---
date: 2021-10-11 17:52
modification date: Tuesday 19th October 2021 21:55:47
title: Geometrization
aliases: [geometrization, geometrizable]

---

Tags: #todo #geomtop/3-manifolds

# Geometrization

## Smooth Category: Geometrization

3-manifolds: Thurston's Geometrization
- Define a **Geometric structure**: a diffeo $M\cong \tilde M/\Gamma$ where $\Gamma$ is a discrete [Lie group](Lie%20group.md) acting freely/transitively on $X$.
- Oriented prime [3-manifolds](Unsorted/Three-manifolds%20MOC.md) can be decomposed into geometric "pieces" of 8 possible types:
	- Spherical $\sim S^3$
	- Euclidean $\sim \RR^3$
	- Hyperbolic $\sim \HH^3$
	- $S^2\cross \RR$
	- $\HH^2\cross \RR$
	- $\tilde{\SL(2, \RR)}$
	- "Nil"
	- "Sol"
- Proved by Perelman 2003, [Ricci flow](Ricci%20flow.md) with [surgery](surgery.md).
- 4-manifolds: classified in the topological category by surgery, but not in the smooth category
  - Hard! Will examine special cases of [Calabi-Yau](Calabi-Yau.md)
  - Open part of [Poincare conjectures](Unsorted/Poincare%20conjectures.md).
- Dimension $\geq 5$: [surgery theory](surgery%20theory), diffeomorphic  $\iff$ [s-cobordant|](s-cobordism.md)