I'm reading Hatcher's "Vector bundles and K-Theory" (version 2.2, November 2017). In chapter 1, section 1.2, he describes how to construct vector bundles with base space a sphere. I can follow his example (that I will state at the bottom for completeness). But somehow I think he is doing more work than is needed for this construction. So to be concrete, my question is: why is he gluing together two hemispheres? Wouldn't it be equivalent to fix an equator (i.e. some $S^{k-1}$ in $S^k$) and considering a map $f: S^{k-1} \to GL_n(\mathbb{R})$, so that the projection from the quotient of $S^k \times \mathbb{R}^n$ (realised by identifying $(x,v)$ with $(x,f(x)v)$ for $x \in S^{k-1}$) to $S^k$ gives a vector bundle?

Citation of his construction: "Write $S^k$ as the union of its upper and lower hemispheres $D_+^k$ and $D_−^k$ , with $D_+^k \cap D_−^k =S^{k−1}$. Given a map $f : S^{k−1} \to GL_n(\mathbb{R})$, let $E_f$ be the quotient of the disjoint union $D_+^k \times R^n \cup D_−^k \times R^n$ obtained by identifying $(x,v) \in \partial D_-^k \times \mathbb{R}^n$ with $(x,f(x)(v)) \in \partial D_+^k \times \mathbb{R}^n$ . There is then a natural projection $E_f \to S^k$ and this is an $n$ dimensional vector bundle (...)"


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