= Section obstruction for a clutched vector bundle
{title2=$[u\mapsto\gamma(u)e_k]\in\pi_{n-1}(S^{k-1})$}
For an oriented rank-$k$ bundle over $S^n$, with $k\ge2$, normalize its <clutching function> to be based at the identity. A <nowhere-zero section> exists precisely when the displayed <homotopy class> vanishes. If it vanishes, extend $\gamma e_k$ across one hemisphere and use the constant section $e_k$ on the other. Conversely normalize any <nowhere-zero section> using the <fiber metric>. Its two hemisphere maps satisfy $\sigma_-=\gamma\sigma_+$ on the boundary. The first hemisphere makes $\sigma_+$ homotopic to a constant; the second makes $\gamma\sigma_+$ null-homotopic. Thus $\gamma e_k$ is <null-homotopic>. If $k>n$, the lower <homotopy group> of the <sphere> vanishes and a section always exists. Rank one is already trivial because $SO(1)$ is trivial.
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