The clutching construction is the construction of a -principal bundle on an n-sphere from a cocycle in -Cech cohomology given by the covering of the -sphere by two semi--spheres that overlap a bit at the equator, and one single transition function on that equator .
In physics, in gauge theory, the clutching construction plays a central role in the discussion of Yang-Mills instantons, and monopoles (Dirac monopole). Here the discussion is usually given in terms of gauge fields on -dimensional Minkowski spacetime such that they vanish at infinity. Equivalently this means that one has gauge fields on the one-point compactification of Minkowski spacetime, which is the n-sphere. The transition function of the clutching construction then appears as the asymptotic gauge transformation.