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1、Heegaard分解的可约性14(3)(1998).25626Z,一,6SomeReduc.ibilitiesofHeegaardSplittingsH.1d丐约恤LeiFengcl~un(雷逢春)andZhangYing(张影)(Dep口rfofMath～ematics,JillnUn,gchun~,1300A)AbstractInthispaper,wediscuss3-reducibility,weak3-reducibility,A—reducibilityandweakA—redueitilityforHeegaardsplittingsandtheirrelations
2、hips.Inparticular,weextendCasson—Gordon'stheoremonreducLbteHeegaardsplLtfings.AU3-manifoldsandsurfacesareassumedtobecompactandorientable.andallconceptsandnotationsnotdefinedinthepaperarestandard;see,forexample[1,2].AcompressionbodyHisconstructedbyadding2-handlestoS×1alongacollectionofpairwised
3、isjointsimpleclosedcurvesonS×{0),andcappingoffanyresulting2一sphereboundarycomponentswith3一balls,whereSisaconnectedclosedorientablesurface.ThecomponentS×{1)of羽isdenotedbyHandthesurfaceaH—a+H,whichmayormaynotbeconnected,isdenotedbyH.If丑H一,thenHisahandlebody.IfH一H×1.thenHiscalledatrivialcompressi
4、onbody.Notethatacompressionbodyisalwaysirreducible.LetFbeaclosedconnectedsurfaceimbeddedina3一manifoldM.FiacalledasplittingsurfaceforaHeegaardsplittingofMifFdividesMintotwocompressionbodiesH1andH.witha+H1一F—a+H2,TheHeegaardsplittingisdenotedbyHUFH2or(M,F).LetH1UFH2beaHeegaardsplittingofM.Recall
5、thatH1UFH2isreducibleifthereexistessentialproperlyimbeddeddisksDLinH1andD2inH2with319一9D2,andisweaklyreducibleifthereexistessentialproperlyimbeddeddisksDlinHandD8inH.withOD1n0192一,ItisobviousthatareducibleHeegaardsplittingisweaklyreducible.AsapartialCoi3verse,oneofCasson—Gordon'smainresults(se
6、e[33)statesthatifH1UFH2isweaklyreducible,theneitherHLUFH2isreducibleorMcontainsconnectedincompressiblesurfacesofpositivegenusThistheoremhasbeensuccessfullyappliedtoReceivedJune5.199*)ProjectpartiallysupportedbyNSFofChina蛊流一～～州NO3LeiFengchunandZhangYingSOMEREDUCIBILITIESOFHEEGAARDSPLITTINGS257d
7、ealwithmanyproblemsrelatedtoHeegaardsplittingsandincomopressiblesurfacesin3-manifolds,seeforexample[3—6].Inthispaper,wewilldiscussrelationshipsamongreducibilitiesinamoregeneralversion.LetHbeacompressionbody.AproperlyimbeddedannulusAinHi
