This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e., 3-manifolds with. This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e. Thurston’s Geometrization Conjecture (now, a theorem of Perelman) aims to answer the question: How could you describe possible shapes of our universe?.

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Geometrization conjecture – Wikipedia

Thurston’s conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds.

Before stating Thurston’s geometrization conjeccture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decompositionwhich says that every compact three- manifold geomerrization the connected sum of a unique collection of prime three- manifolds.


The second decomposition is the Jaco-Shalen-Johannson torus decompositionwhich states that irreducible orientable compact 3- manifolds have a canonical up to isotopy minimal collection of disjointly embedded incompressible tori such that each component of the 3- manifold removed by the tori is either “atoroidal” or “Seifert-fibered.

geometrization conjecture in nLab

Thurston’s conjecture is that, after you split a three- manifold into its connected sum and the Jaco-Shalen-Johannson torus decompositionthe remaining conjecure each admit exactly one of the following geometries:. The geometry of. The geometry of the universal cover of the Lie group.

Here, is the 2- sphere in a topologist’s sense and is the hyperbolic plane. Thurston shared the Fields Medal for work done in proving that conjecfure conjecture held in a subset of these cases.

Six of these geometries are now well understood, and there has been a great deal of progress with hyperbolic geometry the geometry of constant negative scalar curvature.

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Collection of teaching and learning tools built by Wolfram education experts: Mon Dec 31 Euclidean geometry2. Hyperbolic geometry3.

Spherical geometry4. The geometry of5.

The geometry of6. The geometry of the universal cover of conjeccture Lie group7. Nil geometryor 8. Contact the MathWorld Team.