Hopf sphere as SU(2)/U(1)
The sphere quotient with its specified U(1) embedding and boundary map of winding one.
The right-coset homogeneous space of SU(2) by the embedded torus eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ).
Definition sourcesReference properties
The exact sequence with generators
The relevant segment is 0 → Z → Z → 0. The two displayed Z groups are respectively π₂(S²) and π₁(U(1)). With the generator convention stated here, the first nonzero arrow is multiplication by +1. [sources]
The same base manifold occurs for SO(3)/SO(2), but the embedding and the total group change the boundary homomorphism. The integer here is a fiber winding; converting it to a physical magnetic charge also requires the generator and coupling convention. [sources]
Related objects and maps
- Total group
- Embedded torus
- Normalized subgroup map
- Same S², different fibration
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
The right-coset homogeneous space of SU(2) by the embedded torus eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ).
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#definitionReal dimension
2
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#dimensionManifold
S²
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#manifoldFundamental group π₁
0
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
space:su-2-over-u-1#pi1Second homotopy group π₂
Z
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
space:su-2-over-u-1#pi2Fiber angle period
2π
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
space:su-2-over-u-1#fiber-periodBoundary π₂(S²) → π₁(U(1))
n ↦ n
Assumptions: The U(1) generator is θ:0→2π in diag(eⁱθ,e⁻ⁱθ). · Choose the sphere generator so its boundary is the positive fiber generator.
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
Premises: SU(2)=S³ · π₂(S³)=0 · π₁(S³)=0 · π₁(U(1))=Z
space:su-2-over-u-1#boundary-mapThe exact sequence with generators
The relevant segment is 0 → Z → Z → 0. The two displayed Z groups are respectively π₂(S²) and π₁(U(1)). With the generator convention stated here, the first nonzero arrow is multiplication by +1.
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
Premises: map:u-1-to-su-2 · π₂(S³)=π₁(S³)=0
space:su-2-over-u-1#sequence-1The exact sequence with generators
The same base manifold occurs for SO(3)/SO(2), but the embedding and the total group change the boundary homomorphism. The integer here is a fiber winding; converting it to a physical magnetic charge also requires the generator and coupling convention.
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
space:su-2-over-u-1#sequence-2Total group
group:su-2
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#relation-1Embedded torus
group:u-1
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#relation-2Normalized subgroup map
map:u-1-to-su-2
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#relation-3Same S², different fibration
space:so-3-over-so-2
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
space:su-2-over-u-1#relation-4