Geometrisation of 3-manifolds
type: Item
instance of: scholarly article
title: Geometrisation of 3-manifolds
zbMATH Open document ID: 1244.57003
DOI: 10.4171/082
author: L. Bessières
author: G. Besson
author: S. Maillot
author: Michel Boileau
author: Joan Porti
published in: EMS Tracts in Mathematics
publication date: 2010-10-05T00:00:00Z
review text: The Kneser decomposition and JSJ-splitting theorems imply that every closed, orientable \(3\)-manifold can be canonically split by a (possibly empty) collection of essential \(2\)-spheres and tori into \(3\)-submanifolds that are atoroidal or Seifert fibered. The Thurston Geometrization Conjecture is that (after capping off the boundary spheres with 3-balls) each piece of the Kneser-JSJ splitting is geometric. Here a compact \(3\)-manifold is \textit{geometric}, if its interior is the quotient of a geometry \(X\) (a homogeneous, simply connected, unimodular Riemann manifold) by a free action of a discrete subgroup of the isometry group of \(X\). Thurston showed that there are eight relevant model geometries for compact orientable \(3\)-manifolds (six of these yield Seifert fiber spaces, one is hyperbolic geometry, one is Euclidean geometry) and proved the Conjecture for Haken-manifolds (i.e. the irreducible \(3\)-manifolds that have either non-empty boundary or contain a closed essential surface); in particular, any atoroidal Haken-manifold is hyperbolic or Seifert fibered. The full Geometrization Conjecture was proved by Grigori Perelman in 2003. A consequence is the Elliptization Conjecture, that a closed 3-manifold with finite fundamental group is spherical, with the Poincaré Conjecture as a corollary. Perelman's proof is based on Hamilton's theory of Ricci flow with surgery. The main goal of the book under review is to present a proof that is more attractive to topologists by accepting Thurston's proof for the the case of Haken manifolds and concentrating on the proof of the following Theorem: Let \(M\) be a closed, orientable, irreducible, atoroidal \(3\)-manifold. Then (i) If \(\pi_1 (M)\) is finite, \(M\) is spherical. (ii) If \(\pi_1 (M)\) is infinite, \(M\) is hyperbolic or Seifert fibered. The strategy for the proof is well-outlined in the first chapter of the book with the definitions and proofs occupying the remaining chapters that form parts I--IV of th…
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 5796326
zbMATH Keywords: Geometrization Conjecture
zbMATH Keywords: Perelman
zbMATH Keywords: Ricci Flow
reviewed by: Wolfgang Heil
MaRDI profile type: Publication
OpenAlex ID: W4298067916
description: scientific article; zbMATH DE number 5796326