Construction of a vector bundle from a smooth cocycle

Let $X={\coprod }_{\alpha \in A}(U_{\alpha }\times {\mathbb{R}}^r)$ and declare $(p,v{)}_{\alpha }\sim (q,w{)}_{\beta }$ when $p=q\in U_{\alpha }\cap U_{\beta }$ and $w=g_{\beta \alpha }(p)v$. By [L1], this relation is reflexive, symmetric, and transitive, so it is an equivalence relation on $X$.

Let $E=X\mathrm{/}\sim$ and write $q:X\to E$ for the quotient map. For each $\alpha$, define ${\psi }_{\alpha }:q(U_{\alpha }\times {\mathbb{R}}^r)\to U_{\alpha }\times {\mathbb{R}}^r$ by ${\psi }_{\alpha }([p,v{]}_{\alpha })=(p,v)$. The cocycle relation shows that every class meeting $U_{\alpha }\times {\mathbb{R}}^r$ has a unique representative there, so ${\psi }_{\alpha }$ is bijective. Its domain is open because $q^{-1}(q(U_{\alpha }\times {\mathbb{R}}^r))$ is the union over $\beta$ of the open sets $\{(p,g_{\alpha \beta }(p)w{)}_{\beta }:p\in U_{\alpha }\cap U_{\beta },w\in {\mathbb{R}}^r\}$, whence [F1] makes ${\psi }_{\alpha }$ a homeomorphism onto an open subset.

On overlaps, ${\psi }_{\beta }\circ {\psi }_{\alpha }^{-1}(p,v)=(p,g_{\beta \alpha }(p)v)$, so the chart changes are smooth with smooth inverses. The projection $\pi ([p,v{]}_{\alpha })=p$ is well defined and in chart $\alpha$ is the product projection $U_{\alpha }\times {\mathbb{R}}^r\to U_{\alpha }$, while the fibre maps are linear by construction. Therefore the ${\psi }_{\alpha }$ form a smooth rank-$r$ vector-bundle atlas on $E\to M$.

Because the cover is countable and each chart image has a countable base, these bundle charts give $E$ a countable base. Hausdorffness is local in the charts when two classes are distinct over one base point, and over different base points it comes from Hausdorffness of $M$. Hence $E$ is a smooth manifold, and $\pi :E\to M$ is the required smooth vector bundle.