{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/spaces/so-3-over-so-2/",
  "entity": {
    "id": "space:so-3-over-so-2",
    "slug": "so-3-over-so-2",
    "name": "Rotation sphere as SO(3)/SO(2)",
    "symbol": "SO(3)/SO(2) ≅ S²",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Homogeneous space",
      "Sphere",
      "Exact homotopy sequence"
    ],
    "summary": "The sphere quotient with its axial SO(2) embedding and boundary map of winding two.",
    "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.",
    "definitionEvidence": [
      {
        "source": "etingof-lie-notes",
        "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
      }
    ],
    "properties": [
      {
        "key": "dimension",
        "label": "Real dimension",
        "value": "2",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "manifold",
        "label": "Manifold",
        "value": "S²",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "pi1",
        "label": "Fundamental group π₁",
        "value": "0",
        "evidence": [
          {
            "source": "hatcher-algebraic-topology",
            "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
          }
        ],
        "assumptions": []
      },
      {
        "key": "pi2",
        "label": "Second homotopy group π₂",
        "value": "Z",
        "evidence": [
          {
            "source": "hatcher-algebraic-topology",
            "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
          }
        ],
        "assumptions": []
      },
      {
        "key": "fiber-period",
        "label": "Physical rotation angle period",
        "value": "2π",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          }
        ],
        "assumptions": []
      },
      {
        "key": "inclusion-on-pi1",
        "label": "Inclusion π₁(SO(2)) → π₁(SO(3))",
        "value": "n ↦ n mod 2",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "derivation": {
          "method": "A 2π spatial rotation lifts from I to −I in SU(2), while a 4π rotation closes.",
          "inputs": [
            "map:su-2-to-so-3",
            "Fiber angle is the spatial angle θ with period 2π"
          ]
        },
        "assumptions": []
      },
      {
        "key": "boundary-map",
        "label": "Boundary π₂(S²) → π₁(SO(2))",
        "value": "n ↦ 2n",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          },
          {
            "source": "hatcher-algebraic-topology",
            "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
          }
        ],
        "derivation": {
          "method": "Exactness identifies the boundary image with the kernel 2Z of Z→C₂. The boundary is injective because π₂(SO(3))=0.",
          "inputs": [
            "SO(3)=RP³",
            "π₂(RP³)=π₂(S³)=0",
            "π₁(SO(3))=C₂",
            "Inclusion sends winding to its parity"
          ]
        },
        "assumptions": [
          "The fiber generator is one positive spatial 2π rotation.",
          "Choose the sphere generator to make the boundary coefficient positive."
        ]
      }
    ],
    "sections": [
      {
        "id": "sequence",
        "title": "The exact sequence with generators",
        "paragraphs": [
          {
            "text": "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.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
              },
              {
                "source": "hatcher-algebraic-topology",
                "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
              }
            ],
            "derivation": {
              "method": "Use the fibration SO(2) → SO(3) → S² and the induced map on fundamental groups.",
              "inputs": [
                "map:u-1-to-so-3",
                "π₂(SO(3))=0",
                "π₁(SO(3))=C₂",
                "π₁(S²)=0"
              ]
            },
            "assumptions": []
          },
          {
            "text": "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.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "quotient-of",
        "target": "group:so-3",
        "label": "Total group",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      },
      {
        "type": "has-isotropy",
        "target": "group:u-1",
        "label": "Circle realized as SO(2)",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      },
      {
        "type": "defined-by",
        "target": "map:u-1-to-so-3",
        "label": "Normalized rotational subgroup map",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      },
      {
        "type": "same-manifold-as",
        "target": "space:su-2-over-u-1",
        "label": "Same S², different fibration",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      }
    ],
    "capabilities": [
      "topology"
    ],
    "kind": "space",
    "data": {
      "construction": "Right-coset quotient by the axial rotation subgroup SO(2)",
      "groupId": "group:so-3",
      "subgroupId": "group:u-1",
      "mapId": "map:u-1-to-so-3",
      "capabilityPresets": {
        "topology": {
          "kind": "so3"
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      }
    }
  },
  "claims": [
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      "id": "space:so-3-over-so-2#definition",
      "subject": "space:so-3-over-so-2",
      "key": "definition",
      "label": "Definition",
      "value": "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.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
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        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#dimension",
      "subject": "space:so-3-over-so-2",
      "key": "dimension",
      "label": "Real dimension",
      "value": "2",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#manifold",
      "subject": "space:so-3-over-so-2",
      "key": "manifold",
      "label": "Manifold",
      "value": "S²",
      "evidence": [
        {
          "source": "etingof-lie-notes",
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        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#pi1",
      "subject": "space:so-3-over-so-2",
      "key": "pi1",
      "label": "Fundamental group π₁",
      "value": "0",
      "evidence": [
        {
          "source": "hatcher-algebraic-topology",
          "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#pi2",
      "subject": "space:so-3-over-so-2",
      "key": "pi2",
      "label": "Second homotopy group π₂",
      "value": "Z",
      "evidence": [
        {
          "source": "hatcher-algebraic-topology",
          "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#fiber-period",
      "subject": "space:so-3-over-so-2",
      "key": "fiber-period",
      "label": "Physical rotation angle period",
      "value": "2π",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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    {
      "id": "space:so-3-over-so-2#inclusion-on-pi1",
      "subject": "space:so-3-over-so-2",
      "key": "inclusion-on-pi1",
      "label": "Inclusion π₁(SO(2)) → π₁(SO(3))",
      "value": "n ↦ n mod 2",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "derivation": {
        "method": "A 2π spatial rotation lifts from I to −I in SU(2), while a 4π rotation closes.",
        "inputs": [
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          "Fiber angle is the spatial angle θ with period 2π"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#boundary-map",
      "subject": "space:so-3-over-so-2",
      "key": "boundary-map",
      "label": "Boundary π₂(S²) → π₁(SO(2))",
      "value": "n ↦ 2n",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        },
        {
          "source": "hatcher-algebraic-topology",
          "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
        }
      ],
      "derivation": {
        "method": "Exactness identifies the boundary image with the kernel 2Z of Z→C₂. The boundary is injective because π₂(SO(3))=0.",
        "inputs": [
          "SO(3)=RP³",
          "π₂(RP³)=π₂(S³)=0",
          "π₁(SO(3))=C₂",
          "Inclusion sends winding to its parity"
        ]
      },
      "assumptions": [
        "The fiber generator is one positive spatial 2π rotation.",
        "Choose the sphere generator to make the boundary coefficient positive."
      ],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "id": "space:so-3-over-so-2#sequence-1",
      "subject": "space:so-3-over-so-2",
      "key": "sequence-1",
      "label": "The exact sequence with generators",
      "value": "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.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        },
        {
          "source": "hatcher-algebraic-topology",
          "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
        }
      ],
      "derivation": {
        "method": "Use the fibration SO(2) → SO(3) → S² and the induced map on fundamental groups.",
        "inputs": [
          "map:u-1-to-so-3",
          "π₂(SO(3))=0",
          "π₁(SO(3))=C₂",
          "π₁(S²)=0"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#sequence-2",
      "subject": "space:so-3-over-so-2",
      "key": "sequence-2",
      "label": "The exact sequence with generators",
      "value": "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.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:so-3-over-so-2#relation-1",
      "subject": "space:so-3-over-so-2",
      "key": "relation-1",
      "label": "Total group",
      "value": "group:so-3",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "relationship": "quotient-of",
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      "revision": "2026.11-workbench.4"
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      "id": "space:so-3-over-so-2#relation-2",
      "subject": "space:so-3-over-so-2",
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      "label": "Circle realized as SO(2)",
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          "source": "etingof-lie-notes",
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      "relationship": "has-isotropy",
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          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
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      "value": "space:su-2-over-u-1",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
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      "id": "conrad",
      "title": "Conrad and Landesman · Compact Lie groups",
      "authors": [
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        "Aaron Landesman"
      ],
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      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
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      "licenseNote": "Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes."
    },
    {
      "id": "hatcher-algebraic-topology",
      "title": "Algebraic Topology",
      "authors": [
        "Allen Hatcher"
      ],
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      "kind": "book",
      "locatorHint": "Section 4.2, Theorem 4.41 (printed page 376) and the preceding Hopf-fibration example.",
      "licenseNote": "Linked author-hosted textbook. Exact sequences below are independently derived from the stated theorem."
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