Let $n \geq 2$ and $k \geq 1$. Let $D_+^n$ and $D_-^n$ denote the closed upper and lower hemispheres of $S^n$, with common boundary $D_+^n \cap D_-^n = S^{n-1}$. For a continuous map
let $E_g \to S^n$ be the rank $k$ real vector bundle obtained by gluing the trivial bundles $D_+^n \times \mathbb{R}^k$ and $D_-^n \times \mathbb{R}^k$ along $S^{n-1}$ by
Then two such bundles $E_g$ and $E_h$ are isomorphic if and only if there exist constants $a_+,a_- \in GL(k,\mathbb{R})$ such that $h$ is homotopic to the pointwise product
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\begin{align*}
x \mapsto a_-g(x)a_+^{-1}.
\end{align*}
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Equivalently, the set of isomorphism classes of rank $k$ real vector bundles over $S^n$ obtained by the hemisphere clutching construction is
where the two factors act by left and right multiplication.
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For complex vector bundles, with $GL(k,\mathbb{C})$ in place of $GL(k,\mathbb{R})$, the corresponding clutching map construction gives a bijection between isomorphism classes of rank $k$ complex vector bundles over $S^n$ and the ordinary homotopy set
When $n=1$, the same construction uses clutching functions $S^0 \to GL(k,\mathbb{R})$ or $S^0 \to GL(k,\mathbb{C})$, but the [equivalence relation](/page/Equivalence%20Relation) is instead given by changes of trivialization on the two intervals whose endpoint restrictions extend over those intervals.