Definition
Consider the sphere as the union of the upper and lower hemispheres and along their intersection, the equator, an .
Given trivialized fiber bundles with fiber and structure group over the two disks, then given a map (called the clutching map), glue the two trivial bundles together via f.
Formally, it is the coequalizer of the inclusions via and : glue the two bundles together on the boundary, with a twist.
Thus we have a map : clutching information on the equator yields a fiber bundle on the total space.
In the case of vector bundles, this yields, and indeed this map is an isomorphism (under connect sum of spheres on the right).
Read more about this topic: Clutching Construction
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