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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Thurston’s conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. Before stating Thurston’s geometrization conjecture 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 is 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 geometrizwtion the tori is either “atoroidal” or “Seifert-fibered.
 Completion of the Proof of the Geometrization Conjecture
Thurston’s conjecture is that, after you split a three- manifold into its connected sum and geommetrization Jaco-Shalen-Johannson torus decompositionthe remaining components 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.
The Geometrization Conjecture
Thurston shared the Fields Medal for work done in proving that the 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. Publish or Conjecutre Press, p.
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Practice online or make geometrziation printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: Mon Dec 31 Euclidean geometry2.
Hyperbolic geometry3. Spherical geometry4.
Thurston’s Geometrization Conjecture — from Wolfram MathWorld
The geometry of5. The geometry of6. The geometry of the universal cover of the Lie group7.
Nil geometryor 8. Contact the MathWorld Team.