Editing William Thurston (section)

=== The geometrization conjecture ===
{{Main|Geometrization conjecture}}
{{See also|Thurston's 24 questions}}
{{Refimprove section|date=March 2022}}
His later work, starting around the mid-1970s, revealed that [[hyperbolic geometry]] played a far more important role in the general theory of [[3-manifold]]s than was previously realised.  Prior to Thurston, there were only a handful of known examples of [[hyperbolic 3-manifold]]s of finite volume, such as the [[Seifert–Weber space]].  The independent and distinct approaches of [[Robert Riley (mathematician)|Robert Riley]] and [[Troels Jørgensen]] in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the [[figure-eight knot (mathematics)|figure-eight knot]] [[knot complement|complement]] was [[hyperbolic link|hyperbolic]].  This was the first example of a [[hyperbolic knot]].

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the [[figure-eight knot]] complement.  He showed that the figure-eight knot complement could be [[manifold decomposition|decomposed]] as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure-eight knot complement.  By utilizing [[Wolfgang Haken|Haken]]'s [[normal surface]] techniques, he classified the [[incompressible surface]]s in the knot complement.  Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 [[Dehn surgery|Dehn surgeries]] on the figure-eight knot resulted in [[irreducibility (mathematics)|irreducible]], non-[[Haken manifold|Haken]] non-[[Seifert fiber space|Seifert-fibered]] 3-manifolds.  These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken.  These examples were actually hyperbolic and motivated his next theorem.

Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted  in hyperbolic 3-manifolds.  This is his celebrated [[hyperbolic Dehn surgery]] theorem.

To complete the picture, Thurston proved a [[hyperbolization theorem]] for [[Haken manifold]]s.  A particularly important corollary is that many knots and links are in fact hyperbolic.  Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.

The hyperbolization theorem for Haken manifolds has been called ''Thurston's Monster Theorem,'' due to the length and difficulty of the proof.<ref>{{Cite journal |last=Friedl |first=Stefan |date=2014-12-01 |title=Thurston's Vision and the Virtual Fibering Theorem for 3-Manifolds |url=https://link.springer.com/article/10.1365/s13291-014-0102-x |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |language=en |volume=116 |issue=4 |pages=223–241 |doi=10.1365/s13291-014-0102-x |issn=1869-7135}}</ref>  Complete proofs were not written up until almost 20 years later.  The proof involves a number of deep and original insights which have linked many apparently disparate fields to [[3-manifold]]s.

Thurston was next led to formulate his [[geometrization conjecture]].  This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries.  Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. The conjecture was proved by [[Grigori Perelman]] in 2002–2003.<ref>{{cite arXiv|last=Perelman|first=Grisha|date=2003-03-10|title=Ricci flow with surgery on three-manifolds|eprint=math/0303109}}</ref><ref>{{Cite journal|last1=Kleiner|first1=Bruce|last2=Lott|first2=John|date=2008-11-06|title=Notes on Perelman's papers|arxiv=math/0605667|journal=Geometry & Topology|volume=12|issue=5|pages=2587–2855|doi=10.2140/gt.2008.12.2587|issn=1364-0380|doi-access=free}}</ref>