Editing Geometrization conjecture (section)

==History==
The [[Fields Medal]] was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for [[Haken manifold]]s.

In 1982, [[Richard S. Hamilton]] showed that given a closed 3-manifold with a metric of positive [[Ricci curvature]], the [[Ricci flow]] would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse.  He later developed a program to prove the geometrization conjecture by [[Ricci flow with surgery]]. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries '''S'''<sup>3</sup> and '''S'''<sup>2</sup> × '''R''', while what is left at large times should have a [[thick-thin decomposition|thick–thin decomposition]] into a "thick" piece with hyperbolic geometry and a "thin" [[graph manifold]].

In 2003, [[Grigori Perelman]] announced a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above. 

One component of Perelman's proof was a novel collapsing theorem in Riemannian geometry. Perelman did not release any details on the proof of this result (Theorem 7.4 in the preprint 'Ricci flow with surgery on three-manifolds'). Beginning with Shioya and Yamaguchi, there are now several different proofs of Perelman's collapsing theorem, or variants thereof.<ref>{{cite journal |first1=T. |last1=Shioya |first2=T. |last2=Yamaguchi |title=Volume collapsed three-manifolds with a lower curvature bound |journal=Math. Ann. |volume=333 |year=2005 |issue=1 |pages=131–155 |doi=10.1007/s00208-005-0667-x |arxiv=math/0304472 |s2cid=119481 }}</ref>{{sfnm|1a1=Morgan|1a2=Tian|1y=2014}}<ref>{{cite journal |last1=Kleiner |first1=Bruce |last2=Lott |first2=John |title=Locally collapsed 3-manifolds |journal=Astérisque |date=2014 |volume=365 |issue=7–99}}</ref><ref>{{cite journal |last1=Cao |first1=Jianguo |last2=Ge |first2=Jian |title=A simple proof of Perelman's collapsing theorem for 3-manifolds |journal=J. Geom. Anal. |date=2011 |volume=21 |issue=4 |pages=807–869|doi=10.1007/s12220-010-9169-5 |arxiv=1003.2215 |s2cid=514106 }}</ref> Shioya and Yamaguchi's formulation was used in the first fully detailed formulations of Perelman's work.{{sfnm|1a1=Cao|1a2=Zhu|1y=2006|2a1=Kleiner|2a2=Lott|2y=2008}}

A second route to the last part of Perelman's proof of geometrization is the method of [[Laurent Bessières]] and co-authors,<ref>{{cite arXiv |first1=L. |last1=Bessieres |first2=G. |last2=Besson |first3=M. |last3=Boileau |first4=S. |last4=Maillot |first5=J. |last5=Porti |title=Weak collapsing and geometrization of aspherical 3-manifolds |eprint=0706.2065 |class=math.GT |year=2007 }}</ref><ref>{{cite journal |first1=L. |last1=Bessieres |first2=G. |last2=Besson |first3=M. |last3=Boileau |first4=S. |last4=Maillot |first5=J. |last5=Porti |title=Collapsing irreducible 3-manifolds with nontrivial fundamental group |journal=[[Inventiones Mathematicae|Invent. Math.]] |volume=179 |issue=2 |year=2010 |pages=435–460 |doi=10.1007/s00222-009-0222-6 |bibcode=2010InMat.179..435B |s2cid=119436601 }}</ref> which uses Thurston's hyperbolization theorem for Haken manifolds and [[Mikhail Leonidovich Gromov|Gromov]]'s [[norm (mathematics)|norm]] for 3-manifolds.<ref>{{cite book |first=J.-P. |last=Otal |chapter=Thurston's hyperbolization of Haken manifolds |title=Surveys in differential geometry |volume=III |location=Cambridge, MA |pages=77–194 |publisher=Int. Press |year=1998 |isbn=1-57146-067-5 }}</ref><ref>{{cite journal |author-link=Mikhail Leonidovich Gromov |first=M. |last=Gromov |title=Volume and bounded cohomology |journal=Inst. Hautes Études Sci. Publ. Math. |issue=56 |pages=5–99 |year=1983 }}</ref> A book by the same authors with complete details of their version of the proof has been published by the [[European Mathematical Society]].<ref>L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. Available at https://www-fourier.ujf-grenoble.fr/~besson/book.pdf</ref>