# Compl\'ement to the Thurston 3D-Geometrization

## TL;DR

This paper extends Thurston's Geometrization for 3-manifolds by incorporating affine gluing respecting a new regional Lie generated geometry, ultimately addressing a long-standing question related to the Poincaré Conjecture.

## Contribution

It introduces a novel affine gluing method compatible with regional Lie generated geometry, providing a new perspective on Thurston's Geometrization and solving a 45-year-old problem.

## Key findings

- Develops a new notion of regional Lie generated geometry.
- Ensures affine gluing respects the metric geometry interpretation.
- Addresses a long-standing question related to the Poincaré Conjecture.

## Abstract

Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four particular Thurston geometries" (see Introduction and Figure 1). These geometric components have finite volume for the hyperbolic geometries (the H labeled vertices). They also have finite volume for each of the two geometries appearing as Seifert fibrations (the S labeled vertices). The remaining pieces (the I labeled vertices) have Euclidean geometries of linear volume growth. Then these vertex geometries are combined topologically to recover the original manifold. This, by cutting off the toroidal ends and then gluing the torus boundaries by affine mappings (indicated by the labeled edges in Figure 1). The point of this work is to make the affine gluing respect an interpretation of the metric geometry in terms of a new notion of `` regional Lie generated geometry". The vertex regions use four geometries in Lie form combined in the overlap edge regions via affine geometry. The Theorem solves, using Geometrization, a 45 year old question/approach to the Poincar\'{e} Conjecture. This was described in a '76 Princeton Math dept. preprint and finally documented in the 1983 reference by Thurston and the second author.