Geometrization and Poincaré Conjecture
The path toward the Poincaré Conjecture proof was initially “started” by the work of Thurston’s project, in which he aimed to classify and study in-depth low-dimensional manifolds.
The path can be seen as:
- Studying 2-dimensional manifolds and trying to go deep into 3-dimensional manifolds by using it.
- Discovering the decomposability of 3-manifolds.
- The Geometrization Conjecture stated.
- The Elliptization conjecture stated, which is a special case of the Geometrization Conjecture.
- Poincaré Conjecture is a trivial case of the Elliptization conjecture.
There is a great paper by Thurston about 3-manifolds: How to See 3-manifolds.
Loosely & Roughly Speaking
Any closed, orientable 2-dimensional manifold is known to hold one of the three geometric structures: Euclidean $\mathbb{E}$, Spherical $\mathbb{S}$, Hyperbolic $\mathbb{H}$, but what about 3-dimensional manifolds? What type of geometry they can hold?
Well, the process to deduce the answer is complicated. However, it can be simplified as follow:
- 3-dimensional manifolds can be decomposed into sums of prime 3-manifolds, as The Prime Decomposition Theorem states.
- Which means, 3-manifolds can be decomposed into connected sum.
- The Geometrization Conjecture: Thurston conjectured that all the components of the connected sum holds a geometric structure. So, any 3-manifold component has exactly one of these geometric structures:
- Euclidean geometry $\mathbb{E}$.
- Hyperbolic geometry $\mathbb{H}$.
- Spherical geometry $\mathbb{S}$.
- The geometry of $\mathbb{S}^2×\mathbb{R}$.
- The geometry of $\mathbb{H}^2×\mathbb{R}$.
- The geometry of the universal cover ${\displaystyle {\widetilde {\rm {SL}}}(2,\mathbb {R} )}$ of the Lie group ${\rm SL(2,\mathbb{R})}$.
- Nil geometry.
- Sol geometry.
- The geometric structure of a 3-manifold can be characterized by The Fundamental Group.
- The Elliptization Conjecture: Thurston conjectured that a closed 3-manifold with *finite Fundamental Group has $\mathbb{S}^3$ geometric structure.
- The Poincaré Conjecture: Any closed simply connected (aka. with trivial fundamental group) 3-manifold has the $\mathbb{S}^3$ geometric structure.
As the path, The Geometrization Conjecture helps in proving The Elliptization Conjecture, using the fact that if the fundamental group is finite, then the geometric structure is $\mathbb{S}$.
Therefore, The Poincaré Conjecture follows immediately.
About The Proof
The proof was done by Grisha Perelman in three papers that appeared on Arxiv [1][2][3], using surgery technique with Ricci Flow. To take an overview of the proof, you can read Tao presentation.