nLab clutching construction

Idea

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 hemi-n-spheres that overlap a bit at the equator, and one single transition function on that equator $S^{n-1} \to G$.

Examples

Basic example

The Möbius strip is the result of the single non-trivial clutching construction for real line bundle over the circle.

In physics

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.

Related concepts

• QCD instanton, Yang-Mills instanton

Literature

Review:

• Allen Hatcher, page 22 of: Vector bundles and K-theory (web)

• Klaus Wirthmüller, p. 17-28 of: Vector bundles and K-theory, 2012 (pdf)

• Wikipedia, Clutching construction