Fundamental groups of manifolds with positive isotropic curvature

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1、AnnalsofMathematics,158(2003),345–354FundamentalgroupsofmanifoldswithpositiveisotropiccurvatureByAilanaM.Fraser*AbstractAcentralthemeinRiemanniangeometryisunderstandingtherelation-shipsbetweenthecurvatureandthetopologyofaRiemannianmanifold.Positiveisotropiccurvature(PIC)isanaturalandmuchstudiedcu

2、rvatureconditionwhichincludesmanifoldswithpointwisequarter-pinchedsectionalcurvaturesandmanifoldswithpositivecurvatureoperator.BytheresultsofMicallefandMoorethereisonlyonetopologicaltypeofcompactsimplycon-nectedmanifoldwithPIC;namelyanysuchmanifoldmustbehomeomorphictothesphere.Ontheotherhand,ther

3、eisalargeclassofnonsimplyconnectedmanifoldswithPIC.AnimportantopenproblemhasbeentounderstandthefundamentalgroupsofmanifoldswithPIC.Inthispaperweproveanewresultinthisdirection.WeshowthatthefundamentalgroupofacompactmanifoldMnwithPIC,n≥5,doesnotcontainasubgroupisomorphictoZ⊕Z.Thetechniquesusedinvol

4、veminimalsurfaces.1.IntroductionItisafundamentalproblemingeometrytodeterminetherelationshipsbetweenthecurvatureandtopologyofmanifolds.Inthispaperwestudythefundamentalgroupsofcompactmanifoldswithpositiveisotropiccurvature(PIC).PICisanaturalandmuchstudiedcurvaturecondition,whichfirstarXiv:math/04033

5、48v1[math.DG]22Mar2004deriveditsimportancefromthefollowingbeautifultheoremofMicallefandMoore[MM]:Theorem1.1(Micallef-Moore).LetMbeacompactn-dimensionalRie-mannianmanifoldwithPIC,n≥4.Thenπ(M)=0fork=2,...,[n](wherek2[·]denotestheintegerpartofthenumber).Inparticular,ifMissimplycon-nected,thenMishome

6、omorphictoasphere.∗TheauthorwaspartiallysupportedbyNSFGrantDMS-9971927.346AILANAM.FRASERIfMisann-dimensionalRiemannianmanifold,onecanconsiderthecom-plexificationTpM⊗CofthetangentspaceTpMatthepointp.Theinnerprod-uctonthetangentspaceTpMcanbeextendedtothecomplexifiedtangentspaceTpM⊗Casacomplexbilinear

7、form(·,·)orasaHermitianinnerprod-ucth·,·i.Therelationshipbetweentheseextensionsisgivenbyhv,wi=(v,w¯)forv,w∈TpM⊗C.Thecurvaturetensorextendstocomplexvectorsbylinearity,andthecomplexsectionalcurvatureofatwo-dimensionalsub