group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The clutching construction is the construction of a $G$-principal bundle on an n-sphere from a cocycle in $G$-Cech cohomology given by the covering of the $n$-sphere by two semi-$n$-spheres that overlap a bit at the equator, and one single transition function on that equator $S^{n-1} \to G$.
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 $n$-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.
Reviews include
Allen Hatcher, page 22 of Vector bundles and K-theory (web)
Wikipedia, Clutching construction