0.1 Thurston’s Geometrization Theorem and Conjecture

0.1.1 History

Haken manifolds are named after Wolfgang Haken (b.1928) because their definition involves the key concept of incompressible surfaces, and also because Wolfgang Haken has pioneered the use of incompressible surfaces in topology.

At the University of Illinois at Urbana-Champaign, Haken and his colleague Kenneth Appel, solved in 1976 one of the most famous problems in mathematics by proving the four-color theorem. He also proved that Haken manifolds have a hierarchy. The hierarchy facilitates the proof of several theorems about about Haken manifolds through logical/mathematical induction: first, one proves a theorem for 3-balls, then one proves that if a theorem is true for pieces resulting by “cutting” a Haken manifold, then it is also true for the whole Haken manifold; it is essential that such “cutting” is selected along a surface that was incompressible. This makes possible the proof by induction in most cases by proceeding from one induction step to the next. Thus, Haken showed that there is a finite procedure to find an incompressible surface if the 3-manifold had one; 20 years later, Jaco and Oertel showed again by utilizing induction that there was an algorithm to determine if a 3-manifold was Haken. Because according to Jaco and Oertel there exists an algorithm to find out if a 3-manifold is Haken, the fundamental topological problem of recognizing 3-manifolds can be considered to be solved for the category of Haken manifolds. Furthermore, Friedhelm Waldhausen proved that closed Haken manifolds are topologically rigid, and thus such 3-manifolds are completely determined by their fundamental group. One can also conjecture that,in general, CW–complexes C_wh of closed Haken manifolds, M_cH, are completely determined by the fundamental groupoid functor ℱ_G associated with the category of C_wh and CW-homeomorphisms.

0.2 Thurston’s Geometrization Theorem

William Thurston reported his proof of the geometrization theorem in 1980, and several complete proofs have been published since. Furthermore, in 2003, Grigori (Grisha) Perelman sketched a proof of the general, full geometrization conjecture using Ricci flows with surgery; his proof of the full geometrization conjecture –as reported by specialized mathematicians in 2008– is said to be essentially correct.

Thurston’s geometrization theorem, also called “the Hyperbolization Theorem”, is stated as follows:

In essence, the Thurston geometrization theorem (TGT) stated as above was proven by him as a proof of his geometrization conjecture just for the special case of Haken manifolds. A very important corollary of TGT is that many knots and links are in fact hyperbolic. Taken together with his hyperbolic Dehn surgery theorem, the TGT corollary showed that closed hyperbolic 3-manifolds abound. The (TGT) geometrization theorem is sometimes called in mathematical circles “Thurston’s Monster Theorem”, both because of the length and the difficulty of its proof. Complete proofs of TGT were published only 20 years later than the initial report by Thurston in 1980. Such proofs involve several original, profound insights that link several apparently distinct fields of mathematics to 3-manifolds.

In 1981, Thurston announced the orbifold theorem, which is an extension of his geometrization theorem in the setting of 3-orbifolds instead of Haken manifolds. Twenty years later, two teams of mathematicians suceeded to complete a proof of Thurston’s orbifold theorem that was in essence built upon Thurston’s lectures presented in Princeton in 1980 involving his original proof that relied partially on Richard Hamilton’s work on the Ricci flow.

0.3 Thurston’s Geometrization Conjectures

Thurston proposed this more general, geometrization conjecture in 1982 after proving his geometrization theorem. The same year, he was awarded the Fields Medal “for the depth and originality of his contributions to mathematics.”

This geometrization conjecture can be simply stated as follows:

Thurston Conjecture: Compact 3-manifolds can be decomposed into submanifolds that have geometric structures.

The geometrization conjecture (GC) can be considered as a 3-manifold analogue of the uniformization theorem for 2D-surfaces; GC indicates that all 3-manifolds admit a certain kind of geometric decomposition involving eight special geometries, now called Thurston model geometries; the hyperbolic geometry is perhaps the most important of the eight model geometries and seems to raise the most complex problems in this context.

Thurston’s geometrization conjecture implies several other conjectures, such as, for example, Thurston’s elliptization conjecture, and also the Poincaré conjecture. The Poincaré Conjecture that aimed at a topological characterization of the 3-sphere, has been for over 100 years one of the central unresolved questions in topology; since its formulation in 1904, Poincaré conjecture has been repeatedly approached, without success, using various topological methods. Because of its importance and difficulty it was chosen by the Clay Research Institute as one of the seven “Clay Millennium Problems” in Mathematics).

The Poincaré Conjecture:

If a space is homotopically equivalent to a three-dimensional sphere then it is also homeomorphic to the three-sphere.

Thurston’s gemetrization conjecture– that was proven later by Grigori Perelman– solved in the affirmative Poincaré ’s 1904 conjecture. Perelman was awarded in August 2006 the Fields Medal for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”. (Perelman, however, declined to either accept the Fields Medal award or to appear at the congress where it was supposed to be presented to him.)

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