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Spherical space form conjecture
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{{Infobox mathematical statement | name = Spherical space form conjecture | image = | caption = | field = [[Geometric topology]] | conjectured by = [[Heinz Hopf]] | conjecture date = 1926 | first proof by = [[Grigori Perelman]] | first proof date = 2006 | implied by = [[Geometrization conjecture]] | equivalent to = [[Poincaré conjecture]]<br>[[Thurston elliptization conjecture]] | generalizations = | consequences = }} In [[geometric topology]], the '''spherical space form conjecture''' (now a theorem) states that a [[finite group]] acting on the [[3-sphere]] is conjugate to a [[group of isometries]] of the 3-sphere. ==History== The conjecture was posed by [[Heinz Hopf]] in 1926 after determining the fundamental groups of three-dimensional spherical space forms as a generalization of the [[Poincaré conjecture]] to the non-simply connected case.<ref>{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Zum Clifford-Kleinschen Raumproblem | doi=10.1007/BF01206614 | year=1926 | journal=[[Mathematische Annalen]] | volume=95 |issue=1 | pages=313–339}}</ref><ref>{{Citation | last1=Hambleton| first1=Ian | title=Handbook of Group Actions | publisher=ALM |location=Beijing-Boston | series=Clay Math. Proc. | year=2015 | volume=3 | chapter=Topological spherical space forms | pages=151–172}}</ref> ==Status== The conjecture is implied by [[William Thurston|Thurston]]'s [[geometrization conjecture]], which was proven by [[Grigori Perelman]] in 2003. The conjecture was independently proven for groups whose actions have [[Fixed point (mathematics)|fixed points]]—this special case is known as the [[Smith conjecture]]. It is also proven for various groups acting without fixed points, such as [[cyclic group]]s whose orders are a power of two (George Livesay, Robert Myers) and cyclic groups of order 3 ([[J. Hyam Rubinstein]]).<ref>{{Citation | last1=Hass | first1=Joel | authorlink = Joel Hass | title=Global theory of minimal surfaces | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Clay Math. Proc. |mr=2167285 | year=2005 | volume=2 | chapter=Minimal surfaces and the topology of three-manifolds | pages=705–724}}</ref> ==See also== *[[Killing–Hopf theorem]] ==References== {{reflist}} {{DEFAULTSORT:Spherical Space Form Conjecture}} [[Category:Conjectures that have been proved]] [[Category:Geometric topology]] {{topology-stub}}
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