# nLab clutching construction

cohomology

### Theorems

#### Topology

topology

algebraic topology

# Contents

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

## Applications

### 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.

## References

Reviews include

Revised on March 7, 2014 00:35:50 by Urs Schreiber (89.204.135.60)