4SIDDHARTHAGADGILToconstructnonorientable3manifoldsonegluesnonorientablehandlebodiesofthesamegenusalongtheirboundariesAfundamentaltheoremassertsthattheseconstructionsgiveall3manifoldsTheorem2E ID: 176974
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2SIDDHARTHAGADGILMoregenerally,weshallgluesocalledhandlebodiesalongtheirboundaries.Weshallseethatallclosed3-manifoldscanbeobtainedinthismanner.First,welookmorecloselyatthesimplestofthesehandlebodiestogetournextexample.2.Abasicexample2.1.Thesolidtorus.ThesolidtorusistheproductD2S1.Itisusefultolookatitfromacoupleofotherpointsofview.Firstly,onecanobtainasolidtorusfromB3byattachinga1-handle.Namely,toapairofdisjointdiscsinS2=@B3,gluetheboundaryf0;1gD2of[0;1]D2,insuchawayastoobtainanorientablemanifold.ThereisanotherwaytoobtainthesolidtorusfromB3.Takeadiameter inthesolidtorus.Deletetheinteriorofaregularneighbourhoodofthisarc.ItiseasytoseethatB3nint(N( ))=D2S1.Moregenerally,wecantakeanyproperly-embeddedunknottedarc B3,i.e.,anarc suchthatthereexistsanarcembeddedinS2=@B3andanembeddeddiscEB3suchthat@E= [.Ondeletingtheinteriorofaneighbourhoodofthisarc,wegetasolidtorus.Boththesedescriptionsplayakeyroleinwhatfollows.2.2.S3oncemore.WenowgetamoreinterestingexampleofaHeegardsplitting.Namely,asbeforeS3=B1[B2,wheretheBiare3-ballsgluedtogetheralongtheirboundaries.Nowlet beanunknottedproperlyembeddedarcinB1.LetH1=B1nint(N( ))andH2=B2[N( ).SinceH1istheresultofdeletinganopenneighbourhoodofanunknottedarcfroma3-ball,itisasolidtorus.Ontheotherhand,H2isobtainedbyaddinga1-handletoa3-ball,andishencealsoasolidtorus.Thus,S3istheunionoftwosolidtori,gluedalongtheirboundary.ThisisanotherexampleofaHeegardsplittingofS3.IfonethinksofS3astheone-pointcompacticationofR3,thenH1isjusttheregularneighbourhoodofanunknot,andH2istheclosureofitscomplement.Itisoftenuseful,in3-manifoldtopology,tothinkofS3astheunitballinC2.Onecanseetheabovedecompositionformthispointofview.Exercise2.LetHiS3C2,i=1;2,begivenbyHi=f(z1;z2)2S2:jzij21=2g.ShowthatthisisadecompositionofS3intosolidtori.Wecanalsogluesolidtoritogetheralongotherdi eomorphismsoftheirbound-aries.OneobtainsinthismannerS2S1aswellasthelensspaces.Weshalltreattheseindetaillater. 4SIDDHARTHAGADGILToconstructnon-orientable3-manifolds,onegluesnon-orientablehandlebodiesofthesamegenusalongtheirboundaries.Afundamentaltheoremassertsthattheseconstructionsgiveall3-manifolds.Theorem2.Everytriangulated3-manifoldMhasaHeegardsplitting.Proof.LetMbea3-manifoldwithagiventriangulationT.ThetwohandlebodiesW1andW2intheHeegardsplittingweconstructwillberegularneighbourhoodsofthe1-skeletonTandthe1-skeletonofitsdualtriangulationT0.Consideraregularneighbourhoodofthe1-skeleton�,andcallitsboundaryF.ThenFseparatesMintotwopieces.Oneoftheseisaregularneighbourhoodofthe1-skeleton,andhenceisahandlebody.ByconsideringtheintersectionofFwitheachtetrahedron,itiseasytoseethattheothermanifoldobtainedisaregularneighbourhoodofthe1-skeletonofthedualtriangulation,andhenceisalsoahandlebody.ThuswehaveaHeegardsplitting.Remark3.ByatheoremofMoiseandBing,all3-manifoldshavetriangulations.Thustheaboveresultholdsforall3-manifolds.Example4.1.(HeegardsplittingsofS3)S3hasHeegardsplittingsofallgenera.Namely,werstexpressS3=B1[B2.Takegproperlyembeddedarcs 1;1iginB1thatareunknottedandunlinked.LetH1=B1n[gi=1int(N( i))andH2=B2[[gi=1N( i).ThenH1andH2arehandlebodies.TheseHeegardsplittingsofS3arecalledthestandardHeegardsplittingsofS3.ItisnaturaltoaskwhetherthereareanyHeegardsplittingsofS3di erentfromthese.ThenaturalequivalenceonHeegardsplittingsisisotopy.WesaythattwoHee-gardsplittingsofS3areisotopicifthecorrespondingHeegardsurfacesareisotopic.AtheoremofWaldhausenclassiesHeegardsplittingsofS3.Fortheproof,wereferto?Theorem4(Waldhausen).AnyHeegardsplittingofS3isisotopictoastandardone.Exercise7.ShowthatthestandardHeegardsplittingdoesnotdepend,uptoiso-topy,onthechoiceofthearcs i. 6SIDDHARTHAGADGILNotethatifF:W1!W1isahomeomorphismofthehandlebody,thentheimageofaHeegarddiagramunderFgivesanotherHeegarddiagramforthesamemanifold.Aparticularlyusefulsuchhomeomorphismistheso-calledDehntwist.IfDisaproperlyembeddeddiscinahandlebodyW,thenthisisthemapwhichisequaltotheidentityoutsideaneighbourhoodofD,andconsistsofafulltwistinaneighbourhoodofD.Denition5.1.SupposeDisaproperlyembeddeddiscinahandlebodyW.ThenaDehntwistaboutWisahomeomorphismthatistheidentityoutsideaneighborhoodD2[�1;1]ofD2andisisotopicto(z;t)7!(ze(t+1);t)onthisneighborhood.Example5.2.InthecaseofthesolidtorusD2S1,aDehntwistabouttheproperlyembeddeddiscD2f1gxesthemeridianandmaps7!1.Nowconsiderthegenus1HeegarddiagramforS3,withacurveboundingadiscgivenby.ByapplyingDehntwists,weseethatotherHeegarddiagramsofthespherearegivenbyattachingadisctothecurve+k;k2Z.Similarly,manyoftheHeegarddiagramsconstructedabovecorrespondtothesamelensspace.Exercise8.ShowthatL(p;q)=L(p;q+kp)Sofar,wehavenotevenshownthatthelensspacesarenotallS3.Todothis,wecomputethefundamentalgroupintermsofaHeegarddiagram.Recallthatthefundamentalgroupofahandlebodyofgenusgisafreegrouponggenerators i.EachofthecurvesinaHeegarddiagramrepresentawordriinthesegenerators,welldeneduptoconjugation.Proposition6.Apresentationof1(M)isgivenbyh 1;:::; g;r1;:::;rgi.Proof.BytheSeifert-VanKampentheorem,attachinga3-balldoesnotalterthefundamentalgroup.Thus,itsucestoconsiderthemanifold^M=MnB3.Themanifold^Misobtainedbyattachingtoaballg1-handlesandg2-handles.Itiseasytoseethat^Mdeformationretractsintoa2-complex,withauniquevertexcorrespondingtotheball,andedgeforeach1-handleanda2-cellforeachdisc.Thevertexandedgesformawedgeofgcircles,andthushaveasfundamentalgroupthefreegrouponggenerators.Each2-cellgivesarelation.itiseasytoseethattherelationsareasabove.Thefollowingimmediatecorollaryimplies,forinstance,thatZ4isnota3-manifoldgroup(see[3]or[1]). 8SIDDHARTHAGADGILfromauniqueminimalHeegardsplitting,i.e.,onethatcannotbeobtainedbystabilisingaHeegardsplittingoflowergenus.ItwasunknowntillfairlyrecentlywhethereverymanifoldhasauniquesuchHeegardsplitting.ThiswasshownnottobesobyCassonandGordon.Theorem10(Casson-Gordon).Thereisa3-manifoldMwhichhasnon-isotopic,minimalHeegardsplittings.ItturnsoutthattheseHeegardsplittingsbecomeisotopicafterasinglestabili-sation.Indeed,thereisnoknowncasewheremorethanonestabilisationisneeded.Thisshouldnotberegardedasanindicationofwhatistrue,butratherofourignorance.Inthecaseofsocallednon-Hakenmanifolds,thereisaboundonthenumberofstabilisationsrequired,linearinthegeneraoftheHeegardsplittings,byatheoremofRubinsteinandScharlemann.AnotherinterestingquestionistheminimumgenusamongHeegardsplittingsofagivenmanifoldM.SinceaHeegarddiagramgivesapresentationof1(M),weseethatthismustbeatleasttherankof1(M).BoileauandZieschanghaveshownthattherearemanifoldswhoseHeegardgenusisgreaterthantherankof1(M).6.MoreonlensspacesWeconcludebytakinganotherlookatlensspaces.Exercise11.AmoresuccinctdescriptionofthelensspaceL(p;q)isasthequotientofS32C2bytheactiongeneratedby(z1;z2)7!(z1e2i=p;z2e2iq=p).ShowthatthesolidtoriHi=f(z1;z2)2S2:jzij21=2gareinvariantunderthisaction,andtheirimagesgiveaHeegardsplittingforL(p;q).Exercise12.Showthat(z1;z2)7!(z2;z1)isahomeomorphismbetweenL(p;q)andL(p;q�1),whereqq�11(modp)Exercise13.AswithS3,lensspaceshaveuniqueminimalHeegardsplittings.Thus,iff:L(p;q)!L(p0;q0)isahomeomorphism,thentheimageofaHeegardsurfaceunderfisisotopictoaHeegardsurface.Usethistoshowthatasorientedmanifolds,L(p;q)=L(p0;q0)i p=p0andq0q1(modp).References[1]Epstein,D.B.A.Finitepresentationsofgroupsand3-manifoldsQuart.J.Math.OxfordSer.(2)12(1961),205{212.[2]Hempel,J.P.(1976)3-manifolds,Ann.ofMath.Stud.86,PrincetonUniversityPress.