Thurston's Geometrization Theorem

Thurston's Geometrization Theorem is a central result in the field of 3-manifold topology. It describes a way to decompose any closed, orientable 3-manifold into geometric pieces. Originally, it was known as the Geometrization Conjecture, proposed by William Thurston in the late 1970s, but it became a theorem after Grigori Perelman's proof in 2003.

The theorem states:

"Every closed, orientable 3-manifold can be decomposed into a union of prime 3-manifolds which further decompose into irreducible 3-manifolds with geometric structures."

Here is what it means:

• A 3-manifold is a space that locally looks like ordinary three-dimensional space.

• A manifold is said to be closed if it is compact and has no boundary.

• A manifold is orientable if it has a consistent 'orientation' or 'direction'.

• A prime 3-manifold is a 3-manifold that can't be represented as a non-trivial connected sum of other 3-manifolds.

• An irreducible 3-manifold is one that can't be reduced any further in terms of simple connect sums or decomposition. In other words, any embedded 2-sphere bounds an embedded 3-ball.

• Geometric structures are specific types of differential structures that a manifold can have. Thurston identified eight types of geometric structures for 3-manifolds:

  1. Euclidean 3-space (E^3)
  2. Hyperbolic 3-space (H^3)
  3. Spherical 3-space (S^3)
  4. The product of the plane and the real line (E^2 x R)
  5. The product of the hyperbolic plane and the real line (H^2 x R)
  6. The product of the sphere and the real line (S^2 x R)
  7. The universal cover of SL(2,R) (SL(2,R))
  8. The Heisenberg group (also known as Nil)

The Geometrization Theorem, along with its corollary the Poincaré Conjecture, brought profound impacts to the field of 3-manifold topology and led to an increased understanding of the structures and classifications of 3-manifolds.