nLab
clutching construction
Contents
Context
Bundles
bundles

Context
Classes of bundles
covering space

retractive space

fiber bundle , fiber ∞-bundle

numerable bundle

principal bundle , principal ∞-bundle

associated bundle , associated ∞-bundle

vector bundle , 2-vector bundle , (∞,1)-vector bundle

real , complex /holomorphic , quaternionic

topological , differentiable , algebraic

with connection

bundle of spectra

natural bundle

equivariant bundle

Universal bundles
Presentations
Examples
Constructions
Topology
topology (point-set topology , point-free topology )

see also differential topology , algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , 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

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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 .

Literature
Review:

Last revised on October 22, 2021 at 13:26:23.
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