Clutching construction
= Clutching construction
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A <vector bundle> over $S^n$ can be constructed by gluing two trivial bundles over its hemispheres with a transition map $\gamma:S^{n-1}\to GL(k,\mathbb R)$. Extend this map constantly in an equatorial collar to obtain local <vector bundle trivializations>. A transition map into the <special orthogonal group> gives compatible fiber <orientations> and a <fiber metric>. Homotopies of transition maps give isomorphic bundles.