Rotation sphere as SO(3)/SO(2)
The sphere quotient with its axial SO(2) embedding and boundary map of winding two.
The homogeneous space of rotations modulo the subgroup fixing a chosen axis, with SO(2) identified with U(1) by the actual spatial rotation angle.
Definition sourcesReference properties
The exact sequence with generators
The relevant segment is 0 → Z → Z → C₂ → 0. The arrow between the two infinite cyclic groups is multiplication by +2, and the following arrow reduces an integer modulo two. [sources]
The subgroup record is U(1), used through the specified isomorphism U(1) ≅ SO(2). This avoids creating a second abstract circle-group record while retaining its concrete rotational action. [sources]
Related objects and maps
- Total group
- Circle realized as SO(2)
- Normalized rotational 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 homogeneous space of rotations modulo the subgroup fixing a chosen axis, with SO(2) identified with U(1) by the actual spatial rotation angle.
- 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:so-3-over-so-2#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:so-3-over-so-2#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:so-3-over-so-2#manifoldFundamental group π₁
0
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
space:so-3-over-so-2#pi1Second homotopy group π₂
Z
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
space:so-3-over-so-2#pi2Physical rotation angle period
2π
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
space:so-3-over-so-2#fiber-periodInclusion π₁(SO(2)) → π₁(SO(3))
n ↦ n mod 2
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
- 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
Premises: map:su-2-to-so-3 · Fiber angle is the spatial angle θ with period 2π
space:so-3-over-so-2#inclusion-on-pi1Boundary π₂(S²) → π₁(SO(2))
n ↦ 2n
Assumptions: The fiber generator is one positive spatial 2π rotation. · Choose the sphere generator to make the boundary coefficient positive.
- 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: SO(3)=RP³ · π₂(RP³)=π₂(S³)=0 · π₁(SO(3))=C₂ · Inclusion sends winding to its parity
space:so-3-over-so-2#boundary-mapThe exact sequence with generators
The relevant segment is 0 → Z → Z → C₂ → 0. The arrow between the two infinite cyclic groups is multiplication by +2, and the following arrow reduces an integer modulo two.
- 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-so-3 · π₂(SO(3))=0 · π₁(SO(3))=C₂ · π₁(S²)=0
space:so-3-over-so-2#sequence-1The exact sequence with generators
The subgroup record is U(1), used through the specified isomorphism U(1) ≅ SO(2). This avoids creating a second abstract circle-group record while retaining its concrete rotational action.
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
space:so-3-over-so-2#sequence-2Total group
group:so-3
- 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:so-3-over-so-2#relation-1Circle realized as SO(2)
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:so-3-over-so-2#relation-2Normalized rotational subgroup map
map:u-1-to-so-3
- 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:so-3-over-so-2#relation-3Same S², different fibration
space:su-2-over-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:so-3-over-so-2#relation-4