I'm studing this notes, and I have a question about the contruction in the proposition 23.5, page 185. The statement is:
Let $\{g_{αβ} \}$ be a cocycle subordinates to an open cover ${U_α }$ of $M$ . There exists a vector bundle $ξ = (π, E, M )$, that admits a trivialization $\{φ_α \}$ for which the transition functions are the $\{g_{αβ} \}$.
In the proof, he claims:
$E=\cup_{\alpha \in A}(U_\alpha\times\mathbb{R}^r)\Big/\sim$ is a differentiable manifold with the relation gives by $(p,v)\sim (q,w)$ iff $p=q$ and $\exists$ $\alpha, \beta \in A$ such that $g_{αβ}w=v$.
My question is: Why $E$ is a manifold? Does Anyone have some hints about how can I find the charts?
Sure. Let $V^{\prime}=\coprod_{\alpha} (\alpha \times U_\alpha \times \mathbb R^k)/\sim$ where $(\beta,x, \mathbb R^k) \sim (\alpha,x,g_{\alpha\beta})$ and $$q:\coprod_{\alpha} (\alpha \times U_\alpha \times \mathbb R^k) \to V^{\prime} $$ be the quotient map.
Note that $q$ is a continuous open map and the restriction to each $(\alpha \times U_\alpha \times \mathbb R^k)$ denoted $q_\alpha$ is injective, we have that $$\left(q_\alpha(\alpha \times U_\alpha \times \mathbb R^k),q_\alpha^{-1}\right)$$ gives a chart.
details can actually be found in these set of notes actually (same notation is used.)