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John M lee, Introduction to smooth manifolds, Exercise 5-4:

Let $M$ be a smooth manifold and let $\{U_α\}_{α∈A}$ be an open cover of $M$. Suppose for each $α, β ∈ A$ we are given a smooth map $τ_{αβ} : U_α∩U_β → GL(k,\mathbb{R})$ such that $τ_{αβ}(p)τ_{βγ}(p) = τ_{αγ}(p)$, $p∈ U_α ∩ U_β ∩ U_γ$ is satisfied for all $α, β, γ ∈ A$. Show that there is a smooth rank $k$ vector bundle $E → M$ with smooth local trivializations $Φ_α : π^{−1}(U_α) → U_α×\mathbb{R}^k$ whose transition functions are the given maps $τ_{αβ}$. [Hint: Define an appropriate equivalence relation on $\coprod_{α∈A}(U_α × \mathbb{R}^k)$, and use the bundle construction lemma.]

But the bundle construction lemma has a different hypothesis. It uses $E= \coprod _{p \in M} E_p$ where $E_p$ is a real vector space. How can I use the lemma?

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The obvious equivalence relation should turn the union of trivializations into a union of vector spaces indexed by $M$ (transition isomorphisms will make sure of the vector space structure) so that you can use the bundle construction lemma. –  Neal Dec 2 '12 at 5:43

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Let $X=\coprod_{\alpha\in A}(U_\alpha\times\mathbf R^k)$. You want to find a relation ${\sim}$ on $X$ such that $X/{\sim}$ and $E$ are in bijection. The obvious candidate for a bijection is the one defined on each $U_\alpha\times\mathbf R^k$ by $\phi:(p,v)\in U_\alpha\times\mathbf R^k \mapsto (p,v)\in E_p$. This $\phi$ is surjective, but not injective: if $p$ is in $U_\alpha\cap U_\beta$, then both $(p,v)\in U_\alpha\times\mathbf R^k$ and $(p,v)\in U_\beta\times\mathbf R^k$ are sent to $(p,v)\in E_p$. The obvious solution is to define the relation ${\sim}$ by: for $(p,v)\in U_\alpha\times\mathbf R^k$ and $(q,w)\in U_\beta\times\mathbf R^k$, say that $(p,v)\sim(q,w)$ when $p=q$ and $w=\tau_{\alpha\beta}(p)v$. Then $\phi$ gives a bijection $X/{\sim}\to E$, and you can use it to build the bijections $\Phi_\alpha$ in the construction lemma.

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