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gptkbp:instanceOf
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gptkb:mathematical_concept
gptkb:topology
gptkb:Riemannian_manifold
gptkb:Lie_group
homogeneous space
symmetric space
parallelizable manifold
exotic sphere (in higher dimensions)
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gptkbp:canBeEmbeddedIn
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gptkb:4-dimensional_Euclidean_space
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gptkbp:canBeParametrizedBy
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gptkb:stereographic_projection
Hopf coordinates
quaternions of unit norm
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gptkbp:compact
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true
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gptkbp:connects
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true
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gptkbp:dimensions
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3
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gptkbp:Euler_characteristic
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0
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gptkbp:fundamentalGroup
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trivial
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gptkbp:generalizes
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gptkb:circle_(S^1)
gptkb:2-sphere_(S^2)
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gptkbp:groupIsomorphism
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gptkb:SU(2)
unit quaternions
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gptkbp:groupStructure
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gptkb:Lie_group
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gptkbp:hasNoBoundary
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true
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gptkbp:homologyGroup
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H_0(S^3) = Z
H_1(S^3) = 0
H_2(S^3) = 0
H_3(S^3) = Z
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gptkbp:homotopyGroup
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π_3(S^3) = Z
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https://www.w3.org/2000/01/rdf-schema#label
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3-sphere (S^3)
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gptkbp:isBoundaryOf
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4-ball (B^4)
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gptkbp:notation
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S^3
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gptkbp:orientable
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true
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gptkbp:relatedTo
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gptkb:Poincaré_conjecture
gptkb:SU(2)
gptkb:Hopf_fibration
unit quaternions
quaternionic projective line
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gptkbp:simplyConnected
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true
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gptkbp:surface
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2π^2 (unit radius)
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gptkbp:universalCover
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real projective 3-space (RP^3)
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gptkbp:usedIn
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gptkb:cosmology
gptkb:geometry
gptkb:topology
gptkb:string_theory
physics
quantum mechanics
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gptkbp:volume
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2π^2 (unit radius)
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gptkbp:bfsParent
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gptkb:SU(2)
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gptkbp:bfsLayer
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6
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