Poincare Conjecture and Elliptization Conjecture

1. Poincare Conjecture

Theorem 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
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 $S^3/\Gamma$, 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.
4. Papers of Perelman
  • 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