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 $S^{n-1} \to G$.

More generally, if $X$ is a space and $C_+X,C_-X$ are the two cones forming the suspension $\Sigma X$, a map $X\to G$ (called then a clutching map) provides a way to glue the trivial $G$-bundles on $C_\pm X$ to obtain a $G$-bundle on $\Sigma X$. To relate this construction with the classification of $G$-bundles on $\Sigma X$, we can start with a pointed space $(X,x)$ and require that the clutching map sends the basepoint $x$ to the identity $e\in G$. In this case, the clutching construction is the map

$$[X,G]_* \cong [\Sigma X,B G]_* \to [\Sigma X,B G] = \mathrm{Bun}_G(\Sigma X)$$

sending pointed homotopy classes of pointed clutching functions to $G$-bundles over $\Sigma X$. By the path-connectedness of $B G$, the map is surjective, but the map can fail to be injective if $G$ is not connected. In general, the left hand side carries a trivialization of the fibre over the basepoint.

Examples

Basic examples

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

The real plane bundles on the sphere $S^2$ come from clutching functions $S^1\to O(2)$. The pointed maps are $[S^1,O(2)]_*\cong\pi_1O(2)\cong\mathbb{Z}$ and the passage to the corresponding bundle

$$\mathbb{Z} \cong [S^1,O(2)]_* \cong [S^2,BO(2)]_* \to [S^2,BO(2)] \cong \mathbb{N}$$

identifies $n$ with $-n$.

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:

• Dale Husemöller, chapter 7 of: Fibre bundles, Springer (1966, 1975 , 1994) [doi:10.1007/978-1-4757-2261-1, pdf, pdf, pdf]

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

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

See also:

• Wikipedia, Clutching construction