|
gptkbp:instanceOf
|
gptkb:topology
gptkb:Lie_group
abelian group
compact group
|
|
gptkbp:automorphismGroup
|
gptkb:Z/2Z
|
|
gptkbp:centralTo
|
itself
|
|
gptkbp:compact
|
true
|
|
gptkbp:connects
|
true
|
|
gptkbp:containsElement
|
complex numbers of unit modulus
z in C such that |z|=1
|
|
gptkbp:coveringMap
|
exp: R → S^1, t ↦ exp(2πit)
|
|
gptkbp:dimensions
|
1
|
|
gptkbp:dualGroup
|
Z
|
|
gptkbp:firstCohomologyGroup
|
Z
|
|
gptkbp:firstHomologyGroup
|
Z
|
|
gptkbp:fundamentalGroup
|
Z
|
|
gptkbp:hasSubgroup
|
gptkb:GL(1,C)
complex numbers C^*
|
|
gptkbp:Hausdorff
|
true
|
|
gptkbp:homotopyType
|
gptkb:butter
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
circle group S^1
|
|
gptkbp:isomorphicTo
|
gptkb:U(1)
|
|
gptkbp:locallyCompact
|
true
|
|
gptkbp:manifoldType
|
1-dimensional manifold
|
|
gptkbp:parameter
|
z = exp(iθ), θ in [0,2π)
|
|
gptkbp:pathConnected
|
true
|
|
gptkbp:realization
|
gptkb:unit_circle_in_complex_plane
|
|
gptkbp:relatedGroup
|
complex multiplication
|
|
gptkbp:representationTheory
|
characters are integer powers
|
|
gptkbp:simplyConnected
|
false
|
|
gptkbp:universalCover
|
gptkb:real_line_R
|
|
gptkbp:usedIn
|
gptkb:gauge_theory
gptkb:topology
Fourier analysis
quantum mechanics
representation theory
|
|
gptkbp:bfsParent
|
gptkb:torus_group
|
|
gptkbp:bfsLayer
|
5
|