Theorem 1 (Poincare Conjecture). If a compact three-dimensional manifold M3 has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M3 is homeomorphic to the sphere S3?
2. Thurston Elliptization Conjecture
Theorem 2 (Thurston Elliptization Conjecture). Every closed 3-manifold with finite fundamental group has a metric of constant positive curvature and hence is homeomorphic to a quotient S3/Γ, where Γ⊂SO(4) is a finite group of rotations that acts freely on S3. The Poincare Conjecture corresponds to the special case where the group Γ≅π1(M3) is trivial.
3. History
- 1961, Stephen Smale, n>4;
- 1982
- Michael Freedman, n=4;
- William Thurston, Geometrization conjecture;
- Richard Hamilton, Ricci flow method;
- 2006, Grisha Perelman, proved Geometrization conjecture.
- The entropy formula for the Ricci flow and its geometric applications
- Ricci flow with surgery on three-manifolds
- Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
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