nLab
clutching construction

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Contents

Idea

The clutching construction is the construction of a GG-principal bundle on an n-sphere from a cocycle in GG-Cech cohomology given by the covering of the nn-sphere by two hemi-n-spheres that overlap a bit at the equator, and one single transition function on that equator S n1GS^{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 nn-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 February 10, 2017 06:11:24 by Urs Schreiber (46.183.103.8)