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1、BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume42,Number1,Pages57–78S0273-0979(04)01045-6ArticleelectronicallypublishedonOctober29,2004RECENTPROGRESSONTHEPOINCARECONJECTUREAND´THECLASSIFICATIONOF3-MANIFOLDSJOHNW.MORGANIntroductionMotivatedbywhatwaswell
2、-knownatthetimeaboutsurfaces,andafteralongtopologicalstudyof3-manifolds,attheveryendofhis1904paperPoincar´e[18]statesthatthereremainsonequestiontotreat,namely(reformulatedinmodernlanguage):IfMisaclosed3-manifoldwithtrivialfundamentalgroup,thenisMdiffeomorphic1toS
3、3?ThePoincar´eConjectureisthattheanswertothisquestionis“Yes.”De-velopingtoolstoattackthisproblemformedthebasisformuchoftheworkin3-dimensionaltopologyoverthelastcentury,includingforexample,theproofofDehn’slemmaandthelooptheoremandthestudyofsurgeryonknotsandlinks.
4、Inthe1980’sThurstondevelopedanotherapproachto3-manifolds,see[24]and[4].Heconsidered3-manifoldswithriemannianmetricsofconstantnegativecur-vature−1.Thesemanifolds,whicharelocallyisometrictohyperbolic3-space,arecalledhyperbolicmanifolds.Therearefairlyobviousobstruc
5、tionsshowingthatnotevery3-manifoldcanadmitsuchametric.Thurstonformulatedagen-eralconjecturethatroughlysaysthattheobviousobstructionsaretheonlyones;shouldtheyvanishforaparticular3-manifoldthenthatmanifoldadmitssuchametric.Hisproofofvariousimportantspecialcasesoft
6、hisconjectureledhimtoformulateamoregeneralconjectureabouttheexistenceoflocallyhomoge-neousmetrics,hyperbolicorotherwise,forallmanifolds;thisiscalledThurston’sGeometrizationConjecturefor3-manifolds.Thestatementofthisconjectureissomewhatcomplicated,soitisdeferredu
7、ntilSection2.AnimportantpointisthatThurston’sGeometrizationConjectureincludesthePoincar´eConjectureasaveryspecialcase.Inaddition,Thurston’sconjecturehastwoadvantagesoverthePoincar´eConjecture:•Itappliestoallclosedoriented3-manifolds.•Itpositsacloserelationshipbe
8、tweentopologyandgeometryindimensionthree.ReceivedbytheeditorsJune11,2004,and,inrevisedform,September1,2004.2000MathematicsSubjectClassification.Primary57M50,57M27,58J3