Clutching classifies vector bundles over spheres in the stable range

A finite-rank bundle $E$ over a disk $D$ is trivial without a choice principle. Pull it back along the radial contraction $H:D\times I\to D$. Choose finitely many linear charts covering the compact metric space $D\times I$, shrink them by a Lebesgue-number refinement so that the closures of the shrunken opens lie in the original charts, and normalize the finitely many distance-to-complement functions. Their supports lie in the original charts, so the square-root-weighted chart formula embeds $H^{\ast }E$ in one finite trivial bundle. Let $P(x,t)$ be the orthogonal projection onto the resulting image plane. Uniform continuity on $D\times I$ gives a subdivision $0=t_0<\cdots <t_m=1$ with $\parallel P(x,t_{j+1})-P(x,t_j)\parallel <1$ for every $x$. Projection from $\mathrm{im}P(x,t_j)$ to $\mathrm{im}P(x,t_{j+1})$ is then injective—if $v$ is killed, $\parallel v\parallel =\parallel (P_j-P_{j+1})v\parallel <\parallel v\parallel$—and hence is an isomorphism between equal finite dimensions. These continuous bundle isomorphisms compose to identify the restriction at $t=0$ with the constant restriction at $t=1$. Thus $E$ is trivial. Apply this to the two closed hemispheres of $S^q$. Their trivializations differ on the equator by a continuous $g:S^{q-1}\to {\mathrm{GL}}_n(\mathbb{F})$, and [F1] reconstructs the bundle as $E_g$.

Polar normalization $A\mapsto A(A^{\ast }A{)}^{-1\mathrm{/}2}$ deformation retracts ${\mathrm{GL}}_n(\mathbb{C})$ to $U(n)$ and ${\mathrm{GL}}_n(\mathbb{R})$ to $O(n)$, preserving the two real determinant components. The last-column maps give fibrations $U(n)\to U(n+1)\to S^{2n+1}$ and $O(n)\to O(n+1)\to S^n$. Since the homotopy groups of $S^d$ vanish below $d$, [F2] makes ${\pi }_{q-1}U(n)\to {\pi }_{q-1}U(n+1)$ an isomorphism for $q\le 2n$, and the orthogonal map an isomorphism for $q<n$.

Let $E_g$ and $E_h$ be clutched bundles. An isomorphism, written in the chosen upper and lower trivializations, has matrices $A_{\pm }:D_{\pm }^q\to {\mathrm{GL}}_n(\mathbb{F})$ and compatibility $A_{-}g=h{A}_{+}$ on the equator, hence $h=A_{-}g{A}_{+}^{-1}$. Conversely, any such pair of disk-extending matrices defines compatible isomorphisms on the two trivial bundles and therefore an isomorphism of the quotients. This proves both directions of the left/right gauge criterion.

A homotopy $G:S^{q-1}\times I\to {\mathrm{GL}}_n(\mathbb{F})$ clutches a bundle over $S^q\times I$. The finite compact version of the endpoint transport in step 1.1 makes its endpoint restrictions $E_{G_0}$ and $E_{G_1}$ isomorphic. Conversely, after fixing hemisphere trivializations, step 2.1 shows that all ambiguity is precisely homotopy together with disk-extending left and right gauges. Thus the stated equivalence classes classify the bundles.

For complex bundles and $q\ge 2$, path-connectedness of ${\mathrm{GL}}_n(\mathbb{C})$ turns the equivalence in step 3.1 into the based group ${\pi }_{q-1}$; disk gauges restrict to nullhomotopic maps, and the fundamental group of a topological group is abelian in the loop case. For real $q>1$, a clutching map lies in one determinant component; choosing an orientation moves it into ${\mathrm{GL}}_n^{+}$, while forgetting orientation takes the orbit under conjugation by a reflection. The stabilization in step 1.2 respects this orbit action, giving the stated real stable range. For $q=1$, maps from $S^0$ retain the separate component/orbit description. If $n=0$, the structure group is a point and [F1] gives the unique rank-zero bundle.