{
  "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/su-2-over-u-1/",
  "entity": {
    "id": "space:su-2-over-u-1",
    "slug": "su-2-over-u-1",
    "name": "Hopf sphere as SU(2)/U(1)",
    "symbol": "SU(2)/U(1) ≅ S²",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Homogeneous space",
      "Sphere",
      "Exact homotopy sequence"
    ],
    "summary": "The sphere quotient with its specified U(1) embedding and boundary map of winding one.",
    "definition": "The right-coset homogeneous space of SU(2) by the embedded torus eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ).",
    "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": "Fiber angle period",
        "value": "2π",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          }
        ],
        "assumptions": []
      },
      {
        "key": "boundary-map",
        "label": "Boundary π₂(S²) → π₁(U(1))",
        "value": "n ↦ n",
        "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 of 0 → π₂(S²) → π₁(U(1)) → 0 forces the boundary to be an isomorphism; orientation fixes its positive sign.",
          "inputs": [
            "SU(2)=S³",
            "π₂(S³)=0",
            "π₁(S³)=0",
            "π₁(U(1))=Z"
          ]
        },
        "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."
        ]
      }
    ],
    "sections": [
      {
        "id": "sequence",
        "title": "The exact sequence with generators",
        "paragraphs": [
          {
            "text": "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.",
            "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": "Apply the fibration homotopy sequence to U(1) → S³ → S².",
              "inputs": [
                "map:u-1-to-su-2",
                "π₂(S³)=π₁(S³)=0"
              ]
            },
            "assumptions": []
          },
          {
            "text": "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.",
            "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"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "quotient-of",
        "target": "group:su-2",
        "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": "Embedded torus",
        "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-su-2",
        "label": "Normalized 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:so-3-over-so-2",
        "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 explicitly embedded maximal torus",
      "groupId": "group:su-2",
      "subgroupId": "group:u-1",
      "mapId": "map:u-1-to-su-2",
      "capabilityPresets": {
        "topology": {
          "kind": "su2"
        }
      }
    }
  },
  "claims": [
    {
      "id": "space:su-2-over-u-1#definition",
      "subject": "space:su-2-over-u-1",
      "key": "definition",
      "label": "Definition",
      "value": "The right-coset homogeneous space of SU(2) by the embedded torus eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ).",
      "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:su-2-over-u-1#dimension",
      "subject": "space:su-2-over-u-1",
      "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:su-2-over-u-1#manifold",
      "subject": "space:su-2-over-u-1",
      "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": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#pi1",
      "subject": "space:su-2-over-u-1",
      "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:su-2-over-u-1#pi2",
      "subject": "space:su-2-over-u-1",
      "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:su-2-over-u-1#fiber-period",
      "subject": "space:su-2-over-u-1",
      "key": "fiber-period",
      "label": "Fiber 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"
    },
    {
      "id": "space:su-2-over-u-1#boundary-map",
      "subject": "space:su-2-over-u-1",
      "key": "boundary-map",
      "label": "Boundary π₂(S²) → π₁(U(1))",
      "value": "n ↦ n",
      "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 of 0 → π₂(S²) → π₁(U(1)) → 0 forces the boundary to be an isomorphism; orientation fixes its positive sign.",
        "inputs": [
          "SU(2)=S³",
          "π₂(S³)=0",
          "π₁(S³)=0",
          "π₁(U(1))=Z"
        ]
      },
      "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."
      ],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#sequence-1",
      "subject": "space:su-2-over-u-1",
      "key": "sequence-1",
      "label": "The exact sequence with generators",
      "value": "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.",
      "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": "Apply the fibration homotopy sequence to U(1) → S³ → S².",
        "inputs": [
          "map:u-1-to-su-2",
          "π₂(S³)=π₁(S³)=0"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#sequence-2",
      "subject": "space:su-2-over-u-1",
      "key": "sequence-2",
      "label": "The exact sequence with generators",
      "value": "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.",
      "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"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#relation-1",
      "subject": "space:su-2-over-u-1",
      "key": "relation-1",
      "label": "Total group",
      "value": "group:su-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": [],
      "relationship": "quotient-of",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#relation-2",
      "subject": "space:su-2-over-u-1",
      "key": "relation-2",
      "label": "Embedded torus",
      "value": "group: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"
        }
      ],
      "assumptions": [],
      "relationship": "has-isotropy",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#relation-3",
      "subject": "space:su-2-over-u-1",
      "key": "relation-3",
      "label": "Normalized subgroup map",
      "value": "map:u-1-to-su-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": [],
      "relationship": "defined-by",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "space:su-2-over-u-1#relation-4",
      "subject": "space:su-2-over-u-1",
      "key": "relation-4",
      "label": "Same S², different fibration",
      "value": "space:so-3-over-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"
        }
      ],
      "assumptions": [],
      "relationship": "same-manifold-as",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf",
      "kind": "lecture-notes",
      "locatorHint": "MIT 18.755, Spring 2024. Cite section numbers and printed pages.",
      "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"
      ],
      "url": "https://pi.math.cornell.edu/~hatcher/AT/AT.pdf",
      "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."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "e85cbe263b58d3675aeee181c4955f1e014eb1bb7be5291f4ff99ac1afcb1e8b",
    "reviewers": []
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}