Clutching construction for bundles over a suspension

Let $(A,a_0)$ be a well-pointed based compact CGWH space and let $g:A\to {\mathrm{GL}}_n(\mathbb{F})$ be continuous. Write the suspension from Reduced cone suspension and cofiber sequence as the union of its upper and lower cones $C_{+}A$ and $C_{-}A$. The clutched bundle $E_g$ is the quotient of

$(C_{+}A\times {\mathbb{F}}^n)⨿(C_{-}A\times {\mathbb{F}}^n)$

by the equatorial identifications

$(a,v{)}_{+}\sim (a,g(a)v{)}_{-}\mathrm{.}$

Using collar neighborhoods of the equator gives two open product charts whose transition is the same $g$ on every collar slice. Thus Vector bundles are glued from transition cocycles makes $E_g\to \mathrm{\Sigma }A$ a rank-$n$ vector bundle.

This upper-to-lower convention is fixed throughout the page: under the affine coordinates used in the companion calculation, $g(z)=z$ gives the chosen tautological Hopf line over $S^2={\mathbb{C}\mathbb{P}}^1$. Interchanging the upper and lower charts reverses the relation and replaces $g$ by $g^{-1}$. For $n=0$, ${\mathrm{GL}}_0(\mathbb{F})$ is a point and the construction gives the unique rank-zero bundle.