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 acting freely/transitively on \(X\). - Oriented prime 3-manifolds can be decomposed into geometric “pieces” of 8 possible types: - Spherical \(\sim S^3\) - Euclidean \(\sim {\mathbf{R}}^3\) - Hyperbolic \(\sim {\mathbb{H}}^3\) - \(S^2\times{\mathbf{R}}\) - \({\mathbb{H}}^2\times{\mathbf{R}}\) - \(\tilde{{\operatorname{SL}}(2, {\mathbf{R}})}\) - “Nil” - “Sol” - Proved by Perelman 2003, Ricci flow with surgery. - 4-manifolds: classified in the topological category by surgery, but not in the smooth category - Hard! Will examine special cases of Calabi-Yau - Open part of Poincare conjectures. - Dimension \(\geq 5\): surgery theory, diffeomorphic \(\iff\) s-cobordant|