Clutching construction

Summary

The clutching construction builds a vector bundle over a sphere from two trivial bundles over the hemispheres. They are glued along a map from the equator to invertible matrices. Homotopic gluing maps give isomorphic bundles, and every bundle over a sphere arises in this way. So vector bundles over spheres are classified by homotopy classes of maps into the general linear group. This links the K-theory of spheres to homotopy groups of unitary and orthogonal groups.