nLab
clutching construction
Context
Cohomology
cohomology

Special and general types
Special notions
Variants
Operations
Theorems
Topology
topology (point-set topology )

see also algebraic topology , functional analysis and homotopy theory

Introduction

Basic concepts
Constructions
nice topological space

metric space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space (sequentially compact , countably compact , paracompact , countably paracompact , locally compact , strongly compact )

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

topological vector space , Banach space , Hilbert space

topological manifold

CW-complex

Examples
empty space , point space

discrete space , codiscrete space

Euclidean space

sphere , ball ,

circle , torus , annulus

polytope , polyhedron

projective space (real , complex )

classifying space

mapping space , loop space , path space

Zariski topology

Cantor space , Sierpinski space

long line , line with two origins

K-topology , Dowker space

Warsaw circle

Peano curve

Basic statements
Theorems
Theorems

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 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 .

References
Reviews include

Revised on February 10, 2017 06:11:24
by

Urs Schreiber
(46.183.103.8)