book
Algebraic Topology
Allen Hatcher
Use in this reference: Linked author-hosted textbook. Exact sequences below are independently derived from the stated theorem.
Relevant material: Section 4.2, Theorem 4.41 (printed page 376) and the preceding Hopf-fibration example.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| U(2) | Global form | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| U(2) | Global structure and topology | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| U(1) ↪ SU(2) | Global and representation consequences | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| U(1) ≅ SO(2) ↪ SO(3) | Global and representation consequences | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SU(2)/U(1) ≅ S² | Fundamental group π₁ | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SU(2)/U(1) ≅ S² | Second homotopy group π₂ | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SU(2)/U(1) ≅ S² | Boundary π₂(S²) → π₁(U(1)) | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SU(2)/U(1) ≅ S² | The exact sequence with generators | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SU(2)/U(1) ≅ S² | The exact sequence with generators | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SO(3)/SO(2) ≅ S² | Fundamental group π₁ | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SO(3)/SO(2) ≅ S² | Second homotopy group π₂ | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SO(3)/SO(2) ≅ S² | Boundary π₂(S²) → π₁(SO(2)) | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |
| SO(3)/SO(2) ≅ S² | The exact sequence with generators | Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375 |