Clutching construction

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.

A clutching function is the equatorial transition map used in the clutching construction. For an oriented real vector bundle one may choose a fiber metric and orthonormal trivializations, so its clutching function takes values in the special orthogonal group.

For an oriented rank-$k$ bundle over $S^n$, with $k\ge 2$, 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.