times _{\GL _n(\C )}\C ^n\to E\) is an isomorphism, and the map \[ P\longrightarrow \mathrm {Fr}(P\times _{\GL _n(\C )}\C ^n), \qquad p\longmapsto \big (v\mapsto [p,v]\big ) \] is an isomorphism of principal bundles. Thus the two constructions are inverse on isomorphism classes. □
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