Editing Geometrization conjecture (section)

==Higher dimensions==
In four dimensions, only a rather restricted class of closed [[4-manifold]]s admit a geometric decomposition.<ref name=hillman>{{cite arXiv|last=Hillman|first=Jonathan|author-link=|date=13 November 2022|title=Four-manifolds, geometries and knots|eprint=math/0212142|class=}}<!--section 13.3, example 3 gives a Seifert manifold with no geometric decomposition--></ref> However, lists of maximal model geometries can still be given.<ref name=filipkiewicz/>

The four-dimensional maximal model geometries were classified by Richard Filipkiewicz in 1983. They number eighteen, plus one countably infinite family:<ref name=filipkiewicz>{{cite thesis |last=Filipkiewicz |first=Richard |date=1983 |title=Four dimensional geometries |url=https://wrap.warwick.ac.uk/954/ |degree=PhD |chapter= |publisher=University of Warwick |docket= |oclc= |access-date=31 January 2024}}</ref> their usual names are '''E'''<sup>4</sup>, Nil<sup>4</sup>, {{nowrap|Nil<sup>3</sup> × '''E'''<sup>1</sup>}}, {{nowrap|Sol{{su|p=4|b=''m'',''n''}}}} (a countably infinite family), {{nowrap|Sol{{su|p=4|b=0}}}}, {{nowrap|Sol{{su|p=4|b=1}}}}, {{nowrap|'''H'''<sup>3</sup> × '''E'''<sup>1</sup>}}, {{nowrap|<math>{\widetilde{\rm{SL}}}</math> × '''E'''<sup>1</sup>}}, {{nowrap|'''H'''<sup>2</sup> × '''E'''<sup>2</sup>}}, {{nowrap|'''H'''<sup>2</sup> × '''H'''<sup>2</sup>}}, '''H'''<sup>4</sup>, '''H'''<sup>2</sup>('''C''') (a [[complex hyperbolic space]]), '''F'''<sup>4</sup> (the [[tangent bundle]] of the hyperbolic plane), '''S'''<sup>2</sup> × '''E'''<sup>2</sup>, {{nowrap|'''S'''<sup>2</sup> × '''H'''<sup>2</sup>}}, {{nowrap|'''S'''<sup>3</sup> × '''E'''<sup>1</sup>}}, '''S'''<sup>4</sup>, '''CP'''<sup>2</sup> (the [[complex projective plane]]), and {{nowrap|'''S'''<sup>2</sup> × '''S'''<sup>2</sup>}}.<ref name=hillman/> No closed manifold admits the geometry '''F'''<sup>4</sup>, but there are manifolds with proper decomposition including an '''F'''<sup>4</sup> piece.<ref name=hillman/>

The five-dimensional maximal model geometries were classified by Andrew Geng in 2016. There are 53 individual geometries and six infinite families. Some new phenomena not observed in lower dimensions occur, including two uncountable families of geometries and geometries with no compact quotients.<ref name=geng>{{cite arXiv|last=Geng|first=Andrew|author-link=|date=9 June 2016|title=5-dimensional geometries I: the general classification|eprint=1605.07545|class=math.GT}}</ref>