Recent progress on the Poincar&amp#233; conjecture and the classification of 3-manifolds (1).PDF

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1、BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume42,Number1,Pages57–78S0273-0979(04)01045-6ArticleelectronicallypublishedonOctober29,2004RECENTPROGRESSONTHEPOINCARECONJECTUREAND´THECLASSIFICATIONOF3-MANIFOLDSJOHNW.MORGANIntroductionMotivatedbywhatw

2、aswell-knownatthetimeaboutsurfaces,andafteralongtopologicalstudyof3-manifolds,attheveryendofhis1904paperPoincar´e[18]statesthatthereremainsonequestiontotreat,namely(reformulatedinmodernlanguage):IfMisaclosed3-manifoldwithtrivialfundamentalgroup,thenisMdiffe

3、omorphic1toS3?ThePoincar´eConjectureisthattheanswertothisquestionis“Yes.”De-velopingtoolstoattackthisproblemformedthebasisformuchoftheworkin3-dimensionaltopologyoverthelastcentury,includingforexample,theproofofDehn’slemmaandthelooptheoremandthestudyofsurge

4、ryonknotsandlinks.Inthe1980’sThurstondevelopedanotherapproachto3-manifolds,see[24]and[4].Heconsidered3-manifoldswithriemannianmetricsofconstantnegativecur-vature−1.Thesemanifolds,whicharelocallyisometrictohyperbolic3-space,arecalledhyperbolicmanifolds.Ther

5、earefairlyobviousobstructionsshowingthatnotevery3-manifoldcanadmitsuchametric.Thurstonformulatedagen-eralconjecturethatroughlysaysthattheobviousobstructionsaretheonlyones;shouldtheyvanishforaparticular3-manifoldthenthatmanifoldadmitssuchametric.Hisproofofv

6、ariousimportantspecialcasesofthisconjectureledhimtoformulateamoregeneralconjectureabouttheexistenceoflocallyhomoge-neousmetrics,hyperbolicorotherwise,forallmanifolds;thisiscalledThurston’sGeometrizationConjecturefor3-manifolds.Thestatementofthisconjecturei

7、ssomewhatcomplicated,soitisdeferreduntilSection2.AnimportantpointisthatThurston’sGeometrizationConjectureincludesthePoincar´eConjectureasaveryspecialcase.Inaddition,Thurston’sconjecturehastwoadvantagesoverthePoincar´eConjecture:•Itappliestoallclosedoriente

8、d3-manifolds.•Itpositsacloserelationshipbetweentopologyandgeometryindimensionthree.ReceivedbytheeditorsJune11,2004,and,inrevisedform,September1,2004.2000MathematicsSubjectClassification.Primary57M50,57M27,58J3