Completion of the Proof of the Geometrization Conjecture

TL;DR

This paper completes the proof of the Geometrization Conjecture for all compact, orientable three-manifolds by analyzing Ricci flow with surgery and using Alexandrov space theory to understand collapsed regions.

Contribution

It provides a comprehensive proof of the Geometrization Conjecture, building on Ricci flow techniques and Alexandrov space theory to handle collapsed regions in three-manifolds.

Findings

Proved the Geometrization Conjecture for all compact, orientable 3-manifolds.

Showed collapsed parts are graph manifolds with incompressible boundary.

Established local models for collapsed regions using Alexandrov spaces.

Abstract

This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov spaces. This theory gives local models for the collapsed part of the manifold. These local models can be glued together to prove that the collapsed part of the manifold is a graph manifold with incompressible boundary. From this and previous results, geometrization follows easily.