定理 1 (Poincare Conjecture). If a compact three-dimensional manifold $M^3$ has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that $M^3$ is homeomorphic to the sphere $S^3$?
2. Thurston Elliptization Conjecture
定理 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 $S^3/\Gamma$, where $\mathrm{\Gamma} \subset \mathrm{SO(4)}$ is a finite group of rotations that acts freely on $S^3$. The Poincare Conjecture corresponds to the special case where the group $\Gamma \cong \pi_1(M^3)$ 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
