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Singularities of Smooth Maps

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iii LISTOFFIGURES ................................ v ABSTRACT .................................... vi 1INTRODUCTION .............................. 1 1.1RegularPointsofSmoothMappings ................. 1 1.2SingularPointsofSmoothMappings ................ 2 1.3MappingsofSurfacesinto3-dimensionalSpaces .......... 2 1.4MappingsBetweenSurfaces. ..................... 3 2THOM-BOARDMANSINGULARITIES .................. 7 2.1NaiveDenition ............................ 7 2.2FiniteJetSpace ............................ 8 2.3InniteJetSpace ........................... 10 2.4Thom-BoardmanSingularities .................... 11 3THOMPOLYNOMIALS ........................... 14 4SINGULARITIESOFMAPPINGSM4!N3 ............... 17 4.1NormalFormsofSingularities .................... 17 4.2TopologicalDescriptionofSingularities ............... 17 4.3BehavioroftheMappinginaNeighborhoodof .......... 19 4.4BifurcationsofSingularSets ..................... 24 4.5Surgeryoff ............................ 26 5THEINVARIANTef ........................... 29 5.1FirstDenition ............................ 29 5.2SecondDenition ........................... 30 5.3InvarianceoftheSecondaryObstructionUnderHomotopy .... 31 6H-PRINCIPLE ................................ 32 6.1DierentialRelations ......................... 32 6.2H-Principle .............................. 32 6.3MorinSingularities. .......................... 33 6.4Ando-EliashbergTheorem ...................... 34 iv

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7UNIVERSALJETBUNDLES ........................ 36 7.1Construction1 ............................. 36 7.2Construction2 ............................. 37 7.3Construction3 ............................. 38 7.4CorollaryofAndo-EliashbergTheorem ............... 39 8SECONDARYOBSTRUCTION ...................... 40 8.1DenitionoftheSecondaryObstruction ............... 40 8.2NecessaryCondition ......................... 40 8.3SucientCondition .......................... 41 9COMPUTATIONOFTHESECONDARYOBSTRUCTION ....... 48 10SIMPLYCONNECTEDCASE ....................... 56 11FINALREMARKS .............................. 59 REFERENCELIST ................................ 60 BIOGRAPHICALSKETCH ............................ 63 v

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LISTOFFIGURES Figure page 1{1Whitneyumbrella. ............................ 3 1{2Foldandcuspsingularpoints ...................... 4 4{1Theimageofaswallowtailsingularpoint. ............... 18 vi

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CHAPTER1INTRODUCTION 1.1 RegularPointsofSmoothMappingsGivenasmoothmappingfofamanifoldMofdimensionmintoasmoothmanifoldNofdimensionn,thedierentialdfxofthemappingfatapointxofMisalinearmapfromthetangentspaceTxMofMatxtothetangentspaceTfxNofNatfx,dfx:TxM!TfxN:Wesaythatx2Misaregularpointofthemappingfiftherankrkxfofthedierentialdfxisexactlymaxm;n.Otherwisewesaythatthepointxisasingularpointofthemappingf.Weobservethatthesetofregularpointsformsanopensubmanifoldofthesourcemanifold.Indeed,ifahomomorphismhofvectorspacessendsasetfeigofindependentvectorsintoasetofindependentvectors,theneveryhomomorphismsucientlyclosetohalsosendsthevectorsfeigintoindependentones.Consequently,iffisasmoothmappingandxisapointofthesourcemanifold,thentherankofthedierentialdfxatxisnotgreaterthantherankofthedierentialdfyatanypointysucientlyclosetox.Inparticular,inasmallneighborhoodofaaregularpoint,themappingfhasnosingularpoints.Theregularpointsofamappinghaveasimpledescription.Inthecaseofapositivecodimension,n)]TJ/F22 11.955 Tf 12.484 0 Td[(m>0,theregularpointsarepreciselythepointsinaneighborhoodofwhichthemappingfisanembedding.Ifthemappingfisofanon-positivecodimension,i.e.,n)]TJ/F22 11.955 Tf 12.145 0 Td[(m0,thentheregularpointsarethepointsofthesourcemanifoldinaneighborhoodofwhichthemappingfisasubmersion. 1

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2 1.2 SingularPointsofSmoothMappingsWestudysingularitiesofsmoothmappingsuptoanequivalencerelation.Denition.Giventwomappingsfi:Mi!Ni,i=1;2,wesaythatthepointsx12M1andx22M2areofthesamesingularitytypewithrespecttotheright-leftequivalenceifthereareneighborhoodsUicontainingxi,neighborhoodsVicontainingfixianddieomorphismsg:U1!U2,h:V1!V2thattintothecommutativediagramU1g)306()222()306(!U2f1jU1??yf2jU2??yV1h)306()222()306(!V2;wherethemappingsfijUiaretherestrictionsofthemappingsfitoUi.Itisconvenienttodescribearight-leftsingularitytype,say,bychoosinganormalform,i.e.,amappingg:Rm!Rnwithsingularityattheorigin.Oncethenormalformischosen,wesaythatamappingf:M!Nhas-singularityatapointx2MifinsomelocalcoordinateneighborhoodsofxinMandfxinN,themappingfhastheformg.Inthetwosubsequentsectionswewillconsiderexamplesofsingularitiesinthecasesofmappingsofmanifoldsofsmalldimensions. 1.3 MappingsofSurfacesinto3-dimensionalSpacesOneofthesingularitiesofmappingsfromasurfaceintoa3-manifoldistheWhitneyumbrella.InaneighborhoodofaWhitneyumbrella,insomecoordinates,themappingfhastheformfu;v=uv;u;v2: Theorem1.3.1Whitney Everymappingofasurfaceintoa3-manifoldcanbeapproximatedbyamappingwithsingularitiesofonlyWhitneyumbrellatype.

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4 Figure1{2:Foldandcuspsingularpoints andfptherearecoordinatesinwhichthemappingfhastheformfx;y=x;y2orfx;y=x3+xy;yrespectivelyseegure 1{2 .Asitfollowsfromthenormalformsofsingularities,thesetofsingularpointsSofamappingfwithonlyfoldandcuspsingularpointsformsasmoothcurveinthesourcemanifold.Thesetofcusppointsisdiscrete,whilethesetoffoldpointsisthe1-dimensionalcomplementtothecusppointsinS.Wenotethattherankofthedierentialofthemappingfis1bothatapointofthefoldtypeandatapointofthecusptype.Todistinguishafoldsingularpointfromacuspsingularpoint,WhitneyconsideredtherestrictionofthemappingftothesmoothcurveofsingularpointsSandobservedthatthecusppointsoffareexactlythesingularpointsoffjSseegure 1{2 .Thecuspsingularpointsareessentialevenifweconsidermappingsuptohomotopy.Forexample,theprojectiveplaneRP2doesnotadmitamappingintoR2withonlyfoldsingularpoints[ 42 ].Letussketchaproofthatmotivatesthenotionhomologyobstruction."WenotethatanytwomappingsintoR2arehomotopic.ThustoprovetheclaimitsucestoconstructamappingRP2!R2withfoldandcuspsingularitiesandthentoshowthatthecuspsingularpointscannotbeeliminatedbyhomotopy.

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5 Construction.Foldingatrianglealongthethreelinesthatjointhethreemiddlepointsofthesidesofdenesa4-to-1mappingf0fromtothetriangle1=4thatis4timessmallerthan.Letussaythattwoboundarypointsx;yofareoppositeifthesegment[x;y]goesthroughthecenterof.Weobservethatthefoldmappingf0sendstheoppositepointsx;yoftheboundarytothesamepointf0x=f0y.Thus,themappingf0denesamapping~f0oftheprojectiveplaneobtainedfrombyidentifyingtheoppositeboundarypoints.Slightlyperturbingthemapping~f0andcomposingitwithanembeddingofthetriangle1=4intoR2,weobtainamappingffromRP2intoR2withonlyfoldand3cuspsingularpoints.Toprovethatthecuspsingularpointsoffcannotbeeliminatedbyhomo-topy,weneedtoconsiderthebifurcationsofthesetofsingularpointsSthatoccurunderhomotopyoff.Everyhomotopyjoiningtwomappingsbetweensurfacescanbeapproximatedbyageneralpositionhomotopythatchangesthesetofsingularsetbyisotopyexceptforthenitelymanypointswhenoneofthefollowingbifurcationstakesplace[ 23 ].Homotopyunderwhichacircleofsingularpointswithtwocusppointsappear:ftx;y=x3xy2tx;y;t2[)]TJ/F15 11.955 Tf 9.298 0 Td[(1;1];Homotopyunderwhichtwonewcuspsingularpointsappearonacurveoffoldsingularpoints:ftx;y=x4+xy)]TJ/F22 11.955 Tf 11.955 0 Td[(tx2;y;t2[)]TJ/F15 11.955 Tf 9.298 0 Td[(1;1];Homotopyunderwhichacurveofsingularpointswithtwocusppointsdisappear:ftx;y=x3xy2tx;y;t2[)]TJ/F15 11.955 Tf 9.298 0 Td[(1;1];

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6 HomotopyunderwhichtwocuspsingularpointsonacurveSdisappear:ftx;y=x4+xy+tx2;y;t2[)]TJ/F15 11.955 Tf 9.298 0 Td[(1;1]:Thus,undergeneralpositionhomotopythenumberofcuspsingularpointschangesbyamultipleoftwoandthereforeeverymappingofRP2intoR2withonlyfoldandcuspsingularpointshasanoddnumberofcusppoints[ 42 ].

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CHAPTER2THOM-BOARDMANSINGULARITIESTheright-leftequivalencerelationonsingularitiesofsmoothmappingsissonethatthenumberofdierentsingularitytypesofamappingisinniteingeneral.Besides,thebehaviorofthesetofpointsofaright-leftequivalenceclassunderhomotopyofamappinghasnosimpledescription.Toovercomethedicultiesarisinghereonemayconsideracoarserrelationinwhichaclassofequivalenceisaunionofsome,perhapsinnitelymany,right-leftequivalenceclassesofsingularities.OneofsuchrelationsplayingaspecialroleinsingularitytheoryistheThom-Boardmanclassication.EverycontinuousmappingofsmoothmanifoldsadmitsanapproximationbyamappingfwithsingularitiesofonlynitelymanydierentThom-Boardmanclasses.Furthermore,thesetofsingularpointsoffofeachThom-Boardmanclassisasubmanifoldofthesourcemanifold. 2.1 NaiveDenitionLetTMandTNdenotethetangentbundlesofsmoothmanifoldsMandNrespectivelyanddfthedierentialofasmoothmappingf:M!N.ThesetSi=SifisdenedasthesetofpointsxinMatwhichthekernelrankoffiskrxf=i.SupposethatdimM=mn=dimN.Supposethatforeachi,thesetSiisasubmanifoldofM,thenwecanconsidertherestrictionfjSi1offtoSi1anddenethesingularsetSi1;i2asthesubsetSi2fjSi1ofSi1.Again,ifeverysetSi1;i2isasubmanifoldofM,thenthedenitionmaybeiterated.Thus,thesetSi1;:::;ikisdenedbyinductionasSikfjSi1;:::;ik)]TJ/F18 5.978 Tf 5.756 0 Td[(1.TheindexI=i1;:::;ikiscalledthesymbolofthesingularity.WewillwriteSIforSi1;:::;ik. 7

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8 ForexampletheWhitneyfoldsingularpointsofamappingbetweensurfacesandtheWhitneyumbrellaofamappingofasurfaceintoa3-manifoldareThom-BoardmansingularpointsofthetypeS1;0.FromtheWhitneydescriptionofsingularpointsofamappingbetweensurfaces,itfollowsthatthecuspsingularpointsareofthetypeS1;1;0.Certainly,thisnaturaldenitionmakessenseonlyunderheavyrestrictions;thesingularsetSi1;:::;ikcanbedenedonlyifthesingularitystratumSi1;:::;ik)]TJ/F18 5.978 Tf 5.756 0 Td[(1isasubmanifoldofthesourcemanifold.BypassingtojetspacesBoardmanwasabletoextendthedenitionoverallsingularsets. 2.2 FiniteJetSpaceAsingularitytypeofamappingf:M!Natapointx2Mdependsonthebehaviorofthemappingfonlyinasmallneighborhoodofx.So,wepasstogerms.Agermatapointx2Misanequivalencerelationonmappingsunderwhichtwomappingsfi,i=1;2,denedonaneighborhoodofx2Mrepresentthesamegermatxifthereisapossiblysmallerneighborhoodofxwherethemappingsf1,f2coincide.Ak-jetis,bydenition,aclassofk-equivalenceofgerms.Twogermsfandgatxarek-equivalentifatthepointxthemappingsfandghavethesamepartialderivativesoforderk.Aspartialderivativesinvolved,ourdenitionimplicitlyassumescoordinatesystemsinneighborhoodsofxandfx=gx.Itiseasytoverify,however,thatifinsomecoordinatesystemsfandghavethesamepartialderivativesoforderk,thenthesameistrueforanyotherchoiceofthecoordinatesystems.Ifak-jetxisrepresentedbyamappingfatx,thenwealsosaythatxisak-jetofthemappingfatx.Thesetofallk-jetsJkM;Niscalledthek-jetspaceofmappingsofMintoN.Letxbeak-jetatapointx2Mrepresentedbysomemappingf.

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9 Ifacoordinatesysteminaneighborhoodofxandacoordinatesysteminaneighborhoodoffxarexed,thenthek-jetxisdeterminedbytheTaylorpolynomialoffatxoforderk.Initsturn,thesetofpolynomialsoforderkisnaturallyisomorphictothenitedimensionalEuclideanspaceaseachpolynomialischaracterizedbythesetofitscoecients.So,thek-jetspacehasanaturalstructureofasmoothmanifold.Formally,letUandVbecoordinatecoversofthemanifoldsMandNrespectively.ForeachopensetU2UandanopensetV2VwedeneasubsetWUVofthek-jetspaceasthesetofthejetsx,x2U,representedbymappingssendingxintoV.NotethatthesubsetsW=fWUVg,whereUandVrangeoverelementsofUandV,coverthespaceofk-jets.Also,beingisomorphictothesetofpolynomialsoforderk,eachofWUVisisomorphictoaEuclideanspace.Theseisomorphismsinducetopologies,oneforeachWUV,thatcoincideonintersectionsWUVWU0V0,U02U;V02V.ThusthereisanaturaltopologyonJkM;N.Moreover,thecoverWtogetherwithhomeomorphismsfromWUVintotheEuclideanspace,U2U;V2V,denesasmoothstructureonJkM;N.InfactthecoverWnotonlyhelpstointroduceasmoothstructureonJkM;NbutalsoallowsustointroduceonJkM;NastructureofasmoothlocallytrivialbundleoverMN.Indeed,JkM;NiscoveredbythesetsWUV2WeachofwhichisatrivialbundleoverUV.WenotethatthebundleprojectionJkM;N!MNsendsak-jetxrepresentedbyamappingfintothepointxfx.Example.Ifk=1,thenthek-jetofamappingfatx2Misthedierentialdfx:TxM!TfxN.Thusthespaceof1-jetsisthelocallytrivialbundleHOMM;NoverMNtheberoverxyofwhichconsistsoflinearhomomorphismsTxM!TyN.

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10 ForthedenitionofThom-BoardmansingularitiesweneedtopasstoaninnitedimensionalspaceJ1M;Nwhichwedenenext. 2.3 InniteJetSpaceAk-jetatapointxdeterminesanl-jetforeachl
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12 Notethatthemapdjf:TM!TJ1M;Nisinjective.TheunionofimagesofdjfoverallmappingsM!NiscalledthetotaltangentbundleofthejetspaceandisdenotedbyD.Givenajetjfx,wewillusetheinjectivehomomorphismdjfjTxtoidentifytheplaneTxMwithDjjfx.Every1-jetatapointx2MdeterminesahomomorphismTxM!TfxN,wherefisagermatxrepresentingthejet.LetybeapointofthejetbundleandKyDythekernelofthehomomorphismdenedbythe1-jetcomponentofy.Boardmanprovedthatforeveryi1theseti1=fy2J1M;NjdimKy=i1gisasubmanifoldofJ1M;N.LetIrdenotethesetofrintegersi1;:::;irsuchthati1ir.SupposethatthesubmanifoldIr)]TJ/F18 5.978 Tf 5.756 0 Td[(1hasbeenalreadydened.ThendeneIr=fy2Ir)]TJ/F18 5.978 Tf 5.756 0 Td[(1jdimKyTIr)]TJ/F18 5.978 Tf 5.756 0 Td[(1=irg:BoardmanprovedthatforeverysymbolIrthesetIrisasubmanifoldofJ1M;N.Amappingfiscalledageneralpositionmappingifthesectionjfistransver-saltoeverysubmanifoldI.BytheThomStrongTransversalityTheoremseeArnoldetal.[ 5 ]orBoardman[ 6 ],everymappingcanbeapproximatedbyageneralpositionmapping.Givenamappingf:M!N,apointx2MisasingularityoftypeIiftheimagejfxisinI.Ashasbeenmentioned,forgeneralpositionmappings,thedenitionofsingularitytypesgivenbyBoardmancoincideswiththenaivedenitiongiveninsection 2.1 .Examplesofgeneralpositionmappings.Formappingsfromasurfaceintoa3-manifold,generalpositionmappingsarethosewithsingularitiesofonlyWhitney

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13 umbrellatype.Formappingsbetweensurfaces,generalpositionmappingsareexactlythemappingswithonlyfoldandcuspsingularpoints.

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15 ItisremarkablethatthecohomologyclassdualtothehomologyobstructionisalwaysapolynomialintermsofStiefel-WhitneyclassesofthebundlesTMandfTN[ 14 ],[ 17 ].ForexampleinthecaseofcuspsingularitiesofmappingsfromasurfaceintoR2,thehomologyobstructionisdualtotheStiefel-Whitneyclassw2.ThepolynomialintermsofStiefel-WhitneyclassesdualtothehomologyobstructioniscalledThompolynomial.ManypapersaredevotedtocalculationofThompolynomialsofcertainsingularities.SomeofthepolynomialswerealreadycalculatedbyThom[ 39 ].Forthelistofresultsof1970-1980werefertothesurvey[ 5 ]ofV.Vasiliev.Recently,therewasasignicantprogressincalculationofThompolynomialsbyL.Feher,M.KazarianandR.Rimanyi[ 11 ],[ 18 ],[ 26 ],[ 27 ].Thehomologyobstructionisnotcompleteingeneral.Forexample,everymappingofa2-dimensionalsphereintoR2hasacurveoffoldsingularpoints,thoughthehomologyobstructionbelongstothetrivialgroupH2S2;Z.Exceptthetrivialcaseoffoldsingularpointsthatappearasthepointscorrespondingtotheboundaryoftheimage,thequestionwhetherthehomologyobstructionprovideacompleteobstructionisnotsimple.Letuslistsomecaseswherethehomologyobstructionisknowntobecom-plete:cuspsingularitiesformappingsfromm-manifoldintoanorientablesurfaceH.Levine[ 20 ];Y.Eliashberg[ 8 ];anystablesingularitytypeexceptfoldoneformappingsbetweenorientable4-manifoldsO.Saeki,K.Sakuma[ 35 ],Y.Ando[ 2 ],[ 3 ];non-foldsingularitiesofmappingsofnon-positivecodimensionbetweenstablyparalellizablemanifoldsY.Eliashberg[ 8 ],[ 9 ];zerodimensionalsingularitiesofmappingsofnon-positivecodimensionbetweenorentablemanifoldsofdimensions2Y.Ando[ 3 ];

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16 anyMorinsingularityexceptoffoldandcusptypesofmappingsMm!Nnwithm)]TJ/F22 11.955 Tf 11.955 0 Td[(n>0oddR.Sadykov[ 29 ];In1992O.Saeki[ 30 ]presentedanexampleofmanifoldsM;Nandasingu-laritytypesuchthatthehomologyobstruction[]totheexistenceofa-freemappingM!NistrivialandneverthelessthemanifoldMdoesnotadmitamappingintoNwithoutsingularities.IntheexampleofO.SaekiMisa4-manifoldhomotopyequivalenttoCP2,NisR3andisthecuspsingularity.Theclosureofthecuspsingularitiesofamappingfroma4-manifoldintoa3-manifoldisaunionofcircles.ItshomologyclassistrivialasthemanifoldMissimplyconnected.OntheotherhandO.SaekiprovedthatthemanifoldMdoesnotadmitamappingintoNwithoutcuspsingularpoints.Inthelattersectionswedeterminethesecondaryobstructionfortheexistenceofmappingsof4-manifoldsinto3-manifoldswithoutcuspsingularitiesandprovethatthesecondaryobstructioniscomplete.

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CHAPTER4SINGULARITIESOFMAPPINGSM4!N3 4.1 NormalFormsofSingularitiesIngeneralpositionwithrespecttoThom-Boardmansingularities,asmoothmappingfofa4-manifoldintoa3-manifoldhasonlyfold,cuspandswallowtailsingularities[ 5 ].Infact,onemaydeneageneralpositionmappingofa4-manifoldintoa3-manifoldasamappingwithsingularitiesofonlythesetypes.Inaneighborhoodofanysingularpointofamappingf,therearelocalcoordinates,calledspecial,inwhichthemappingftakesoneofthenormalforms:DenitefoldsingularitytypeS+T1=t1;T2=t2;Z=q21+q22.IndenitefoldsingularitytypeS)]TJ/F22 11.955 Tf -165.718 -22.115 Td[(T1=t1;T2=t2;Z=q21)]TJ/F22 11.955 Tf 11.955 0 Td[(q22.CuspsingularitytypeT1=t1;T2=t2;Z=q21+t1x+x3.SwallowtailsingularitytypeT1=t1;T2=t2;Z=q21+t1x+t2x2+x4. 4.2 TopologicalDescriptionofSingularitiesDirectlyfromthenormalformsofsingularities,itfollowsthatthesetofallsingularpointsofageneralpositionmappingf:M!Nofa4-manifoldintoa3-manifoldisaclosed,perhapsnon-orientable,2-dimensionalsubmanifoldSofM.However,theimageofthesingularsetfSisnotasubmanifoldofNingeneral;therestrictionofthemappingftoS,maynotbeanimmersion.ThesingularsetoffjSisexactlythesetofcuspandswallowtailsingularpointsoff.Theimageof 17

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18 thesingularsetunderthemappingfjSinaneighborhoodofaswallowtailsingularpointisdepictedonthegurebelow. Figure4{1:Theimageofaswallowtailsingularpoint. Weseethattheimageoftheswallowtailsingularpointscanbeidentiedwiththeendpointsoftheself-intersectioncurveoffS.Inparticular,thehomologyobstructiontoeliminatingswallowtailsingularpointsbyhomotopyoffisalwaystrivial.Infact,bytheAndotheorem[ 3 ],ifthemanifoldsMandNareorientable,thenthehomotopyeliminatingtheswallowtailsingularpointsoffexists.Weareinterestedinthequestionwhetherthereexistsahomotopyeliminatingcuspsingularpointsf.Thesetofcuspsingularpointsconstituteaunionofcurveswithboundaryattheswallowtailsingularpointssothattheclosure fcusppointsg=fcusppointsg[fswallowtailpointsgisaunionofcirclesembeddedintoM.Itiswell-knownthatifMandNareorientable,thenthehomologyclass[ f]2H2M;Zistrivial.Infactastrongerassertiontakesplace. Lemma4.2.1 SupposethatthemanifoldNisorientable.Thenthecurve fboundsanorientablesurfaceinM.Proof.Letusprovethatforanorientablesurfaceboundedby f,wemaychosethesetofdenitefoldsingularpointsS+f.

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19 Firstly,weneedtoshowthat@S+f= f.Thisfollowsdirectlyfromthenormalformsofthecuspandswallowtailsingularpoints;thecurve fcutsthesetofsingularpointsSf=S+f[ f[S)]TJ/F15 11.955 Tf 7.085 1.793 Td[(fintothesetofdenitefoldpointsS+fandindenitefoldpointsS)]TJ/F15 11.955 Tf 7.085 1.793 Td[(fsothateachcomponentof fbelongstotheboundaryofexactlyonecomponentofS+fandonecomponentofS)]TJ/F15 11.955 Tf 7.084 1.793 Td[(f.Secondly,weneedtoshowthatthesetsetS+fisorientable.Thisfollowsfromthenormalformoffinaneighborhoodofadenitesingularpoint.Indeed,inaneighborhoodofadenitefoldpoint,themappingfcanbewrittenastheproductidR2moftheidentitymappingid:R2!R2andtheMorsefunctionm:R2!Rdenedbymq1;q2=q21+q22.TheMorsefunctionmgivesrisetoanorientationofRastheimageofminthesetofnon-negativereals.HencetheimagefS+alsohasanaturalco-orientation.TogetherwithorietationofN,theco-orientationoffS+denesanorientationofS+.Remark.NotethatincontrastwithS+f,thesetS)]TJ/F15 11.955 Tf 7.085 1.794 Td[(fisnon-orientableingeneral.Letusrecallthatavectorvpatacusppointpiscalledacharacteristicvectorifthereisastandardcoordinateneighborhoodofpsuchthatvp=@ @t1p.AvectoreldonthecurvefinM4iscalledacharacteristicvectoreldifitconsistsofcharacteristicvectors.Theexistenceofacharacteristicvectoreldforanarbitrarycuspmappingofanorientable4-manifoldintoanorientable3-manifoldwillfollowfromLemma 4.3.2 bellow. 4.3 BehavioroftheMappinginaNeighborhoodofInthissectionwedescribethebehaviorofthemappingfinaneighborhoodofthecurveofcuspsingularpoints.Forconvenienceofexposition,weassumethatisconnected.

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20 Notethatinaneighborhoodofeachcuspsingularpoint,themappingfcanbewrittenastheproductoftheidentitymappingid[0;1]:[0;1]![0;1]andthestandardcuspmappingg:D3!D2whichinspecialcoordinatesft;q;xgofD3andspecialcoordinatesofD2hastheformgt;q;x=t;q2+tx+x3:Ouraimistoprovethatinaneighborhoodof,fhastheproductstructure.Astherststepoftheproof,letusshowthatwecandecomposeaneighbor-hoodofthecurveintotheproductS1D3,=S1f0g,andaneighborhoodofthecurvefintotheproductS1D2,f=S1f0g,withrespecttowhichthemappingfistheproductoftheidentitymappingidS1:S1!S1andthestandardcuspmappingg,i.e.,foreachpointx2S1therestrictionfjfxgD3mapsthe3-discintothecorresponding2-discffxgD2and,moreover,insomecoordinatesfjfxgD3:fxgD3)167(!fxgD2isthestandardcuspmapping.WestartwithanarbitrarytubularneighborhoodofthecurvefinNandchooseitsdecompositionintotheproductS1D2sothat,rst,coincideswithS1f0gand,second,eachdiscfxgD2,x2S1,intersectsthecurvetransversally.Inasucientlysmallneighborhoodof,themappingfcomposedwiththeprojectionS1D2!S1f0gontothecurvehasrank1.Hence,bytheInverseFunctionTheorem,thereisaproductneighborhoodS1D3ofeachdiscfxgD3,x2S1,ofwhichmapsunderfintothecorrespondingdiscffxgD2oftheneighborhoodoff.

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21 Tocompletetheproofoftheassertionthatinaneighborhoodof,themappingfhasaproductstructure,weneedtoshowthatforeachx2S1themappingfjfxgD3isthestandardcuspmapping.Forapointx2,letD3xandD2xdenotethediscsoftheproductstructuresofandfthatgothroughthepointsxandfxrespectively. Lemma4.3.1 Foreverypointx2,therestrictionfx=fjD3xisageneralpositionmapping.ProofofLemma 4.3.1 .Intermsofjetspaces,itsucestoshowthatundertheassumptionsofthelemma,rst,thejetextensionj1fxofthemappingfxsendsD3xtotheset2J1D3x;D2xand,second,thatthemappingj1fxistransversaltothesubmanifoldsof2and2;1-points.WewillwriteJ1[)]TJ/F21 7.97 Tf 6.586 0 Td[(1;1]forJ1[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]D3x;[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]D2xandJ1f0gfortherestrictionofJ1[)]TJ/F21 7.97 Tf 6.586 0 Td[(1;1]tof0gD3x.Lets:J1D3x;D2x!J1[)]TJ/F21 7.97 Tf 6.586 0 Td[(1;1]betheembeddingthatrelatesthejetsectionofamappingg:D3x!D2xwiththejetsectionofthemappingidg:[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]D3x![)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]D2xrestrictedtof0gD3x,whereidstandsfortheidentitymappingof[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1].Moreprecisely,bydenition,sisauniqueembeddingthatmakesthediagramJ1D3x;D2xs)306()222()306(!J1[)]TJ/F21 7.97 Tf 6.586 0 Td[(1;1]jgx??jgidx??D3x)306()222()306(![)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]D3x

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22 commutativeseealsothedenitionbeforeLemma 9.0.9 .HerethebottommappingistheembeddingidentifyingD3xwiththediscf0gD3x.Letuswrite0IforthesetofI-pointsinJ1[)]TJ/F21 7.97 Tf 6.586 0 Td[(1;1]and~IforthesetofI-pointsinJ1D3x;D2x.Weclaimthats)]TJ/F21 7.97 Tf 6.587 0 Td[(10I=~I.WewillprovetheclaimbyinductionoverthelengthofthesymbolI.ThemappingsdenesahomomorphismsofthetangentbundlesofJ1D3x;D2xandJ1[)]TJ/F21 7.97 Tf 6.587 0 Td[(1;1].Notethatforeveryy2J1D3x;D2x,thehomomorphismsbijectivelysendsthekernelKydenedinsection 2.4 intothekernelKsy.ThisimpliestheclaimforsymbolsIoflength1.Iftheclaimholdsforallsymbolsoflengthk,thenforeachIkoflengthk,sKyT~Ik=KsyT0Ik:Henceeachset~Ik+1withsymboloflengthk+1mapsundersintothecorrespond-ingset0Ik+1.Thiscompletestheinduction.Inparticular,theinverseimagesofthesetsof2,2;1and2-pointsofJ1[)]TJ/F21 7.97 Tf 6.587 0 Td[(1;1]underthemappingsarethecorrespondingsetsinJ1D3x;D2x.By[ 28 ,Lemma4.3],themappingsistransversalto2J1f0gandto2;1J1f0ginJ1f0g.Bydenitions 7.1 and 7.2 ,thesets2and2;1aretransversaltotheberJ1f0ginJ1[)]TJ/F21 7.97 Tf 6.587 0 Td[(1;1].Thereforethemappingsistransversalto2and2;1inJ1[0;1].IdentifyingD3xwithf0gD3x[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]D3xS1D3,weobtainjfjD3x=sjfx.Thus,toproveLemma 4.3.1 itsucestoshowthatC1theimageofjfjD3xisin2J1[)]TJ/F21 7.97 Tf 6.587 0 Td[(1;1]andC2jfjD3xistransversaltothesetsof2and2;1-points.TheconditionC1holdssinceImjf2.LetusproveC2.Thedierentialofthejetsectionjfatxsplitsintothesumofhomomorphismsdjf=djfjTxD3x+djfjTx;.1

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23 whereTxD3xandTxarethetangentspacesofD3xandatxrespectively.Thedif-ferentialdjfjTxsendsTxintothetangentspaceofthe2;1-points.Now,sincefisingeneralposition,theequation 4.1 impliesthatdjfjTxD3xistransversaltothesetsof2;1and2-points.ThiscompletestheproofofLemma 4.3.1 Lemma4.3.2 ThereareproductneighborhoodsS1D3ofandS1D2offsuchthatfrestrictedtoS1D3istheproductoftheidentitymappingofS1andthestandardcuspmappingg.Proof.InviewofLemma 4.3.1 ,thecollectionofthemappingsffxgindexedbythepointsofanintervalofcanbeviewedasahomotopyofthestandardcuspmapping,whichisknowntobehomotopicallystableforexampleseeTheorem7.1in[ 12 ].Therefore,thereisacoverfIgofthecurvebyintervalssuchthatforeachintervalIofthecover,themappingfrestrictedtoID3isequivalenttotheproductidgoftheidentitymappingofIandthestandardcuspmappingg.ThetrivializationsfID3gandfID2gleadtobundlestructuresofS1D3andS1D2overS1withcommoncoverfIgofS1andwithtransitionmappingsconsistentwithf.Thelattermeansthateachpair;ofthecorrespondingtransitionmappingsbelongstothegroupstabilizingthestandardcuspmappingg.SincethenormalbundlesofinM4andoffinN3areorientable,thetransitionmappingsareelementsofthegroupAutg=f;'2Di+D3;0Di+D2;0j'g)]TJ/F21 7.97 Tf 6.586 0 Td[(1=gg;whereDi+D2;0andDi+D3;0standforthegroupsoforientationpreservingauto-dieomorphismsofD2;0andD3;0respectively.ThegroupAutgreducestoamaximalsubgroupMCAutgconjugatetoalinearcompactsubgroup[ 16 ],[ 40 ].ToproveLemma 4.3.2 ,itremainstoshowthatthegroupMCAutgistrivial.

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24 LetKdenotetheorientableversionofthecontactgroup,i.e.Kisasemiprod-uctofDi+D2;0andofthegroupofgermsD2;0!Di+D1;0.ThegroupAutKhofagermh:D2;0!D1;0isdenedbyAutKh=f'2Kj'h=hg:Thestandardcuspmappinggisaminiversalunfoldingofthegermg0:D2;0!D1;0,denedbyg0q;x=q2+x3.Henceby[ 40 ,Proposition3.2],thegroupMCAutgisisomorphictoMCAutKg0.ThelattergroupisisomorphictoacompactsubgroupHofthegroupAutQg0ofautomorphismsofthelocalalgebraofg0[ 25 ,Theorem1.4.6].Let2GL+bealinearautomorphismofR2thatpreservesorientation.ItdenesanactionongermsR2;0!R;0bysendingagermhintoh.SupposethatthisactionfactorsthroughanactiononlocalalgebraAutQg0.ThendenesanelementinAutQg0.Bytheproofof[ 25 ,Theorem1.4.6],wemayassumethateachelementofH0.Hence,thegroupHiscontractible.ThiscompletestheproofofLemma 4.3.2 4.4 BifurcationsofSingularSetsWesaythatahomotopyofamappingfisingeneralpositionifitisageneralpositionmapping.

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25 Lemma4.4.1 Everyhomotopyjoiningtwogeneralpositionmappingscanbeapproximatedbyageneralpositionhomotopy.Proof.Letfi:M!N,i=0;1,betwogeneralpositionmappings,andF:M[0;1]!N[0;1]beahomotopyfromf0f0gtof1f1g.Wemayassumethatforasmallnumber",thehomotopyFdoesnotchangethemappinginintervals[0;"[0;1]and)]TJ/F22 11.955 Tf 11.956 0 Td[(";1][0;1].Asamapping,thehomotopyFadmitsaC1-closeapproximationbyageneralpositionmapping.Moreover,wemaychooseanapproximation~FthatcoincideswithFonthe"intervals[0;"and)]TJ/F22 11.955 Tf 11.955 0 Td[(";1]Since~FisC1-closetoF,thecompositionp~F:M[0;1]![0;1],wherep:N[0;1]![0;1]istheprojectionontothesecondfactor,hasnosingularpoints.Therefore,foreverymomentt02[0;1],theinverseimage~F)]TJ/F21 7.97 Tf 6.586 0 Td[(1Nft0gisdieomorphictoM.Thus,~Fcanbeconsideredasanewhomotopyjoiningf0andf1,whichasamappingisingeneralposition.Lemma 4.4.1 guaranteesthatanytwohomotopicgeneralpositionmappingscanbejoinedbyahomotopyingeneralposition.Thatiswhywerestrictourattentiontobifurcationsthatoccuronlyundergeneralpositionhomotopy.Intherestofthesectionwedescribebifurcationsofhomotopiesofmappingsfroma4-manifoldintoa3-manifold.Bydimensionalreasoningssee,forexample,Boardman[ 6 ]andAndo[ 4 ]ageneralpositionmappingffroma5-manifoldintoa4-manifoldhasonlyMorinsingularities,andD4singularitieswithsymbolI=;2;0.InaneighborhoodofaMorinsingularpoint,insomelocalcoordinates,themappingfcanbewrittenasft1;t2;t3;q;x=t1;t2;t3;q2+k)]TJ/F21 7.97 Tf 6.587 0 Td[(1Xi=1tixi+xk+1;k=1;2;3;4:.2

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26 ForaD4pointoff,therearelocalcoordinatesinwhichthemappingftakestheformft0;t1;t2;u;v=t0;t1;t2;u2vv3+t0u+t1v+t2v2:.3SupposethatF:M4[0;1]!N3[0;1]isageneralpositionhomotopyjoiningtwogeneralpositionmappingsf0andf1.Fromthenormalforms 4.2 and 4.3 ,itiseasytoverifythatthesingularsetofageneralpositionhomotopyF:M4[0;1]!N3[0;1]isasubmanifoldofM4[0;1].Therefore,SFdenesanembeddedbordismbetweenthesingularsetsoff0andf1. 4.5 SurgeryoffInthissectionwestudyasurgeryofthesingularsetofacuspmappingf:M4!N3andndasucientconditionfortheexistenceofahomotopyoffrealizingagivensurgery.Letf:M4!N3beacuspmappingofanorientable4-manifoldintoanorientable3-manifold.Ingeneral,therestrictionofftothecurveofcuspsingularpointsisanimmersion.Tosimplifyarguments,wemakefjanembeddingbyaslightperturbationoffinaneighborhoodof.Letbeaeldofcharacteristicvectorson.WesaythatanorientablesurfaceHisabasisofsurgeryseeEliashberg[ 9 ],if 1. @H=, 2. thevectoreldistangenttoHandhasaninwarddirection, 3. Hn@HdoesnotintersectSf,and 4. therestrictionfjHisanimmersion.WewillshowthatifabasisofsurgeryHexists,thenwecanreducebymodicationoffinaneighborhoodofH.Weassumethatisconnectedandfhasnoothercuspsingularpoints.Theproofinthegeneralcaseissimilar.Weneedonemorepreliminaryobservation.LetIdenotetheclosedinterval[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]andg:ID2!IIbethestandardcuspmappingdenedby

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27 gt;q;x=t;q2+tx+x3,wheretandq;xarethecoordinatesoftherstandthesecondfactorsofthedomainID2respectively.Thengcanbeconsideredasahomotopygt:D2!I;t2I;denedbygtq;x=q2+tx+x3.Notethatg)]TJ/F21 7.97 Tf 6.586 0 Td[(1:D2!IisaMorsefunction.SinceMorsefunctionsarehomotopicallystable,therearecoordinatesinwhichgs=g)]TJ/F21 7.97 Tf 6.587 0 Td[(1foreachs2[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;)]TJ/F15 11.955 Tf 9.298 0 Td[(1+". Lemma4.5.1 SupposethatthereisabasisofsurgeryH1.Thenthereisafoldmapping~f:M4!N3,whichdiersfromfonlyinaneighborhoodofH1.Proof.LetU=S1D3beaneighborhoodofinM4givenbyLemma 4.3.2 .Lett2beacycliccoordinateonthecircleS1,t1;q;xbecoordinatesonD3andT1;T2;ZbecoordinatesinaneighborhoodoffwithT2cyclicsuchthatfjUisgivenbyT1=t1;T2=t2;Z=q2+t1x+x3:WemayassumethatH1U=ft1;t2;q;xjx=q=0;t12[0;1]g:WedeneH0=ft1;t2;q;xjx=q=0;t12[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;0]gandsetH=H0[H1.Weregardatubularneighborhoodofasubmanifoldasadiscbundle.Thepropertiesand4ofthedenitionofabasisofsurgeryguarantiesthatthesubmanifoldHhasatubularneighborhoodAsuchthattherestrictionofAtoHUisinU,theintersectionSf@AisintherestrictionofthebundleAto@HandthesetAnUcontainsnosingularpointsofthemappingf.TosimplifyexplanationsweassumethatfjHisanembedding.ThentheimageofA,whichwedenotebyB,isalinebundleoverfH.Inthefollowing,foramanifoldXwithboundary@X,letCXdenoteacollarneighborhoodof@XinX,andletIdenote[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1].

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28 First,bytheremarkprecedingthelemma,themanifoldsB1=BjfCHandA1=f)]TJ/F21 7.97 Tf 6.586 0 Td[(1B1AhaveproductstructuresA1=CHIIandB1=fCHIsuchthatfjA1isaproductofadieomorphismCH!fCHandaMorsefunction.WemayassumethattherestrictionofthisMorsefunctiontoCII[ICIistheprojectionontothefactorcorrespondingtothecoordinatex,CII!I,ICI!CI,andthatA1=AjCH.Next,weextendtheproductstructureofB1toaproductstructurefHIofB.ThenwerestrictthisproductstructuretoB2=fHCIanddeneA2=f)]TJ/F21 7.97 Tf 6.587 0 Td[(1B2A.ThemappingfjA2isregularandthereforewemayassumethatA2=HICIisatriviallinebundleoverB2withprojectionfjA2alongthesecondfactor.Finally,wecanndA3AandaproductstructureHCIIofA3suchthatfjA3isatrivialCI-bundleoverfHIandA1[A2[A3isacollarneighborhoodof@A.TheconnectedcomponentsofA2andA3areorientable1-dimensionalbundleswithbundlemappingsgivenbytherestrictionsoff.Sincethestructuregroupoforientablelinebundlesreducestothetrivialgroup,wecanmakethethirdcoordinatesofA1;A2andA3agreeonintersections.WexanextensionoftheproductstructuresofA1,A2andA3toaproductstructureHIIofA.Letp2@HandfpdenotetherestrictionofftotheberofthebundleAoverp.Welet~fx=fxforx2M4nAand~fu;v;w=fufpv;wforx=u;v;w2A=HII:Itiseasilyveriedthat~fisasmoothmappingand~fsatisestherequirementsofthelemma.

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CHAPTER5THEINVARIANTefInthissectionwegiveseveraldenitionsofasecondaryobstructiontoeliminatingthecuspsingularpointsofageneralpositionmappingofaclosedorientable4-manifoldintoanorientable3-manifold;anddiscusssomeofitsproperties. 5.1 FirstDenitionGivenageneralpositionmappingfofa4-manifoldMintoanorientable3-manifold,todenethehomologyobsturction,i.e.,theprimaryobsruction,weconsidertheclosureofthecuspsingularpointsandtakeitsfundamentalhomologyclassinthegroupH1M;Z2.Todenethesecondaryobstruction,considerthesetofallsingularpointsSfofthemappingf.Thisisaclosed2-dimensionalsubmanifoldofM.WemayslightlyperturbSfsothatthenewsurfaceS0fintersectstheoriginalsurfaceSfatnitelymanypoints.Witheachintersectionpointweassociateanumber1,thesignoftheintersectionpoint;andthendenethesecondaryobstructionasthesumoftheassociatednumbers.Todenethesignoftheintersectionpointpp02SfS0fwechooseanorientationofthemanifoldMandanorientationofaneighborhoodofpinSf.Thelatterorientationdeterminesanorientationofaneighborhoodofp0inS0f.Indeed,foranychoiceofaRiemannianmetriconMtheprojectionofthetangentplaneofS0fatp0intothetangentplaneofSfatpisanisomorphismprovidedthattheperturbationS0fisC1-closetoSf.TheorientationofSfatpandtheorientationofS0fatp0determineanorientationofMatp=p0.IfthelatterorientationcoincideswithonechosenforM,thentheintersectionpointpisassigned+1,otherwise)]TJ/F15 11.955 Tf 9.298 0 Td[(1. 29

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31 Forageneralpositionmappingffromanorientedclosed4-manifoldM4intoa3-manifoldN3,wedenethesecondaryobstructionefasthenormalEulernumberoftheembedded2-dimensionalsubmanifoldSf.Thisdenitionisconsistentwiththerstdenitionofthesecondaryobstruc-tion. 5.3 InvarianceoftheSecondaryObstructionUnderHomotopyLetusshowthatthesecondaryobstructionefisinvariantunderhomotopyofthemappingf:M4!N3. Lemma5.3.1 Supposethatf0andf1aretwohomotopicgeneralpositionmap-pings.Thenef1=ef2.Proof.LetF:M4[0;1]!N3[0;1]beageneralpositionhomotopyjoiningf0:M4f0g!N3f0gandf1:M4f1g!N3f1g.TheboundaryofthesingularsetBofFistheunionofthesingularsetsB0off0andB1off1.Letit:Bt!B,t=0;1,denotetheinclusion,FtheorientationsystemoflocalcoecientsonB,andlete2H2B;FbetheEulerclassofthenormalbundleofBinM4[0;1].Thenef0)]TJ/F22 11.955 Tf 11.956 0 Td[(ef1=i0e;[B0])]TJ/F15 11.955 Tf 11.955 0 Td[(i1e;[B1]=e;i0[B0])]TJ/F22 11.955 Tf 11.955 0 Td[(i1[B1]=0;sincei0[B0])]TJ/F22 11.955 Tf 12.123 0 Td[(i1[B1]correspondstotheboundaryofBandvanishesinH2B;F.

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33 IffisasolutionofadierentialrelationR,thenitsjetextensionjfisasectionofthecompositionM:R)167(!Jk)167(!E)167(!Moftheinclusion,projectionmapofthejetbundleand.TheH-principlesuggeststhestatementconversetothegivenoneuptohomotopy:H-Principle.EverysectionofthemappingM:R!Mishomotopicthroughsectionstoasectionthatarisesasthejetextensionofasectionof.Wereferto[ 13 ],[ 10 ]and[ 38 ]fornumerousexamplesofdierentialrelationsforwhichtheh-principleholds.Letusconcludethesectionwithillustrationoftheh-principlebyinterpretingtheSmale-Hirshtheorem[ 15 ]intermsofjetspaces.TheSmale-HirschtheoremassertsthatasmoothmappingfofamanifoldMofdimensionmintoamanifoldNofdimensionn,n
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34 insomecoordinateneighborhoodofwhichthemappingfhastheformfx1;:::;xm=x1;:::;xn)]TJ/F21 7.97 Tf 6.586 0 Td[(1;x2nx2m:WewilldenoteitsThom-BoardmansymbolI=m)]TJ/F22 11.955 Tf 12.18 0 Td[(n+1;0byI1.ThemildestsingularitiesoffoforderkarecalledMorinsingularities.ThenormalformoffinaneighborhoodofaMorinsingularityoforderkisTi=ti;i=1;2;:::;n)]TJ/F22 11.955 Tf 11.956 0 Td[(r;Li=li;i=2;3;:::;r; .1 Z=Q+rXt=2ltkt)]TJ/F21 7.97 Tf 6.587 0 Td[(1+kr+1;Q=k21:::k2m)]TJ/F23 7.97 Tf 6.586 0 Td[(n;ItsThom-BoardmansymbolisI=m)]TJ/F22 11.955 Tf 12.313 0 Td[(n+1;1;:::;1;0oflengthr+1andwillbedenotedbyIr.WenotethattheMorinsingularitywithsymbolI2isthecuspsingularityandtheMorinsingularitywithsymbolI3istheswallowtailsingularity.Returningtothejetspaces,let0=k0JkM;Nbethesubsetcorre-spondingtothesetofregularpointsandforr0,r=krdenotesthesubsetcorrespondingtotheThom-BoardmansingularitieswithsymbolsIt,tr. Theorem6.3.1Morin[ 23 ] Supposethatthejetextensionofageneralpositionmappingf:M!NtakesMintotheset1r.TheneachsingularpointofthemappingfisMorin.WeemphasizethatintheMorintheoremtherequirementthatthemappingfisingeneralposition,i.e.,thatthejetextensionjfistransversaltoThom-Boardmanstratum,isessential. 6.4 Ando-EliashbergTheoremTheAndo-EliashbergtheoremassertsthatifdimM>dimN2,thenthedierentialrelationsrabidebytheh-principle.TheMorintheoremallowstoformulatetheAndo-Eliashbergtheoremasfollows.

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35 Theorem6.4.1 Ando{Eliashberg[ 3 ]LetMandNbeorientablemanifolds.LetdimN2.Thenforanycontinuoussections:M!rthereexistsaMorinmapg:M!Nsuchthatjg:M!rbecomesasectionber-wisehomotopictosinr.TheproofofTheorem 6.4.1 in[ 3 ]showsthattherelativeversionofTheorem 6.4.1 isvalidaswell.Inotherwords,supposethatUisanopensetinMands:M!risasectionsuchthattherestrictionofstoaneighborhoodofMnUisthejetsectionjginducedbyaMorinmappingg:MnU!N.ThengadmitsanextensiontoaMorinmapping~g:M!Nwhosejetsectionj~gisber-wisehomotopicinrtosbyhomotopyconstantoverMnU.InparticulartheAndo{Eliashbergtheoremreducesthequestionoftheexistenceofafoldmappingofanorientable4-manifoldMintoanorientable3-manifoldNtotheproblemofndingacontinuoussectionofthemappingM:1)167(!J2M;N)167(!MN)167(!M:

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CHAPTER7UNIVERSALJETBUNDLESLetJk0Rm;Rnbethevectorspaceofk-jetsofgermsRm;0!Rn;0,i.e.,theberofthejetbundleJkRm;Rnoverthepoint;02RmRn.Auniversalk-jetbundleJm;nisabundlewithberJk0Rm;RnoverauniversalspaceBGsuchthatforanymanifoldsMandN,thek-jetbundleJkM;NcanbeinducedfromJm;nbyanappropriatemappingofthebasespaceMN!BG.Theuniversalk-jetbundlesturnsouttobeverypowerfulinstudyofsingulari-tiesofsmoothmappings.Usingtheuniversalk-jetbundles,HaeigerandKosinskiprovedthatthehomologyobstructionsarePoincaredualtopolynomialsintermsofStiefel-Whitneyclasses.Recently,Kazarianconstructedanddescribedspectralsequencesassociatedwithclassicationsofsingularitieswhichencodenotonlythefundamentalclassesofsingularitiesbutalsotheadjacencyofclassesofdierentsingularities. 7.1 Construction1LetESOk!BSOkbeauniversalvectorbundleclassifyingtheorientablevectorbundlesofdimensionk.ForeachmanifoldXofdimensioni,letX:X!BSOidenotethemappingclassifyingthetangentbundleofX.LetxybeapointofBSOmBSOn,RmxbetheberofESOmoverx,RnxbetheberofESOnovery.ThentheberofthebundleESOmESOnoverxyisRmxRny.WedenethesetJkxym;nasthesetofk-jetsofmappingsRm;0!Rn;0andthendeneauniversalk-jetspaceJkm;nastheunion[Jkxym;nthatrangesoverthepairsxyofBSOmBSOn.Notethattheuniversalk-jetspacehasanaturalstructureofabundleoverBSOmBSOn. 36

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37 Letusprovethatforanym-manifoldMandn-manifoldNthebundleinducedfromtheuniversalk-jetbundlebythemappingMN:MN!BSOmBSOnisisomorphictothek-jetbundleJkM;N.Indeed,let JkM;Nbethespaceofk-jetsofmappingsTxM!TyN,wherexyrangesoverthepointsofMN.Byconstruction,thespace JkM;NisabundleoverMNisomorphictothebundleinducedfromtheuniversalk-jetbundlebyMN. Lemma7.1.1 LetMandNbeRiemannianmanifolds.Thenthek-jetbundleJkM;Niscanonicallyisomorphicto JkM;N.Proof.ForeachpointxofMthereisaneighborhoodUxoftheorigininTxMdieomorphicundertheexponentialmappingtoaneighborhoodVxofxinM,ExpMx:Ux!Vx:LetfExpNxgbethesetofexponentialmappingsforthemanifoldN.Thenthemapping JkM;N!JkM;Nthattakesthek-jetoverxyrepresentedbyamappingf:TxM!TyNintothek-jetofthemappingExpNyfExpMx)]TJ/F21 7.97 Tf 6.586 0 Td[(1isacanonicalisomorphism. 7.2 Construction2InthissectionwesketchtheconstructionbyKazarjan.Letkbeapositiveinteger.ThegroupG=Jk0DiRmJk0DiRn;whereDiRmandDiRnstandfortheorientationpreservingauto-dieomorphismgroupsofRm;0andRn;0respectively,isanitedimensionalLiegroupwithanaturalactiononthevectorspaceJk0Rm;Rn.ThenaturalactionofGisdened

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38 sothatanelementofGrepresentedbyapairofauto-dieomorphismsdmdntakesthek-jetrepresentedbyagermfintothek-jetofthegermdnfdm.LetE!BGbetheuniversalG-bundleoverBG.ThenthespaceJkm;nisdenedasthebundleJkm;n=EGJk0Rm;Rn;overBGassociatedwiththeuniversalbundleE.SincethegroupGishomotopyequivalenttoSOmSOn,thespaceBGishomotopyequivalenttoBSOmBSOn.ItiseasilyveriedthatthebundleJkm;nistheuniversaljetbundle. 7.3 Construction3Ifisavectorspace,thenr=denotesthevectorspacedenedasthevectorspacerfactoredbytherelationofequivalence:v1v2vrw1w2wrifandonlyifthereisapermutationofrelementssuchthatvi=wifori=1;:::;r.Thespaceriscalledthesymmetricr-tensorproductof.AsintheinterpretationofSmale{Hirschtheoremintermsofjetspaces,foreveryr,thebundlesandgiverisetothebundleHOMr;.TheberofHOMr;overapointx2MisthesetofhomomorphismsHomrx;xbetweenthebersrxandxofthebundlesrandrespectivelyoverx.ThespaceHOM;intheformulationoftheSmale{HirschTheoremisgeneralizedbythevectorbundleSr;=HOM;HOM;HOMr;overMseepaper[ 28 ]ofRonga.AsabovewedeneS1;astheinverselimitlim)]TJ/F22 11.955 Tf 18.915 5.83 Td[(Sr;.Letkrgdenotetherankofthekernelofalinearfunctiong.ApointofS1;overapointx2Misasetg=fgigthatconsistsofhomomorphisms

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39 gi2Homix;x.Weset~i1=[x2Mfg2S1x;xjkrg1=i1g:.1LetKhandChrespectivelydenotethekernelandcokernelofahomomorphismh2Hom;.ThecompositionofnaturalhomomorphismsHom;!Hom;Hom;!HomKg1;HomKg1;Cg1takesthehomomorphismg22Homxx;xintosomehomomorphism~g2.Wedene~i1;i2=[x2Mfg2S1x;xjg2i1andkr~g2=i2g:.2Thisconstructionproceedsbyinduction.Wereferthereaderto[ 6 ],[ 29 ].ThebundleJrM;NisisomorphictothebundleSr;,where=TMand=fTN.Moreover,thereisanisomorphismofbundlesS1;andJ1M;Nthattakeseach~IisomorphicallyontoI[ 29 ]. 7.4 CorollaryofAndo-EliashbergTheoremAsacorollaryoftheAndo{Eliashbergtheorem,weobtainthattheexistenceofahomotopyeliminatingthecuspsingularpointsofamappingfintoanorientable3-dimensionalmanifoldisindependentoff. Corollary7.4.1 Thehomotopyclassoff:Mm!N3;m3;containsafoldmappingifandonlyifthereisafoldmappingg:Mm!R3.Proof.ByAndo{Eliashbergtheorem,thehomotopyclassoffcontainsafoldmappingifandonlyifthereisasectionMm!1J2TMm;fTN3.ThelatterdoesnotdependonforTN3sincethetangentbundleofanorientable3-manifoldistrivial.

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CHAPTER8SECONDARYOBSTRUCTION 8.1 DenitionoftheSecondaryObstructionLetMbeaclosedconnectedoriented4-manifold.TheintersectionformdenedonthefreepartofH2M4;Zisaquadraticform.LetQM4denotethesetofintegerstakenonbythisquadraticform.WenotethatthesetQM4andthenormalEulernumberofageneralpositionmappingdependonthechoiceoforientationofM4.Itiseasilyveried,however,thatforagivenmappingf:M4!N3,theconditionef2QM4doesnotdependontheorientationofM4.Insections 8.2 and 8.3 wewillprovethemaintheorem 1 Theorem8.1.1 Letf:M4!N3beageneralpositionmappingfromanorientableclosedconnected4-manifoldintoanorientable3-manifold.Thenthehomotopyclassoffcontainsafoldmappingifandonlyifef2QM4. 8.2 NecessaryConditionTheinvariantefallowsustogiveanecessaryconditionfortheexistenceofafoldmappingintoN3.Inthelatersectionswewillprovethatthisconditionisalsosucient.Witheveryorientedclosedconnected4-dimensionalmanifoldM4weassociatethesetQM4ofintegerseachofeachisthenormalEulernumberofanorientablesurfaceinM4. Lemma8.2.1 Iff:M4!N3isafoldmapping,thenef2QM4. 1Afterthepaperwaswritten,O.Saekiinformedtheauthorthatheobtainedsimilarresultsusingadierentapproach[ 32 ]. 40

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41 Proof.ThesingularsetofafoldmappingconsistsofthesurfacesS)]TJ/F15 11.955 Tf 7.085 1.793 Td[(fofindef-initefoldsingularpointsandS+fofdenitefoldsingularpoints.Therefore,ef=eS)]TJ/F15 11.955 Tf 7.084 1.793 Td[(f+eS+f.In[ 30 ],[ 1 ]itisprovedthateS)]TJ/F15 11.955 Tf 7.085 1.793 Td[(f=0.Hence,ef=eS+f.SinceS+fisorientableseeLemma 4.2.1 ,weconcludethatef2QM4. Corollary8.2.1 Supposethatthehomotopyclassofageneralpositionmappingf:M4!N3containsafoldmapping.Thenef2QM4. 8.3 SucientConditionTheobjectiveofthissectionistoprovethattheconditionef2QM4issucientfortheexistenceofafoldmappinghomotopictoageneralpositionmappingf:M4!R3.InviewofCorollaries 4.4.1 and 7.4.1 thiscompletestheproofofTheorem 8.1.1 Lemma8.3.1 Letfbeageneralpositionmappingfromaconnectedclosedoriented4-manifoldM4intoR3.Supposeef2QM4.ThenM4admitsafoldmappingintoR3.Proof.Theconditionef2QM4guaranteestheexistenceofanorientable2-submanifoldSofM4withnormalEulernumberef.LetusprovethatinthecomplementM4nS,thereisanorientablepossiblydisconnectedembeddedsurface~SsuchthatPeveryorientablesurfaceembeddedinM4nSwithnon-trivialnormalbundleintersects~S.IfM4nSadmitsnoorientableembeddedsurfacewithnon-trivialnormalbundle,thenthepropertyPholdsforanyorientableembeddedsurface~S.SupposethatinM4nSthereisanorientableembeddedsurfacewithnon-trivialnormalbundleandthatasurfacewithpropertyPdoesnotexist.ThenforanypositiveintegerkthereisafamilyoforientedembeddedsurfacesfFigi=1;:::;ksuchthateachofthesurfaceshasanon-trivialnormalbundleanddoesnotintersecttheothersurfaces

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42 ofthefamily.LetTorH2M4nS;ZdenotethesubgroupofH2M4nS;Zthatconsistsofallelementsofniteorder.ThegroupH2M4nS;Z=TorH2M4nS;Zisnitelygenerated.Fixasetofgeneratorse1;:::;es.EverysurfaceFirepresentsaclass[Fi]inH2M4nS;Z=TorH2M4nS;Z,whichisnottrivialsinceFihasanon-trivialnormalbundle.Moreover,[Fi][Fi]6=0and[Fi][Fj]=0fori6=j.Ifthenumberkofthesurfacesisgreaterthanthenumbersofthegenerators,thenthereisacombination1[F1]+2[F2]++k[Fk]=0with21++2k6=0.Multiplicationofbothsidesby[Fi],i=1;:::;k,givesi[Fi][Fi]=0.Therefore,i=0foreveryi=1;:::;k.Contradiction.Thusasurface~SwithpropertyPexists.Letusconstructamappingforwhichtheset~S[Sisthepartofthesingularset.Werecallthatweidentifythebaseofavectorbundlewiththezerosection. Lemma8.3.2 Thereisageneralpositionmappingh:NS!R3fromthenormalbundleNSofSinM4suchthatthesetSisthesetofdenitefoldsingularpointsofhandhhasnoothersingularpoints.Proof.TheberofthebundleNSisdieomorphictothestandarddiscD2=fx;y2R2jx2+y2<1g.Letm:D2![)]TJ/F15 11.955 Tf 9.299 0 Td[(2;2]bethemappingdenedbytheformulamx;y=x2+y2.ThenmisaMorsefunctiononD2withonesingularpoint.ForeveryopendiscUinS,therestrictionofthenormalbundleNStoUisatrivialbundleUD2!U.LetI3denotethesegment)]TJ/F15 11.955 Tf 9.298 0 Td[(3;3.Wedenethemappingg:UD2!UI3bygu;z=u;mz,whereu2U,andz2D2.NotethatrotationsofD2donotchangethefunctionm.WemayassumethattheberbundleNS!SisanSO2-bundle.Thenthemappingsggiverisetoa

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43 mappingg:NS!SI3.TheopenorientedmanifoldSI3admitsanimmersionintoR3.Wedeneh:NS!R3asthecompositionofgandthisimmersion. Lemma8.3.3 LetN~Sbethenormalbundleof~SinM4.Thenthereisageneralpositionmapping~h:N~S!R3suchthattheset~Sisthesingularsetof~handeverycomponentof~Shasatleastonecuspsingularpointof~h.Proof.ForacloseddiscD~S,therestrictionofthebundleN~S!~Sto~SnDisatrivialbundle~SnDD2!~SnD,whereD2isthediscasinLemma 8.3.2 .LetI3denotethesegment)]TJ/F15 11.955 Tf 9.298 0 Td[(3;3.Thefunctionm1:D2!I3denedbym1x;y=x2)]TJ/F22 11.955 Tf 12.531 0 Td[(y2isaMorsefunctionwithonesingularpointattheorigin.Letid1:~SnD!~SnDbetheidentitymapping.PutE1=~SnDD2andB1=~SnDI3.Thenid1m1:E1!B1isafoldmapping.SetE2=S1)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1D2andB2=S1)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1I3.Thereisamappingg:E2!B2suchthat1thesetS1)]TJ/F15 11.955 Tf 9.299 0 Td[(1;0f;0gisthesetofallindenitefoldsingularpointsofg,2thesetS1;1f;0gisthesetofalldenitefoldsingularpointsofg,3thecurveS1f0gf;0gisthesetofallcuspsingularpointsofg.LetUdenotetheintersectionof~SnDandacollarneighborhoodof@~SnDin~SnD.ThenUisdieomorphictoS1)]TJ/F15 11.955 Tf 9.299 0 Td[(1;)]TJ/F15 11.955 Tf 9.299 0 Td[(1=2.WecanidentifythesubsetUD2ofE1withthesubsetS1)]TJ/F15 11.955 Tf 9.298 0 Td[(1;)]TJ/F15 11.955 Tf 9.299 0 Td[(1=2D2ofE2andthesubsetUI3ofB1withthesubsetS1)]TJ/F15 11.955 Tf 9.298 0 Td[(1;)]TJ/F15 11.955 Tf 9.299 0 Td[(1=2I3ofB2sothattheobtainedsetsE1[E2andB1[B2aremanifoldsandthemappingid1m1coincideswiththemappinggonthecommonpartofthedomainsE1E2E1[E2.Thus,id1m1andgdeneacuspmappingc:E1[E2!B1[B2.NotethatE1[E2isdieomorphicto~SnDD2andB1[B2isdieomorphicto~SnDI3.Letm3:D2!I3betheMorsefunction,denedbym3x;y=x2+y2,andid3:D!Dbetheidentitymappingoftheopen2-discD=Dn@D.Thenid3m3:DD2!DI3isafoldmapping.LetVbetheintersectionofD

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44 andatubularneighborhoodof@Din~S.ThenVisdieomorphictoS1=2;1.WeidentifythepartVD2ofE3=DD2withthepartS1=2;1D2ofE2E1[E2andthepartVI3ofB3=DI3withthepartS1=2;1I3ofB2B1[B2sothat1theobtainedsetsE=E1[E2[E3andB=B1[B2[B3aremanifolds,2themappingid3m3coincideswithconthecommonpartofthedomains,3themanifoldEisdieomorphictoN~S.Theconditioncanbeachievedsincethemappingm3doesnotchangeunderrotationsoftheberD2.Thenid3m3andcdeneacuspmappingN~S!B.NotethatB~SI3isanopenorientable3-manifold.Thus,itadmitsanimmersionintoR3.ThecompositionofN~S!BandtheimmersionB!R3isacuspmappingsatisfyingtheconditionsofthelemma.WeidentifyNSandN~SwithopentubularneighborhoodsofSand~SinM4respectively.Thereisageneralpositionmappingg:M4!R3whichextendsh:NS!R3and~h:N~S!R3.Ingeneraltheextensionghassomeswallowtailsingularpoints.Letusprovethatwemaychoosegtobeacuspmapping.AndoseeSection5in[ 3 ]showedthattheobstructiontotheexistenceofasectionofthebundle2TM4;TR3overtheorientableclosed4-manifoldM4coincideswiththenumberoftheswallowtailsingularpointsofageneralpositionmappingM4!R3modulo2.AlsoAndocalculatedthatthisobstructionistrivial.Sincethemappingh[~hdoesnothaveswallowtailsingularpoints,theobstructiontotheexistenceofanextensionofthesectionj3h[~h,denedoverNS[N~S,toasectionof2overM4istrivial.Therefore,therelativeversionoftheAndo{Eliashbergtheoremseechapter 7 impliestheexistenceofanextensiontoacuspmappingg:M4!R3.

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45 ThesingularsetSgconsistsofS[~SandprobablyofsomeotherconnectedsubmanifoldsA1;:::;AkofM4.Wehaveef=eSg=eS+e~S+eA1+eA2++eAk: .1 ThenormalEulernumberofthesubmanifoldSequalsef.HencethesumofthenormalEulernumberseA1++eAkequals)]TJ/F22 11.955 Tf 9.298 0 Td[(e~S.LetAtbeacomponentof[Ai.SupposeAtisasurfaceofdenitefoldsingularpoints.ThesurfaceAtisorientableseeLemma 4.2.1 anddoesnotintersectS[~S.Bydenitionof~S,thisimplieseAt=0.SupposeAtisasurfaceofindenitefoldsingularpoints.ThenagaineAt=0[ 30 ],[ 1 ].Therefore,eAtisnontrivialonlyifthesurfaceAtcontainscuspsingularpoints.LetusrecallthattheunionofthosecomponentsofthesingularsubmanifoldSgthatcontaincuspsingularpointsisdenotedbyC=Cg.Theequation 8.1 impliesthateC=e~S+eA1++eAk=0.Itremainstoprovethefollowinglemma. Lemma8.3.4 Ifg:M4!R3isacuspmappingandeC=0,thenthereexistsahomotopyofgeliminatingallcuspsingularpoints.Proof.Ifthecurveofcuspsingularpointsisnotconnected,thenthereexistsahomotopyofgtoamappingwithonecomponentofthecurveofcuspsingularpoints.WemayrequirethatthehomotopypreservesthenumbereC.Weomittheproofofthesefactssincethereasoningsaresimilartothoseinsection 4.5 .WewillassumethatthecurveofcuspsingularpointsgisconnectedandhencesoisCg. Lemma8.3.5 Letxbeacharacteristicvectoreldong.IfeC=0,thenxcanbeextendedonCgasanormalvectoreld.Proof.Forageneralpositionmappingg:M4!R3,thesetF=f)]TJ/F21 7.97 Tf 6.587 0 Td[(1fC)]TJ/F15 11.955 Tf 7.085 1.794 Td[(g#M4

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46 isanimmersed3-manifold.Theself-intersectionpointsofFcorrespondtothepointsofthesurfaceC)]TJ/F15 11.955 Tf 7.085 1.793 Td[(g.Wesaythattwovectorsv1andv2ofavectorspacehavethesamedirectionifv1=v2forsomescalar6=0.Thereisanunorderedpairofdirectionsl1p;l2poverC)]TJ/F15 11.955 Tf 7.085 1.793 Td[(g[ 1 ]withthefollowingproperty.ForeverypointpofC)]TJ/F15 11.955 Tf 7.084 1.794 Td[(gthereareaneighborhoodUaboutpwithcoordinatesx1;x2;x3;x4andacoordinateneighborhoodaboutgpsuchthattherestrictiongjUhastheformx1;x2;x23)]TJ/F22 11.955 Tf 12.444 0 Td[(x24andthedirectionsofthevectors@=@x3and@=@x4coincidewithl1pandl2prespectively.AnL-pairisapairl1p;l2pthatsatisesthisproperty.LetF1C)]TJ/F15 11.955 Tf 7.085 1.794 Td[(gdenotethecomplementofaregularneighborhoodofthecurveginC)]TJ/F15 11.955 Tf 7.084 1.794 Td[(g.TheproofofLemma3in[ 1 ]showsthatthereisavectoreldvpinthenormalbundleoverF1withdirectionsl1p+l2porl1p)]TJ/F22 11.955 Tf 12.286 0 Td[(l2povertheboundary@F1forsomeL-pairl1p;l2p.Wesaythatadirectionatacuspsingularpointisanx-directionifitistangenttothesurfaceSfandtransversaltothecurveg.Notethatforaspecialcoordinateneighborhoodaboutacuspsingularpointthedirectionofthevector@=@xhasanx-direction.ItiseasilyveriedthatforanL-pairl1p;l2p,thedirectionsl1pl2paretangenttoFateverypointpinC)]TJ/F15 11.955 Tf 7.085 1.793 Td[(f.Furthermorethedirectionsl1p+l2pandl1p)]TJ/F22 11.955 Tf 12.509 0 Td[(l2papproachthesamex-directionaspapproachesg.ItimpliesthatthevectoreldvpoverF1hasanextensionto C)]TJ/F15 11.955 Tf 7.084 1.793 Td[(gsuchthatvpistransversaltoC)]TJ/F15 11.955 Tf 7.085 1.794 Td[(gateverypointofC)]TJ/F15 11.955 Tf 7.085 1.794 Td[(gandhasanx-directionateverypointofg.Ifnecessary,wemultiplythevectoreldvpby)]TJ/F15 11.955 Tf 9.298 0 Td[(1togetavectoreldwhichpointstowardC)]TJ/F15 11.955 Tf 7.085 1.793 Td[(goverg.NowthevectoreldvpcanbemodiedinaneighborhoodofgsothatanewvpisnormaltoC)]TJ/F15 11.955 Tf 7.084 1.793 Td[(gateverypointofC)]TJ/F15 11.955 Tf 7.084 1.793 Td[(gandtherestrictionofvptogisthecharacteristicvectoreldp.

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47 TheobstructiontotheexistenceofanextensionofvptoavectoreldoverCgisthenormalEulernumbereC.SinceeC=0,suchanextensionexists.Theendsofthevectorsp,p2C+g,deneanembeddingofanorientablesurfaceHdieomorphictoC+gintoM4.Wemodifytheembeddinginaneigh-borhoodoftheboundary@Hsothatthenewembeddingdenesabasisofsurgery.NowLemma 8.3.4 followsfromLemma 4.5.1 .TheproofofLemma 8.3.1 iscomplete.

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CHAPTER9COMPUTATIONOFTHESECONDARYOBSTRUCTIONInthischapterwewillexpressthesecondaryobstructionintermsofthePontrjaginclassp1M4ofthetangentbundleofM4.Namely,wewillprovetheformula 1 ef=p1M4;[M4],where[M4]isthefundamentalclassofthemanifoldM4. Lemma9.0.6 Letfbeageneralpositionmappingfromaclosedorientedconnected4-manifoldM4intoanorientable3-manifoldN3.Letp1M4denotetherstPontrjaginclassofM4and[M4]thefundamentalclassofM4.Thenef=p1M4;[M4].Itallowsustoformulatethetheorem 8.1.1 intermsofthecohomologyringofM4. Theorem9.0.1 Letf:M4!N3beacontinuousmappingfromanorientableclosedconnected4-manifoldintoanorientable3-manifold.Thenthehomotopyclassoffcontainsafoldmappingifandonlyifthereisacohomologyclassx2H2M4;Zsuchthatp1M4=x2.Asmoothmappingf:M4!N3inducesasectionj2fofthe2-jetbundleJ2TM4;fTN3overM4.TocalculatetheinvariantefweconsidersectionsM4!J2;,whereisanarbitraryorientable4-vectorbundleoverM4andisanarbitraryorientable3-vectorbundleoverM4. 1Afterthepaperwaswritten,theauthorlearnedthatthisequalityisaspecialcaseofaresultobtainedin[ 24 ]. 48

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49 ThesingularsetinthebundleJ2;overM4isamanifoldwithsin-gularities.Bydimensionalreasonings,theimageofageneralpositionsectionj:M4!J2;doesnotcontainsingularpointsofthemanifoldwithsingular-ities.Consequently,thesingularsetj)]TJ/F21 7.97 Tf 6.586 0 Td[(1ofthesectionjisasubmanifoldofM4.WedenethenormalEulernumberejofthesectionjasthenormalEulernumberej)]TJ/F21 7.97 Tf 6.587 0 Td[(1.AregularneighborhoodEofinJ=J2;isanopenmanifold.ThereisasystemoflocalcoecientsFoverE,therestrictionFjofwhichgivesaZ-orientationof.ThePoincarehomomorphismforcohomologiesandhomologieswithtwistedcoecientstakesthefundamentalclass[]ontosomeclass2H2E;En;F.Notethat`isinH4E;En;Z.LetibethecompositionH4E;En;Z!H4J;Jn;Z!H4J;;;Zoftheexcisionisomorphismandthehomomorphisminducedbytheinclusion.Wedeneh;=i`.Thenweclaimthatej=jh;;[M4]:.1 Lemma9.0.7 Foreverygeneralpositionsectionj:M4!J2;,thenormalEulerclassofthesurfacej)]TJ/F21 7.97 Tf 6.587 0 Td[(1isgivenby 9.1 .Inparticular,foreverymappingf:M4!N3,wehaveef=j2fhTM4;fTN3;[M4].Proof.LetAM4denotethesingularsetj)]TJ/F21 7.97 Tf 6.587 0 Td[(1andBdenoteatubularneighborhoodofA.ThetubularneighborhoodsEofandBofAmaybeviewedasvectorbundles.Sincej:B!Eistransversalto,thereisacommutativediagramofvectorbundlesB)306()222()306(!E??y??yAj)306()222()306(!:

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50 TheThomclasspossessesthepropertyofnaturality.HencetheThomclassofthebundleB!Aisj.Wehaveacommutativediagrami:H4E;En;Z)306()222()306(!H4J;Jn;Z)306()222()306(!H4J;Zj??yj??yj??yH4B;BnA;Z)306()222()306(!H4M4;M4nA;Z)306()222()306(!H4M4;Z;whichcompletestheproof.Lemma 9.0.7 showsthatthenumberejdependsonlyonthebundlesand.Thatiswhywewilldenotethisnumberbye;.Inthefollowing,foranarbitrarymanifoldV,wedenotethetriviallinebundleoverVbyVorsimplyby. Lemma9.0.8 Thereisanintegerk6=0suchthatforanyorientable4-vectorbundleoveranyclosedoriented4-manifoldM4,theequalityp1;[M4]=ke;3holds.Proof.Werecallseechapter 7 thatthebundleJ2;3overM4isinducedbyanappropriatemapping:M4!BSO4BSO3fromsomebun-dleJ2E4;E3overBSO4BSO3.AsabovewedeneacohomologyclasshE4;E32H4J2E4;E3;Z.LetbeanarbitrarysectionofthebundleJ2E4;E3.Togetherwith,thesectiondenesasectionj:M4!J2;3suchthatthediagramJ2;3)306()222()306(!J2E4;E3x??jx??M4)306()222()306(!BSO4BSO3commutes.Wehavejh;3=j~hE4;E3=hE4;E3;where~denotestheupperhorizontalhomomorphismofthediagram.Con-sequently,theclassjh;3isinducedbyfromsomeclasshE4;E3in

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51 H4BSO4BSO3;Z.Moreover,since3isatrivialbundle,themappingishomotopictoamappingM4!BSO4ptBSO4BSO3andthereforejh;3isinducedfromsomeclassinH4BSO4;Z.ModulotorsionthegroupH4BSO4;ZisisomorphictoZZandisgeneratedbytherstPontrjaginclassp1andtheEulerclassW4.SinceH4M4;Zistorsionfree,forsomeintegerskandl,wehavejh;3=kp1+lW4:.2Letusapply 9.2 tothe4-sphereS4with=TS4.Thesingularsetofthestandardprojectionf:S4,!R4!R3isa2-spherewithtrivialnormalbundleinS4.Hencejh;3;[S4]=ef=0.Sincep1TS4=0andW4TS4=2,weconcludethatl=0.Finally,k6=0followsfromp16=0forsome.TondthenumberkofLemma 9.0.8 weneedanotherdescriptionoftheinvariantej.Let,andbevectorbundlesoveramanifoldM4.Therearenaturalprojectionspr:r!randinclusioni:!.ApointofJn;isasetofnhomomorphismsfgigi=1;:::;nseechapter 7 .Denetheembeddingsn:Jn;!Jn;bysng1;:::;gk=g1id;ig2p2;:::;ignpn:Thehomomorphismsn,n1,iscalledthestabilizationhomomorphismaordedby. Lemma9.0.9 Ronga[ 28 ] 1. s)]TJ/F21 7.97 Tf 6.587 0 Td[(12i;=i;, 2. s)]TJ/F21 7.97 Tf 6.587 0 Td[(12i;j;=i;j;,and

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52 3. theembeddings2istransversaltothesubmanifoldsi;j;andi;.Letbeanorientable4-vectorbundle,thetrivial3-vectorbundleoverM4,ands2thestabilizationhomomorphismaordedbythetriviallinebundleoverM4.Lemma 9.0.9 allowsustogiveadenitionofefintermsofsomecohomologyclassofH4J2;;Z. Lemma9.0.10 Thereisacohomologyclassh2H4J2;;Zsuchthatforasectionj:M4!J2;,wehaveej=s2jh;[M4].TheproofofLemma 9.0.10 isthesameasthatofLemma 9.0.7 .Letandbethevectorbundlesofdimensions4and3respectivelyoverthestandard4-discD4.ThesetofregularpointsinJ2;ishomotopyequivalenttoSO5.Therefore,eachsectionj:S3!J2;thatsendsS3intothesetofregularpointsdenesanelement~jinthesetofhomotopyclasses[S3;SO5].ThespaceSO5isanH-space;hence~jisanelementof3SO5=Z.SinceJ2;iscontractible,thesectionjadmitstheextensiontoasectionoverD4.Weobtainamappinge:3SO5!Zthatsendsthehomotopyclass~jofasectionjtothenormalEulernumberofthesingularsetofthesectionjextendedoverD4. Lemma9.0.11 Themappinge:3SO5!Zisawelldenedhomomorphism.Proof.Letj1andj2betwosectionsofthebundleJ2;overD4whoserestrictionsj1j@D4andj2j@D4mapS3=@D4intothesetofregularpointsofJ2;andrepresentthesamehomotopyclass~j2[S3;SO5].TheargumentssimilartothoseintheproofofLemma 5.3.1 showthatthenormalEulernumbersofthesubmanifoldsj)]TJ/F21 7.97 Tf 6.586 0 Td[(11andj)]TJ/F21 7.97 Tf 6.586 0 Td[(12,whereisthesingularsetofJ2;,areequal.Therefore,thenumbere~jdoesnotdependonthechoiceofrepresentativeofthehomotopyclass~j.

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54 LetJ1andJ0respectivelydenotethespaceJ1;4andthecomplementtothesingularsetinJ1.ToproveLemma 9.0.12 ,itsucestodeterminethenormalEulernumberofj)]TJ/F21 7.97 Tf 6.586 0 Td[(1foraparticularsectionj:S4!J1.WeregardasectionofJ1asabundlehomomorphism!4.IfjisgivenoverD1,thenthediagramj@D1)306()222()306(!j@D2??yjj@D1??yidjj@D1)]TJ/F18 5.978 Tf 5.757 0 Td[(14j@D1id)306()222()306(!4j@D2showsthatinthetrivializationofoverD2thesectionjj@D2isidjj@D1)]TJ/F21 7.97 Tf 6.586 0 Td[(1,whereidistheidentitymapping.IfwechoosejtobeconstantoverD1,theninthetrivializationofoverD2thesectionjj@D2inducesamappingS3!J0SO5representingthehomotopyclass)]TJ/F15 11.955 Tf 9.298 0 Td[([]23SO5.ThusthenormalEulernumberofjextendedoverD2is#uptosign. Lemma9.0.13 ThereisanintegerqsuchthateTCP2;3=1+q#.Proof.Thereisamapping~fofaregularneighborhoodEofCP1CP2intoR3suchthatthesingularsetof~fisCP1seeLemma 8.3.2 .Letfbeageneralpositionextensionof~fonCP2.ThenumberefisthesumofthenormalEulernumberofCP1andthenormalEulernumberofthesurfaceofsingularpointsthatliesinthediscD4=CP2nE.Thelatternumberisamultipleof#.Henceforsomeq,ef=1+q#.TocalculatetheexactvalueofeTCP2;3weusethenotionoftheconnectedsumoftwobundles.Fori=1;2,letM4ibeaclosedoriented4-manifoldandianorientable4-vectorbundleoverM4i.Identifyingtheberof1oversomepointinM41withtheberof2oversomepointinM42,weobtainabundleoverM41_M42,whichistransferredtoabundleoverM41#M42byanaturalmappingM41#M42!M41_M42.WedenotetheresultingbundleoverM41#M42by1#2.Itfollowsthatthe

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55 additivitypropertiesp11#2;[M41]M42]=p11;[M41]+p12;[M42]ande1#2;3M41]M42=e1;3M41+e2;3M42takeplace. Lemma9.0.14 eTCP2;3=3:Proof.Letbethe4-vectorbundleoverS4withp1=2.Lemma 9.0.12 impliesthate;3=#.ForK=2 CP2,wehavep1TK##3=0,where#3standsfor##.Lemma 9.0.8 showsthateTK##3;3isalsozero.Byadditivity,0=eTK##3;3=)]TJ/F15 11.955 Tf 9.298 0 Td[(21+q#3#:Sinceqand#areintegers,weconcludethateTK;3=6.ThisimpliesthateTCP2;3=3.OntheotherhanditisknownseeSakuma[ 36 ]and[ 37 ]thateTCP2;33mod4.Therefore,eTCP2;3=3.Lemma 9.0.14 showsthattheintegerkinLemma 9.0.8 equals1.Thus,foreveryoriented4-manifoldM4andageneralpositionmappingf:M4!R3,wehaveef=p1M4;[M4].ThiscompletestheproofofLemma 9.0.6 .Theorem 9.0.1 isproved.

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CHAPTER10SIMPLYCONNECTEDCASEThischapterisdevotedtothecasewherethemanifoldM4issimplycon-nected.Weexaminetheequationp1M4=x2anddeterminewhenithasasolution. Theorem10.0.2 SupposethatM4isanorientableclosedconnectedsimplyconnected4-manifoldandN3isanorientable3-manifold.Thenahomotopyclassofamappingf:M4!N3hasnofoldmappingifandonlyifM4ishomotopyequivalenttoCP2orCP2#CP2.Herehomotopyequivalenceisnotsupposedtobeorientationpreserving.Ashasbeenshown,ahomotopyclassofageneralpositionmappingffromaconnectedclosedoriented4-manifoldM4intoanorientable3-manifoldN3hasafoldmappingifandonlyifef=p1M4;[M4]2QM4.Thatisthenumberp1M4;[M4]isavalueoftheintersectionformofM4.First,letusconsiderthecasewheretheintersectionformofM4isindenite.Ifp1M4=0,thenforeveryf,ef=02QM4.Supposep1M46=0. Lemma10.0.15 IftheintersectionformofaclosedsimplyconnectedmanifoldM4withp1M46=0isindeniteodd,thenQM4=Z.Inparticularp1M4;[M4]2QM4.Proof.SincetheintersectionformofM4isodd,inH2M4;Zthereexistsabasisg1;g2;:::;gs;:::;gksuchthatthevalueoftheintersectionformat1e1++ses++kekis21++2s)]TJ/F22 11.955 Tf 11.015 0 Td[(2s+1)-143()]TJ/F22 11.955 Tf 37.695 0 Td[(2k.Therefore,thenumberefisinQM4ifandonlyifefcanberepresentedintheform21++2s)]TJ/F22 11.955 Tf 11.632 0 Td[(2s+1)-195()]TJ/F22 11.955 Tf 39.546 0 Td[(2kforsomeintegersi,i=1;:::;k.Sincetheintersectionformisindenite,thissumhasatleastonepositivesquareandatleastonenegativesquare.Sincethesignature 56

PAGE 65

57 M4=1 3p1M46=0,thenumberkofsquaresisatleast3.Supposethatthenumberefisodd.Thenitcanberepresentedasthedierenceoftwosquares.Supposethatefiseven.Thentheoddnumberef1canberepresentedasthedierenceoftwosquaresandthethirdsquareofthesumcanbeusedtoadd1tothedierencetogetef.HenceQM4=Z.SupposethattheintersectionformofM4isindeniteeven.Beingeven,itisisomorphictoadirectsumofsomecopiesofE8-formandsomecopiesoftheformwithmatrix0B@01101CA.Consequently,thenumberp1M4;[M4]=3M4iseven.Everyevenindeniteintersectionformcontainsasubformisomorphicto0B@01101CA.Sincethissubformtakeseveryevenvalue,wehavep1M4;[M4]2QM4.Thus,everyclosedsimplyconnected4-manifoldwithindeniteortrivialintersectionformadmitsafoldmappingintoR3.TotreatthecasewheretheintersectionformofM4isdenite,weneedtheDonaldsonTheorem.LetkJ,k6=0,denotetheformofrankjkjgivenbythediagonalmatrixwitheigenvalues1ifk>0and)]TJ/F15 11.955 Tf 9.298 0 Td[(1ifk<0. Theorem10.0.3 Donaldson[ 7 ]Everydeniteintersectionformofaclosedorientedsmooth4-manifoldisisomorphictotheformkJforsomeintegerk6=0. Lemma10.0.16 SupposethattheintersectionformofaconnectedclosedsimplyconnectedmanifoldM4withp1M46=0isdenite.Thenp1M4;[M4]2QM4ifandonlyiftheintersectionformisisomorphictokJ,jkj3.Proof.Itsucestoconsideronlythecasewherek>0.Ifk=1,thentheinter-sectionformisisomorphictothatofCP2andp1M4;[M4]=3M4isnotinQM4.Fork=2,thesetQM4consistsonlyofintegersthatcanberepresentedasthesumofatmosttwosquares.Hencethenumberp1M4;[M4]=6isnotinQM4.Ifk=3,thenp1M4;[M4]=92QM4.Finally,bytheLagrange

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58 theorem,everypositiveintegercanberepresentedasasumoffoursquares.Thusfork4,wehavep1M4;[M4]2QM4.InviewofTheorem 8.1.1 ,Lemmas 10.0.15 and 10.0.16 implythatM4admitsafoldmappingintoR3ifandonlyiftheintersectionformofM4isdierentfromJand2J.BytheJ.H.C.WhiteheadTheoremabouttheorientedhomotopytypeofasimplyconnected4-manifold[ 22 ],[ 41 ],thiscompletestheproofofTheorem 10.0.2

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CHAPTER11FINALREMARKSRemark1.IftwomanifoldsM41andM42admitafoldmappingintoR3,thentheconnectedsumM21#M22alsoadmitsafoldmappingintoR3.In[ 34 ]theauthorsconjecturedthattheobstructiontotheexistenceofafoldmappingintoR3isadditivewithrespecttoconnectedsum,andthemanifoldkCP2#l CP2admitsafoldmappingintoR3ifandonlyifk+lisodd.Theorem 10.0.2 solvestheconjectureinthenegative.Remark2.SakumaconjecturedseeRemark2.3in[ 19 ]thataclosedorientablemanifoldwithoddEulercharacteristicdoesnotadmitafoldmappingintoRnforn=3;7.Saeki[ 33 ]presentedanexplicitcounterexampletothisconjecture.Theorem 10.0.2 showsthattherearemanymanifoldswithoddEulercharacteristicadmittingfoldmappingsintoR3.However,itshouldbementionedthatTheorem 10.0.2 doesnotsuggestamethodofanexplicitconstructionandthequestionofanexplicitconstructionofafoldmappingforagivenmanifoldseemsdicult.Remark3.IfthemanifoldsMmandNnareorientableandm)]TJ/F22 11.955 Tf 12.691 0 Td[(nisodd,theneveryMorinmappingf:Mm!Nnishomotopictoamappingwithatmostcuspsingularpoints[ 29 ].Theorem 10.0.2 givesarestrictiononfurthersimplicationofMorinmappingsbyhomotopy. 59

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61 [15] M.W.Hirsch,Immersionsofmanifolds,Trans.Amer.Math.Soc.,9359,242{276. [16] K.Janich,SymmetrypropertiesofsingularitiesofC1-functions,Math.Ann.,238978,147{156. [17] M.Kazarian,ClassifyingspacesofsingularitiesandThompolynomials,NewdevelopmentsinSingularityTheoryCambridge2000,NATOSci.Ser.IIMath.Phys.Chem,21,KluwerAcad.Publ.,Dordrecht,2001,117-134. [18] M.Kazarian,Homepage,http://genesis.mi.ras.ru/kazarian,lastaccessedon04/08/04. [19] S.Kikuchi,O.Saeki,Remarksonthetopologyoffolds,Proc.Amer.Math.Soc.,12395,905{908. [20] H.Levine,Eliminationofcusps,Topology,3suppl.265,263{296. [21] W.S.Massey,ProofofaconjectureofWhitney,PacicJ.Math.,31969,143{156. [22] J.Milnor,Onsimplyconnected4-manifolds,Symp.Int.deTopologiaAlge-braica,Univ.deMexico,1958,122{128. [23] B.Morin,Formescanoniquesdessingularitesd'uneapplicationdierentiable,ComtesRendus,2601965,5662-5665,6503-6506. [24] T.Ohmoto,O.Saeki,K.Sakuma,Self-intersectionclassforsingularitiesanditsapplicationtofoldmaps,Trans.Amer.Math.Soc.,35503,3825{3838. [25] R.Rimanyi,GeneralizedPontrjagin-Thomconstructionforsingularmaps,Doctoraldissertation,EotvosLorandUniversity,Budapest,1996. [26] R.Rimanyi,ComputationoftheThompolynomialof1111viasymmetriesofsingularities,ChapmanandHall/CRCResearchNotesinMathematics412000,110{118. [27] R.Rimanyi,Thompolynomials,symmetriesandincidencesofsingularities,Inv.Math.,14302,499{521. [28] F.Ronga,LecalculdesclassesdualesauxsingularitesdeBoardmand'ordredeux,Comment.Math.Helv.,47972,15{35. [29] R.Sadykov,TheChessconjecture,AlgebraicandGeometricTopology,3003,777{789. [30] O.Saeki,Notesonthetopologyoffolds,J.Math.Soc.Japan,44992,551{566.

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62 [31] O.Saeki,StudyingthetopologyofMorinsingularitiesfromaglobalviewpoint,Math.Proc.Camb.Phil.Soc.,117995,117{223. [32] O.Saeki,Foldmapson4-manifolds,Comment.Math.Helv.,782003,627-647. [33] O.Saeki,Topologyofsingularbersofdierentiablemaps,LectureNotesinMathematics1854,Springer,2004. [34] O.Saeki,K.Sakuma,Eliminationofsingularities:Thompolynomialsandbeyond,LondonMath.Soc.,LectureNotes,Ser.26399,291{304. [35] O.Saeki,K.Sakuma,Stablemapsbetween4-manifoldsandeliminationoftheirsingularities,J.LondonMath.Soc.,5999,1117-1133. [36] K.Sakuma,Onspecialgenericmapsofsimplyconnected2n-manifoldsintoR3,TopologyAppl.,5093,249{261. [37] K.Sakuma,Anoteonnonremovablecuspsingularities,HiroshimaMath.J.,312001,461{465. [38] D.Spring,Convexintegrationtheory:solutionstotheh-principleingeometryandtopology,MonographsinMath.,92,Birkhauser,1998. [39] R.Thom,Lessingularitiesdesapplicationsdierentiables,Ann.Inst.FourierGrenoble,6955{1956,43{87. [40] C.T.C.Wall,Asecondnoteonsymmetryofsingularities,Bull.London.Math.Soc.,12980,347{354. [41] J.H.C.Whitehead,Onsimplyconnected,4-dimensionalpolyhedra,Com-ment.Math.Helv.,22949,48{92. [42] H.Whitney,Onsingularitiesofmappingsofeuclideanspaces,I.Mappingsoftheplaneintotheplane,Ann.Math.,62955,374{410.

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Full Text

SINGULARITIES OF SMOOTH MAPS

By

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2005

by

ACKNOWLEDGMENTS

The problem which I solve in the dissertation appeared in the early 90s in a

paper by Osamu Saeki. I learned about the problem 4 years ago, before I become a

graduate student. Since that time many people helped me.

I wish to express my sincere gratitude to teacher and thesis adviser Alexander

Nikolaevich Dranishnikov, to whom I owe thanks not only for valuable discussions

but also for assisting me at every stage of the work.

I am grateful to Peter Akhmetev for teaching me global singularity theory, and

to Yuli Borisovich Rudyak for explaining ideas of algebraic topology. I am grateful

to my friends, Sergey Melikhov and Yuri Turygin, for their interest in the work and

for discussions of this and closely related topics.

I gave the proofs of the main statements at the graduate topology seminars. I

would like to thank Alexander Nikolaevich-who organized and led the seminars

and the other participants for their help in checking the proofs.

I am happy to thank Osamu Saeki whose papers motivated my interest in

the problem and led to the dissertation. I also grateful to Osamu Saeki for the

invitation and support of my trip to Japan, where I could present the results and

learn about new developments in singularity theory.

Finally, I appreciate the generous help of an .. iril: mous referee, who helped

me to correct erroneous claims and whose numerous comments and recommenda-

tions considerably improved the presentation of results.

page

ACKNOWLEDGMENTS ................... ...... iii

LIST OF FIGURES ................................ v

ABSTRACT ...................... ............. vi

1 INTRODUCTION .................... ....... 1

1.1 Regular Points of Smooth Mappings ......... ...... .. 1
1.2 Singular Points of Smooth Mappings ...... ....... ... 2
1.3 Mappings of Surfaces into 3-dimensional Spaces . . .. 2
1.4 Mappings Between Surfaces. .................. 3

2 THOM-BOARDMAN SINGULARITIES ....... . .... 7

2.1 Naive Definition .................. ... 7
2.2 Finite Jet Space . . . . . . 8
2.3 Infinite Jet Space .................. ........ .. 10
2.4 Thom-Boardman Singularities ............ .. .. .. 11

3 THOM POLYNOMIALS .................. ........ .. 14

4 SINGULARITIES OF MAPPINGS .31 -- N3 ............... 17

4.1 Normal Forms of Singularities ............ .. .. .. 17
4.2 Topological Description of Singularities ..... . . 17
4.3 Behavior of the Mapping in a Neighborhood of 7 . .... 19
4.4 Bifurcations of Singular Sets ................ .. .. 24
4.5 Surgery of 7(f) ................... ....... 26

5 THE INVARIANT e(f) .................. ........ .. 29

5.1 First Definition .................. ....... .. .. 29
5.2 Second Definition ..... . . ..... ........ 30
5.3 Invariance of the Secondary Obstruction Under Homotopy . 31

6 H-PRINCIPLE .................. ............. .. 32

6.1 Differential Relations .................. ..... .. 32
6.2 H-Principle .. .. ... .. .. .. .. ... .. .. .. ... .. .. 32
6.3 Morin Singularities. .................. ....... .. 33
6.4 Ando-Eliashberg Theorem .................. ..... 34

7 UNIVERSAL JET BUNDLES ................... .... 36

7.1 Construction 1 ............... .......... ..36
7.2 Construction 2 ............... .......... ..37
7.3 Construction 3 ............ . . ... 38
7.4 Corollary of Ando-Eliashberg Theorem ..... . . 39

8 SECONDARY OBSTRUCTION .................. ..... 40

8.1 Definition of the Secondary Obstruction . . ..... 40
8.2 Necessary Condition .................. ..... .. 40
8.3 Sufficient Condition .................. .. 41

9 COMPUTATION OF THE SECONDARY OBSTRUCTION ....... 48

10 SIMPLY CONNECTED CASE .................. .. 56

11 FINAL REMARKS .................. ......... .. 59

REFERENCE LIST .................. ............. .. 60

BIOGRAPHICAL SKETCH .................. ......... .. 63

LIST OF FIGURES
Figure page

1-1 Whitney umbrella .................. ........ 3

1-2 Fold and cusp singular points ................ ...... 4

4-1 The image of a swallowtail singular point. .. . ..... 18

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SINGULARITIES OF SMOOTH MAPS

By

August 2005

C'!h In: Alexander N. Dranishnikov
Major Department: Mathematics

A singular point of a smooth mapping f : M -- N of manifolds is a point

at which the rank of f is less than the minimum of dimensions of M and N.

Singularities of smooth mappings have a nice classification, with respect to which

for almost any smooth mapping f, the set of singular points of any type J forms a

smooth submanifold Sj(f) c M. We study those topological properties of the set

Sj(f) that does not change under homotopy of f.

One of the first questions that arises in the singularity theory asks whether a

singularity type J is inessential for a mapping f; in other words, does there exist

a homotopy of f eliminating all the 3-singular points? The primary obstruction

is defined as the cohomology class [Sj(f)] E H*(M; Z2) dual to the closure of

Sj(f). Remarkably, the class [S3(f)] is a polynomial, called Thom polynomial, in

Stiefel-Whitney classes of the tangent bundle TM and the induced bundle f*TN.

The Thom polynomial turns out not to be a complete obstruction; O. Saeki

constructed an example of a mapping from a 4-manifold into a 3-manifold where

the cohomology obstruction corresponding to certain singularities, cusps, is trivial

though a homotopy to a general position mapping without cusp singular points

does not exist.

We consider smooth mappings of 4-manifolds into 3-manifolds, determine

the secondary obstruction, prove its completeness and express it in terms of the

cohomology ring of the source manifold.

Definition A general position mapping of a 4-manifold into 3-manifold without

cusp singular points is called a fold mapping.

Theorem For a closed oriented 4-u,,i,.I.: [ 1', the following conditions are

equivalent:

(1) .1' admits a fold i'".'l'.:. y into R3;

(2) for every orientable 3- I,.;,:..' .1/ N3, every homot('i.,; class of I,,I,.'':,i. of im1
into N3 contains a fold i'l ''l.: l.

(3) there exists a, 1. i,'i,,. ',i/;/ class x E H2(i 1; Z) such that x x x is the first
Pontrjagin class of iF\.

For a simply connected manifold 3 1T, we show that .[1 admits no fold

mappings into N3 if and only if [1T is homotopy equivalent to CP2 or CP2# P2.

CHAPTER 1
INTRODUCTION

1.1 Regular Points of Smooth Mappings

Given a smooth mapping f of a manifold M of dimension m into a smooth

manifold N of dimension n, the differential df(x) of the mapping f at a point x

of M is a linear map from the tangent space T1M of M at x to the tangent space

Tf(x)N of N at f(x),

df(x) : TM Tf(,) N.

We -,-v that x E M is a ,.iJ.'; point of the in'rll.:.l f if the rank rkx(f) of the

differential df(x) is exactly max(m, n). Otherwise we -~v- that the point x is a

S..:!,l.r, point of the iq. '.:,;j f.

We observe that the set of regular points forms an open submanifold of

the source manifold. Indeed, if a homomorphism h of vector spaces sends a

set {ei} of independent vectors into a set of independent vectors, then every

homomorphism sufficiently close to h also sends the vectors {ei} into independent

ones. Consequently, if f is a smooth mapping and x is a point of the source

manifold, then the rank of the differential df(x) at x is not greater than the rank of

the differential df(y) at any point y sufficiently close to x. In particular, in a small

neighborhood of a a regular point, the mapping f has no singular points.

The regular points of a mapping have a simple description. In the case of a

positive codimension, n m > 0, the regular points are precisely the points in a

neighborhood of which the mapping f is an embedding. If the mapping f is of a

non-positive codimension, i.e., n m < 0, then the regular points are the points of

the source manifold in a neighborhood of which the mapping f is a submersion.

1.2 Singular Points of Smooth Mappings

We study singularities of smooth mappings up to an equivalence relation.

Definition. Given two mappings fi : MI -- Ni, i = 1, 2, we i- that the points

xl E M1 and x2 c 31 are of the same singularity type with respect to the right-

left equivalence if there are neighborhoods Ui containing xi, neighborhoods 1V

containing fi(xi) and diffeomorphisms g : U -- 2 U2, h : V1 V2 that fit into the

commutative diagram
U1 U2
filUl f2U2

V, h V2,
where the mappings fiUi are the restrictions of the mappings fi to Ui.

It is convenient to describe a right-left singularity type, 'r, by choosing a

normal form, i.e., a mapping g : R' R' with singularity r at the origin. Once

the normal form is chosen, we -- that a mapping f : M N has r-singularity at

a point x E M if in some local coordinate neighborhoods of x in M and f(x) in N,

the mapping f has the form g.

In the two subsequent sections we will consider examples of singularities in the

cases of mappings of manifolds of small dimensions.

1.3 Mappings of Surfaces into 3-dimensional Spaces

One of the singularities of mappings from a surface into a 3-manifold is the

Whitney umbrella.

In a neighborhood of a Whitney umbrella, in some coordinates, the mapping f

has the form

f(u, ) (uv, u, v2).

Theorem 1.3.1 (Whitney) Every in'jrr,,:', of a ,'ifi..: into a 3-ia,,,.',... 43 can be

approximated by a i'.r'll'.,:. with singularities of only Wiili,. .1 umbrella /.

The set of Whitney umbrellas is a discrete set. In particular, a mapping of a

closed surface may have only finitely many Whitney umbrellas.

Figure 1-1: Whitney umbrella.

In fact, the number of Whitney umbrellas is even. To prove this, we describe

the Whitney umbrellas as the end points of self-intersection curves. If the source

surface is closed, then each connected component of self-intersection points is either

a circle which has no end points or a closed interval which has two end points.

Thus the number of Whitney umbrellas is twice the number of closed intervals of

self-intersection points.

If we consider mappings up to homotopy, then the Whitney umbrellas are no

longer essential; every mapping of a surface into a 3-manifold is homotopic to an

immersion. This follows from the Smale-Hirsch h-principle for immersions, which

we will discuss in later sections.

1.4 Mappings Between Surfaces.

Singularities of mappings between surfaces were studied by Whitney who

proved that every continuous mapping of surfaces can be approximated by a

mapping with only regular points, fold singular points, and cusp singular points.

A j. ,.n point of a mapping f, as it has been defined, is a point in a neighbor-

hood of which the mapping f is a diffeomorphism.

The fold and cusp singular types are defined by normal forms. We v that a

singular point p is of the fold type or the cusp type if in some neighborhoods of p

\Z

Figure 1-2: Fold and cusp singular points

and f(p) there are coordinates in which the mapping f has the form

f(x,y) = (x,y2) or fx, y) = 3 + xy,y)

respectively (see figure 1-2). As it follows from the normal forms of singularities,

the set of singular points S of a mapping f with only fold and cusp singular points

forms a smooth curve in the source manifold. The set of cusp points is discrete,

while the set of fold points is the 1-dimensional complement to the cusp points in

S.

We note that the rank of the differential of the mapping f is 1 both at a

point of the fold type and at a point of the cusp type. To distinguish a fold

singular point from a cusp singular point, Whitney considered the restriction of the

mapping f to the smooth curve of singular points S and observed that the cusp

points of f are exactly the singular points of f S (see figure 1-2).

The cusp singular points are essential even if we consider mappings up to

homotopy. For example, the projective plane RP2 does not admit a mapping into

R2 with only fold singular points [42].

Let us sketch a proof that motivates the notion I, .! .. ,gy obstruction."

We note that any two mappings into R2 are homotopic. Thus to prove the

claim it suffices to construct a mapping RP2 IR2 with fold and cusp singularities

and then to show that the cusp singular points can not be eliminated by homotopy.

Construction. Folding a triangle A along the three lines that join the three

middle points of the sides of A defines a 4-to-1 mapping fo from A to the triangle

A1/4 that is 4 times smaller than A. Let us z- that two boundary points x, y of A
are opposite if the segment [x, y] goes through the center of A. We observe that the

fold mapping fo sends the opposite points x, y of the boundary to the same point

fo(x) = fo(y). Thus, the mapping fo defines a mapping fo of the projective plane
obtained from A by identifying the opposite boundary points. Slightly perturbing

the mapping fo and composing it with an embedding of the triangle A1/4 into R2,

we obtain a mapping f from RP2 into R2 with only fold and 3 cusp singular points.

To prove that the cusp singular points of f can not be eliminated by homo-

topy, we need to consider the bifurcations of the set of singular points S that occur
under homotopy of f.

Every homotopy joining two mappings between surfaces can be approximated

by a general position homotopy that changes the set of singular set by isotopy

except for the finitely many points when one of the following bifurcations takes

place [23].

Homotopy under which a circle of singular points with two cusp points appear:

ft(x, y) = (x3 2 tx, ), t [- 1, ],

Homotopy under which two new cusp singular points appear on a curve of fold

singular points:

ft(x, y) = (+ xy t2,Y), t E [-1,],

Homotopy under which a curve of singular points with two cusp points disappear:

ft(x, y) = (x3 2 t ),t [-1,l],

6

Homotopy under which two cusp singular points on a curve S disappear:

ft (x, y) ( + xy + tx2), t [-,].

Thus, under general position homotopy the number of cusp singular points

changes by a multiple of two and therefore every mapping of RP2 into R2 with only

fold and cusp singular points has an odd number of cusp points [42].

CHAPTER 2
THOM-BOARDMAN SINGULARITIES

The right-left equivalence relation on singularities of smooth mappings is

so fine that the number of different singularity types of a mapping is infinite in

general. Besides, the behavior of the set of points of a right-left equivalence class

under homotopy of a mapping has no simple description.

To overcome the difficulties arising here one may consider a coarser relation in

which a class of equivalence is a union of some, perhaps infinitely many, right-left

equivalence classes of singularities. One of such relations pl wing a special role in

singularity theory is the Thom-Boardman classification.

Every continuous mapping of smooth manifolds admits an approximation by

a mapping f with singularities of only finitely many different Thom-Boardman

classes. Furthermore, the set of singular points of f of each Thom-Boardman class

is a submanifold of the source manifold.

2.1 Naive Definition

Let TM and TN denote the tangent bundles of smooth manifolds M and N

respectively and df the differential of a smooth mapping f : M -- N. The set

Si = S(f) is defined as the set of points x in M at which the kernel rank of f is

krf i. Suppose that dim M = m > n = dim N. Suppose that for each i, the set

Si is a submanifold of M, then we can consider the restriction f Si, of f to Si, and

define the singular set Si,~, as the subset S2,(f Si,) of Si,. Again, if every set Sil,i

is a submanifold of M, then the definition may be iterated. Thus, the set Sil,..., i

is defined by induction as Si,(f lSi,.... -). The index J = (i, ...,ik) is called the

symbol of the iin, Ii H;; We will write Sj for Sil,.... .

For example the Whitney fold singular points of a mapping between surfaces

and the Whitney umbrella of a mapping of a surface into a 3-manifold are Thom-

Boardman singular points of the type Si,o. From the Whitney description of

singular points of a mapping between surfaces, it follows that the cusp singular

points are of the type Si,1,o.

Certainly, this natural definition makes sense only under heavy restrictions;

the singular set Si, .... i can be defined only if the singularity stratum Sil...,4-_ is a

submanifold of the source manifold. By passing to jet spaces Boardman was able to

extend the definition over all singular sets.

2.2 Finite Jet Space

A singularity type of a mapping f : M -- N at a point x E M depends on

the behavior of the mapping f only in a small neighborhood of x. So, we pass to

germs. A germ at a point x E M is an equivalence relation on mappings under

which two mappings f, i = 1, 2, defined on a neighborhood of x E M represent the

same germ at x if there is a possibly smaller neighborhood of x where the mappings

fl, f2 coincide.
A k-jet is, by definition, a class of ~k-equivalence of germs. Two germs f and

g at x are ~k-equivalent if at the point x the mappings f and g have the same

partial derivatives of order < k. As partial derivatives involved, our definition

implicitly assumes coordinate systems in neighborhoods of x and f(x) = g(x). It is

easy to verify, however, that if in some coordinate systems f and g have the same

partial derivatives of order < k, then the same is true for any other choice of the

coordinate systems.

If a k-jet ax is represented by a mapping f at x, then we also -iv that ax is a

k-jet of the mapping f at x.

The set of all k-jets Jk(M, N) is called the k-jet space of mappings of M

into N. Let ax be a k-jet at a point x E M represented by some mapping f.

If a coordinate system in a neighborhood of x and a coordinate system in a

neighborhood of f(x) are fixed, then the k-jet ax is determined by the Taylor

polynomial of f at x of order k. In its turn, the set of polynomials of order k is

naturally isomorphic to the finite dimensional Euclidean space as each polynomial

is characterized by the set of its coefficients. So, the k-jet space has a natural

structure of a smooth manifold. Formally, let U and V be coordinate covers of

the manifolds M and N respectively. For each open set U E U and an open set

V E V we define a subset Wuv of the k-jet space as the set of the jets ax, x E U,

represented by mappings sending x into V. Note that the subsets W = {Wuv},

where U and V range over elements of U and V, cover the space of k-jets. Also,

being isomorphic to the set of polynomials of order k, each of Wuv is isomorphic to

a Euclidean space. These isomorphisms induce topologies, one for each Wuv, that

coincide on intersections Wuv n Wuv,, U' E U, V' c V. Thus there is a natural

topology on Jk (M, N). Moreover, the cover W together with homeomorphisms

from Wuv into the Euclidean space, U E U, V E V, defines a smooth structure on

Jk (M, N).

In fact the cover W not only helps to introduce a smooth structure on

Jk(M, N) but also allows us to introduce on Jk(M, N) a structure of a smooth

locally trivial bundle over M x N. Indeed, Jk(M, N) is covered by the sets

Wuv C W each of which is a trivial bundle over U x V. We note that the bundle

projection Jk(M, N) -- M x N sends a k-jet ac represented by a mapping f into

the point x x f(x).

Example. If k = 1, then the k-jet of a mapping f at x E M is the differential df :

T1M Tf(x)N. Thus the space of 1-jets is the locally trivial bundle HOM(M, N)

over M x N the fiber over x x y of which consists of linear homomorphisms

TM -M TyN.

For the definition of Thom-Boardman singularities we need to pass to an

infinite dimensional space J~ (M, N) which we define next.

2.3 Infinite Jet Space

A k-jet at a point x determines an i-jet for each I < k. Thus for each pair k, I

with I < k we have a natural projection

S: Jk (M, N) J(M, N).

The jet space J~ (M, N) is a topological space defined as the inverse limit of the

system {(Jk(M, N), 7j}. Though the jet space is infinite dimensional and we cannot

define a smooth structure on J"(M, N), still, using projections

k7 : J' (M, N)- Jk (M, N)

we may define on J~"(M, N) smooth functions, tangent vectors and submanifolds.

We -v,- that a function on the jet space is smooth if locally it is the compo-

sition of the projection onto some k-jet space and a smooth function on the k-jet

space. Having defined smooth functions we may define a 'i',.. ,./ vector at a jet a.

The set of germs F(a) of functions on J~(M, N) at the jet a is an algebra over R.

A differential operator D, at a is a correspondence

D, : F(a) -> F(a)

that is linear, i.e.,

D,(af + bg) =aD f(f) + bD,(g), f, g E F(a), a, b E R,

and satisfy the Leibniz rule

D f(fg) = D(f) + Dc(g), f, g E (a).

We define a tangent vector at the jet a as a differential operator D,. We may view

a vector D, as an infinite sequence of vectors D, k E N, respectively tangent to

the jet spaces Jk(M, N) at (ca) such that

(7k), : k D D k >1. (2.1)

Indeed, let ak denote the k-jet 7'"(). By definition of F(a),

F(a) = UkEN-F(ak),

where each F(cak) is identified with a subset of F(cak+1) under the mapping induced

by the projection F(ak+l) --- F(ak). Given a vector D,, its restrictions to F(ak)

produce a sequence of operators D, On the other hand suppose a sequence of

vectors D, satisfies (2.1). Then we define a differential operator D, as follows.

Let f be a smooth function defined in a neighborhood of a. Then, by definition,

for some k, f is lifted by 7r from a smooth function fk on Jk(M, N). We define

D,(f) as the lift by 7r' of the smooth function Dl (fk). It is easily verified that so
defined correspondence Do is a differential operator.

The set of vectors at the jet a is a vector space called a ',",.,'. ,./ space at a.

Finally, we i- that a subset of the jet space is a subrni, .:[. ./.I if it is the

inverse image of a submanifold of some k-jet space.

2.4 Thom-Boardman Singularities

Given a smooth mapping f : M -- N, at each point of the manifold M we

have an infinite jet of f. The correspondence that takes a point x into the jet of f

at x is a mapping

jf: M J(M,N),

called the jet extension of f. Similarly for each k, we define the k-jet extension

jkf of f. If v is a vector at a point x of M, then the sequence of vectors d(jkf)(v)
satisfies the condition 2.1 and therefore defines a vector d(jf)(v) at jf(x).

Note that the map d(jf) : TM TJ"(M, N) is injective. The union of

images of d(jf) over all mappings M -+ N is called the total Ir,'.. ,./I bundle of

the jet space and is denoted by D. Given a jet jf(x), we will use the injective

homomorphism d(jf)lTx to identify the plane TxM with Dl\jf(x).

Every 1-jet at a point x E M determines a homomorphism TxM -- Tf(x)N,

where f is a germ at x representing the jet. Let y be a point of the jet bundle and

Ky C Dy the kernel of the homomorphism defined by the 1-jet component of y.

Boardman proved that for every i1 the set

i, = { ye Jcc(M, N) | dimK = ii }

is a submanifold of J~(M, N). Let T' denote the set of r integers (i, ..., i,) such

that i1 > .. > ir. Suppose that the submanifold Ejr-i has been already defined.

Then define

EZj { y E Ej-1 I dim(Ky n TEj,-) = ir}.

Boardman proved that for every symbol Jr the set EZj is a submanifold of

J (M, N).

A mapping f is called a general position i,,r.lr.:l if the section jf is transver-

sal to every submanifold Ej. By the Thom Strong Transversality Theorem (see

Arnold et al. [5] or Boardman [6]), every mapping can be approximated by a

general position mapping.

Given a mapping f : M -+ N, a point x E M is a singularity of type J if

the image jf(x) is in EZ. As has been mentioned, for general position mappings,

the definition of singularity types given by Boardman coincides with the naive

definition given in section 2.1.

Examples of general position mappings. For mappings from a surface into a

3-manifold, general position mappings are those with singularities of only Whitney

13

umbrella type. For mappings between surfaces, general position mappings are

exactly the mappings with only fold and cusp singular points.

CHAPTER 3
THOM POLYNOMIALS

Two of the main questions in the global singularity theory are

Question 1. Given two i,1r,,..:[..1.14 M, N and a ;i,;.,l.; Ii; type r, does there exists

a general position i.1i'1r.:', f : M N without 7 singularities?

Question 2. Given a continuous i''rr'.:jj f : M N of smooth ,,<,;,4.:4..l., and a

-c.:,,il;.,~I:1, r,T, does there exists a homote(.i' of f to a general position In'1'.:i.j

without 7 singularities.

We already know the answers to these questions in two particular cases. Every

mapping of a surface into R3 admits a homotopy to an immersion; thus the answers

to the Questions 1,2 with 7 standing for Whitney umbrella are positive. The

answers to the Questions 1,2 for mappings between surfaces without cusp singular

points is negative in general. Every general position mapping of the projective

plane into R2 has at least one cusp singular point.

Lemma 3.0.1 (tliashberg [8]) A ,,,,'.:,.1 of a connected closed -i, fi,.-: into an

orientable -,'f, ': is homotopic to a i.irr.':,.j with only fold .:,,g.l,.n points if and

only if the Euler characteristic of the source ,,,,:f.;,l .11 is even.

The obstruction to elimination of the cusp singular points of a mapping f

of a closed surface into an orientable surface can be considered as a homology

obstruction because it is represented by the fundamental class of the cusp singular

points.

In general, given a general position mapping f : M -+ N and a Thom-

Boardman singularity type 7, the closure of the set of r-singular points of f

represents a homology class [r] E H,(M; Z2). Under general position homotopy the

homology class [r] does not change and therefore defines a homology obstruction.

It is remarkable that the cohomology class dual to the homology obstruction

is .1.- l- a polynomial in terms of Stiefel-Whitney classes of the bundles TM and

f*TN [14], [17]. For example in the case of cusp singularities of mappings from a

surface into R2, the homology obstruction is dual to the Stiefel-Whitney class w2.

The polynomial in terms of Stiefel-Whitney classes dual to the homology

obstruction is called Thorn p i 'l ,.;;;,. Many papers are devoted to calculation of

Thom polynomials of certain singularities. Some of the polynomials were already

calculated by Thom [39]. For the list of results of 1970-1980 we refer to the survey

[5] of V. Vasiliev. Recently, there was a significant progress in calculation of Thom

polynomials by L. Feher, M. Kazarian and R. Rimanyi [11], [18], [26], [27].

The homology obstruction is not complete in general. For example, every

mapping of a 2-dimensional sphere into R2 has a curve of fold singular points,

though the homology obstruction belongs to the trivial group H2(S2; Z). Except

the trivial case of fold singular points that appear as the points corresponding to

the boundary of the image, the question whether the homology obstruction provide

a complete obstruction is not simple.

Let us list some cases where the homology obstruction is known to be com-

plete:

(1) cusp singularities for mappings from m-manifold into an orientable surface

(H. Levine [20]; Y. Eliashberg [8]);

(2) any stable singularity type except fold one for mappings between orientable

4-manifolds (0. Saeki, K. Sakuma [35], Y. Ando [2], [3]);

(3) non-fold singularities of mappings of non-positive codimension between

stably paralellizable manifolds (Y. Eliashberg [8], [9]);

(4) zero dimensional singularities of mappings of non-positive codimension

between orentable manifolds of dimensions > 2 (Y. Ando [3]);

(5) any Morin singularity except of fold and cusp types of mappings Mm -'

N1 with m n > 0 odd (R. Sadykov [29]);

In 1992 0. Saeki [30] presented an example of manifolds M, N and a singu-

larity type T such that the homology obstruction [r] to the existence of a r-free

mapping M -- N is trivial and nevertheless the manifold M does not admit a

mapping into N without r singularities.

In the example of O. Saeki M is a 4-manifold homotopy equivalent to CP2,

N is R3 and 7 is the cusp singularity. The closure of the cusp singularities of a

mapping from a 4-manifold into a 3-manifold is a union of circles. Its homology

class is trivial as the manifold M is simply connected. On the other hand O. Saeki

proved that the manifold M does not admit a mapping into N without cusp

singular points.

In the latter sections we determine the secondary obstruction for the existence

of mappings of 4-manifolds into 3-manifolds without cusp singularities and prove

that the secondary obstruction is complete.

CHAPTER 4
SINGULARITIES OF MAPPINGS M3[ -- N3

4.1 Normal Forms of Singularities

In general position with respect to Thom-Boardman singularities, a smooth

mapping f of a 4-manifold into a 3-manifold has only fold, cusp and swallowtail

singularities [5]. In fact, one may define a general position mapping of a 4-manifold

into a 3-manifold as a mapping with singularities of only these types.

In a neighborhood of any singular point of a mapping f, there are local

coordinates, called special, in which the mapping f takes one of the normal forms:

Definite fold singularity type S+

T = t1, T2 t2, Z q +q .

Indefinite fold singularity type S_

Ti t1, T2 t2, Z= q -q .

Cusp singularity type 7

Ti tl, T2 t2, Z q + tx + x3

Swallowtail singularity type

Ti tl, T2 t, Z = q + tlx + t2x2 + x4

4.2 Topological Description of Singularities

Directly from the normal forms of singularities, it follows that the set of all

singular points of a general position mapping f : M -- N of a 4-manifold into a

3-manifold is a closed, perhaps non-orientable, 2-dimensional submanifold S of M.

However, the image of the singular set f(S) is not a submanifold of N in general;

the restriction of the mapping f to S, may not be an immersion. The singular set

of f S is exactly the set of cusp and swallowtail singular points of f. The image of

the singular set under the mapping f S in a neighborhood of a swallowtail singular

point is depicted on the figure below.

Figure 4-1: The image of a swallowtail singular point.

We see that the image of the swallowtail singular points can be identified with

the end points of the self-intersection curve of f(S). In particular, the homology

obstruction to eliminating swallowtail singular points by homotopy of f is ahl--i-

trivial. In fact, by the Ando theorem [3], if the manifolds M and N are orientable,

then the homotopy eliminating the swallowtail singular points of f exists.

We are interested in the question whether there exists a homotopy eliminating

cusp singular points 7(f). The set of cusp singular points constitute a union of

curves with boundary at the swallowtail singular points so that the closure

{cusp points} {cusp points} U {swallowtail points}

is a union of circles embedded into M.

It is well-known that if M and N are orientable, then the homology class

[7(f)] E H2(M;Z ) is trivial. In fact a stronger assertion takes place.
Lemma 4.2.1 Suppose that the I,,;.,[.,1 1N is orientable. Then the curve 7(f)

bounds an orientable -, .i f.. in M.

Proof. Let us prove that for an orientable surface bounded by 7(f), we may chose

the set of definite fold singular points S+(f).

Firstly, we need to show that aS+(f) = 7(f). This follows directly from the
normal forms of the cusp and swallowtail singular points; the curve 7(f) cuts the

set of singular points

S(f)= S+(f)U 7(f) US_(f)

into the set of definite fold points S+(f) and indefinite fold points S_(f) so that

each component of 7(f) belongs to the boundary of exactly one component of

S+(f) and one component of S_(f).
Secondly, we need to show that the set set S+(f) is orientable. This follows

from the normal form of f in a neighborhood of a definite singular point. Indeed,

in a neighborhood of a definite fold point, the mapping f can be written as the

product idR2 x m of the identity mapping id : R2 -- R2 and the Morse function

m : R2 R defined by m(qi, q2) = q2 + q The Morse function m gives rise to

an orientation of R as the image of m in the set of non-negative reals. Hence the

image f(S+) also has a natural co-orientation. Together with orientation of N, the
co-orientation of f(S+) defines an orientation of S+ 0.

Remark. Note that in contrast with S+(f), the set S_(f) is non-orientable in

general.

Let us recall that a vector v(p) at a cusp point p is called a characteristic

vector if there is a standard coordinate neighborhood of p such that v(p) = (p).

A vector field on the curve 7(f) in .1 is called a characteristic vector field if it

consists of characteristic vectors. The existence of a characteristic vector field

for an arbitrary cusp mapping of an orientable 4-manifold into an orientable

3-manifold will follow from Lemma 4.3.2 bellow.

4.3 Behavior of the Mapping in a Neighborhood of 7

In this section we describe the behavior of the mapping f in a neighborhood of

the curve of cusp singular points 7. For convenience of exposition, we assume that

7 is connected.

Note that in a neighborhood of each cusp singular point, the mapping f can

be written as the product of the identity mapping id[o,] : [0, 1] [0, 1] and the

standard cusp i,'.':'hI g : D3 D2 which in special coordinates {t,q, x} of D3 and

special coordinates of D2 has the form

g(t, q,x) =(t,q2+ t +3).

Our aim is to prove that in a neighborhood of 7, f has the product structure.

As the first step of the proof, let us show that we can decompose a neighbor-

hood of the curve 7 into the product S1 x D3, = S1 x {0}, and a neighborhood

of the curve f(7) into the product S1 x D2, f(7) S1 x {0}, with respect to

which the mapping f is the product of the identity mapping idsl : S1 S1 and

the standard cusp mapping g, i.e., for each point x E S1 the restriction f {x} x D3

maps the 3-disc into the corresponding 2-disc {f(x)} x D2 and, moreover, in some

coordinates

f {x} x D: {} x D -x {x} x D2

is the standard cusp mapping.

We start with an arbitrary tubular neighborhood of the curve f(7) in N

and choose its decomposition into the product S1 x D2 so that, first, 7 coincides

with S1 x {0} and, second, each disc {x} x D2, x e S1, intersects the curve 7

transversally.

In a sufficiently small neighborhood of 7, the mapping f composed with the

projection S1 x D2 S1 x {0} onto the curve 7 has rank 1. Hence, by the

Inverse Function Theorem, there is a product neighborhood S1 x D3 of 7 each disc

{x} x D3, x e S1, of which maps under f into the corresponding disc {f(x)} x D2

of the neighborhood of f(7).

To complete the proof of the assertion that in a neighborhood of 7, the

mapping f has a product structure, we need to show that for each x E S1 the

mapping f {x} x D3 is the standard cusp mapping.

For a point x E 7, let D' and D' denote the discs of the product structures of

7 and f(7) that go through the points x and f(x) respectively.

Lemma 4.3.1 For every point x E 7, the restriction fx f D' is a general position

ll l I ": "l,!l

Proof of Lemma 4.3.1. In terms of jet spaces, it suffices to show that under the

assumptions of the lemma, first, the jet extension j"f, of the mapping fx sends Dj

to the set Q2 C J (D', D ) and, second, that the mapping j fx is transversal to

the submanifolds of E2 and E2,1-points.

We will write J] for J([-1, 1] x D [-1, 1] x D ) and Jf for the

restriction of J,, ] to {0} x D Let

s: J (D, n3 D J,,p

be the embedding that relates the jet section of a mapping

g:D D3 D2

with the jet section of the mapping

id x g : [-1, 1] x D [-1, ] x D2

restricted to {0} x D where id stands for the identity mapping of [-1, 1]. More

precisely, by definition, s is a unique embedding that makes the diagram

J g (Dj, (D) J
39 Ij(gx id)

D [-t, t] x D

commutative (see also the definition before Lemma 9.0.9). Here the bottom
mapping is the embedding identifying D' with the disc {0} x Du .

Let us write E' for the set of 3-points in J~,i] and Ej for the set of 3-points in

J~(DD, D'). We claim that s-'(E',) = Ej. We will prove the claim by induction
over the length of the symbol J. The mapping s defines a homomorphism s, of the
tangent bundles of J({D, Dx) and J[,71]. Note that for every y e J<(D, Dj),
the homomorphism s, bijectively sends the kernel Ky defined in section 2.4 into the
kernel K,(y). This implies the claim for symbols J of length 1. If the claim holds for

all symbols of length k, then for each jk of length k,

s,(K n TYZk) =K,(y) n T',.

Hence each set Ejme+ with symbol of length k + 1 maps under s into the correspond-
ing set E',+i. This completes the induction.
In particular, the inverse images of the sets of E2, Z2,1 and 22-points of J[1,,]
under the mapping s are the corresponding sets in J%(D D ).
By [28, Lemma 4.3], the mapping s is transversal to E2 n J' and to E2,1 J

in J By definitions (7.1) and (7.2), the sets E2 and E2,1 are transversal to the
fiber J" in J",,]. Therefore the mapping s is transversal to E2 and E2,1 in J .

Identifying D\$ with {0} x D\$ C [-1,1] x D3 C S1 x D3, we obtain

(jf)D3 s o jf,. Thus, to prove Lemma 4.3.1 it suffices to show that
(C1) the image of (jf)uD is in 2 C [J1,1] and

(C2) (jf)lD3 is transversal to the sets of E2 and E2,1-points.
The condition (C1) holds since Im(if) C Q2. Let us prove (C2). The
differential of the jet section jf at x splits into the sum of homomorphisms

d(jf)= d(f)\lTD + d(jf)lT(y,

(4.1)

where TDD' and T17 are the tangent spaces of D' and 7 at x respectively. The dif-

ferential d(jf)lT., sends Tx7 into the tangent space of the E2,1-points. Now, since

f is in general position, the equation (4.1) implies that d(jf)|lTD3 is transversal to

the sets of E2,1 and E2-points. This completes the proof of Lemma 4.3.1. U

Lemma 4.3.2 There are product neighborhoods S1 x D3 of / and S1 x D2 of f(7)

such that f restricted to S1 x D3 is the product of the :/. 1,'/:/.;/ i'"ll':.'j of S1 and

the standard cusp i'.',:,,:'.' g.

Proof. In view of Lemma 4.3.1, the collection of the mappings {fx} indexed by

the points of an interval of 7 can be viewed as a homotopy of the standard cusp

mapping, which is known to be homotopically stable (for example see Theorem 7.1

in [12]). Therefore, there is a cover {Ia} of the curve 7 by intervals such that for

each interval l, of the cover, the mapping f restricted to a, x D3 is equivalent to

the product ida x g of the identity mapping of I, and the standard cusp mapping

9.
The trivializations {I1 x D3} and {I x D2} lead to bundle structures of S x D3

and S1 x D2 over S1 with common cover {I,} of S1 and with transition mappings

consistent with f. The latter means that each pair (i, Q) of the corresponding

transition mappings belongs to the group stabilizing the standard cusp mapping

g. Since the normal bundles of 7 in .31 and of f(7) in N3 are orientable, the
transition mappings are elements of the group

Aut(g) = {(, 9) e Diff (D3, 0) x Diff+(D2, 0) 0 o go 0-1 g},

where Diff+(D2, 0) and Diff+(D3, 0) stand for the groups of orientation preserving

auto-diffeomorphisms of (D2, 0) and (D3, 0) respectively. The group Aut(g) reduces

to a maximal subgroup MCAut(g) conjugate to a linear compact subgroup [16],

[40]. To prove Lemma 4.3.2, it remains to show that the group MCAut(g) is

trivial.

Let CK denote the orientable version of the contact group, i.e. /K is a semiprod-

uct of Diff+(D2, 0) and of the group of germs (D2, 0) -- Diff+(D1, 0). The group

Autkc(h) of a germ h : (D2, 0) -- (D1, 0) is defined by

A11/. (h)={ cEIC I(h)= h }.

The standard cusp mapping g is a miniversal unfolding of the germ go : (D2, 0)

(D1, 0), defined by go(q, x) = q2 + x3. Hence by [40, Proposition 3.2], the group
MCAut(g) is isomorphic to MC1A,1. (go). The latter group is isomorphic to a

compact subgroup H of the group Aut QgO of automorphisms of the local algebra of

go [25, Theorem 1.4.6].
Let b E GL+(2) be a linear automorphism of R2 that preserves orientation. It

defines an action on germs (R2, 0) -- (R, 0) by sending a germ h into ho Suppose

that this action factors through an action on local algebra Aut QgO. Then b defines

an element in Aut QgO. By the proof of [25, Theorem 1.4.6], we may assume that

each element of H < Aut QgO is induced by a linear map of GL+(2). The linear

maps b E GL+(2) corresponding to elements of H leave invariant the principal

ideal P generated by a germ that in some coordinates has the form (q, x) v-> q2 + 3.

Note that up to a scalar multiple the vector is determined by the property that

for every germ h c P, the partial derivative (0, 0) = 0. Consequently, if a map

ib E GL+(2) corresponds to an element in H, then b leaves the direction of the

vector a- invariant, i.e. &b(a) C= a with a : 0. Moreover, since ib leaves the ideal

P invariant, we conclude that a > 0. Hence, the group H is contractible. This

completes the proof of Lemma 4.3.2. U

4.4 Bifurcations of Singular Sets

We w that a homotopy of a mapping f is in general position if it is a general

position mapping.

Lemma 4.4.1 Every homot(ei', joining two general position in.ry'r,:isl can be

approximated by a general position homot(e'i',

Proof. Let fi : M -- N, i = 0, 1, be two general position mappings, and

F: M x [0,1] N x [0,1]

be a homotopy from fo x {0} to fl x {1}. We may assume that for a small number

E, the homotopy F does not change the mapping in intervals [0, E) C [0, 1] and

(1 1] [0, 1].

As a n lppli.' the homotopy F admits a C1-close approximation by a general

position mapping. Moreover, we may choose an approximation F that coincides

with F on the E intervals [0, E) and (1 E, 1]

Since F is C'-close to F, the composition p o F : Mx [0, 1] [0, 1], where

p : N x [0, 1] [0, 1] is the projection onto the second factor, has no singular

points. Therefore, for every moment to E [0, 1], the inverse image F-1(N x {to}) is

diffeomorphic to M. Thus, F can be considered as a new homotopy joining fo and

fl, which as a mapping is in general position. U
Lemma 4.4.1 guarantees that any two homotopic general position mappings

can be joined by a homotopy in general position. That is why we restrict our

attention to bifurcations that occur only under general position homotopy.

In the rest of the section we describe bifurcations of homotopies of mappings

from a 4-manifold into a 3-manifold.

By dimensional reasoning (see, for example, Boardman [6] and Ando [4]) a

general position mapping f from a 5-manifold into a 4-manifold has only Morin

singularities, and D4 singularities with symbol 3 = (2, 2, 0). In a neighborhood of a

Morin singular point, in some local coordinates, the mapping f can be written as
k-1
f(ti,t2,3,q,x) (tl 2, 3, q2 + tii +xk+l'), k 1,2, 3,4. (4.2)
i= 1

For a D4 point of f, there are local coordinates in which the mapping f takes the

form

f(to, tl, t2, v) = ,t, t2, 2 v3 + tU + t V + t 2). (4.3)

Suppose that F : V1 T x [0, 1] -- N3 x [0,1] is a general position homotopy

joining two general position mappings fo and fl. From the normal forms (4.2)

and (4.3), it is easy to verify that the singular set of a general position homotopy

F : .1T x [0, 1] -- N3 x [0,1] is a submanifold of .1 T x [0, 1]. Therefore, S(F) defines

an embedded bordism between the singular sets of fo and fl.

4.5 Surgery of 7(f)

In this section we study a surgery of the singular set of a cusp mapping

f : 1 T -- N3 and find a sufficient condition for the existence of a homotopy of f

realizing a given surgery.

Let f : 1 T -- N3 be a cusp mapping of an orientable 4-manifold into an

orientable 3-manifold. In general, the restriction of f to the curve of cusp singular

points 7 is an immersion. To simplify arguments, we make f\ an embedding by a

slight perturbation of f in a neighborhood of 7. Let v be a field of characteristic

vectors on 7. We ?- that an orientable surface H is a basis of surgery (see

Eliashberg [9]), if

1. H = 7,

2. the vector field v is tangent to H and has an inward direction,

3. H \ OH does not intersect S(f), and

4. the restriction f H is an immersion.

We will show that if a basis of surgery H exists, then we can reduce 7 by

modification of f in a neighborhood of H. We assume that 7 is connected and f

has no other cusp singular points. The proof in the general case is similar.

We need one more preliminary observation. Let I denote the closed interval

[-1, 1] and g : I x D2 I x I be the standard cusp mapping defined by

g(t, q, x) = (t, q2 + tx + X3), where t and q, x are the coordinates of the first and
the second factors of the domain I x D2 respectively. Then g can be considered

as a homotopy gt : D2 I,t c I, defined by gt(q, x) = q2 + tx + 3. Note that

g-1 : D2 -- I is a Morse function. Since Morse functions are homotopically stable,

there are coordinates in which g8 = g-1 for each s E [-1, -1 + E).

Lemma 4.5.1 Suppose that there is a basis of surgery Hi. Then there is a fold

"''. ':'r.j f : 31 N3, which differs from f only in a neighborhood of Hi.

Proof. Let U(7) = S x D3 be a neighborhood of 7 in 1 given by Lemma 4.3.2.

Let t2 be a cyclic coordinate on the circle S1, (t, q, x) be coordinates on D3 and

(T1, T2, Z) be coordinates in a neighborhood of f(7) with T2 cyclic such that f u(y)

is given by

Ti ti, T2 t2, Z = q2 + t + 3.

We may assume that

H, n U() = { (t,t2,q,x) x q =0, ti c [0,1] }.

We define

Ho = { (ti,t2,q,x) x =q=0, tic [-1,0] }

and set H = Ho U H1.

We regard a tubular neighborhood of a submanifold as a disc bundle. The

properties (3) and (4) of the definition of a basis of surgery guaranties that the

submanifold H has a tubular neighborhood A such that the restriction of A to

H n U(7) is in U(7), the intersection S(f) n OA is in the restriction of the bundle

A to OH and the set A \ U(7) contains no singular points of the mapping f. To

simplify explanations we assume that f H is an embedding. Then the image of A,

which we denote by B, is a line bundle over f(H).

In the following, for a manifold X with boundary aX, let CX denote a collar

neighborhood of aX in X, and let I denote [-1, 1].

First, by the remark preceding the lemma, the manifolds B1 = B f(CH) and

A1 = f-1(BI)nA have product structures A1 = CHxIxI and B1 = f(CH)xI such
that f A1 is a product of a diffeomorphism CH f(CH) and a Morse function.

We may assume that the restriction of this Morse function to CI x I U I x CI

is the projection onto the factor corresponding to the coordinate x, C I x -- I,

I x CI CI, and that A1 = A|CH.

Next, we extend the product structure of B1 to a product structure f(H) x I

of B. Then we restrict this product structure to B2 = f(H) x CI and define

A2 f -(B2) n A. The mapping f A, is regular and therefore we may assume that
A2 = H x I x CI is a trivial line bundle over B2 with projection f A2 along the

second factor.

Finally, we can find A3 C A and a product structure H x CI x I of A3

such that f A3 is a trivial CI-bundle over f(H) x I and A1 U A2 U A3 is a collar

neighborhood of OA.

The connected components of A2 and A3 are orientable 1-dimensional bundles

with bundle mappings given by the restrictions of f. Since the structure group

of orientable line bundles reduces to the trivial group, we can make the third

coordinates of A1, A2 and A3 agree on intersections. We fix an extension of the

product structures of A1, A2 and A3 to a product structure H x I x I of A.

Let p E OH and fp denote the restriction of f to the fiber of the bundle A over

p. We let f(x) = f(x) for x e [1 \ A and

f(u, v, w) f(u) x fp(v, w)

for x = (u, v, w) E A = H x I x I. It is easily verified that f is a smooth mapping

and f satisfies the requirements of the lemma. U

CHAPTER 5
THE INVARIANT e(f)

In this section we give several definitions of a secondary obstruction to

eliminating the cusp singular points of a general position mapping of a closed

orientable 4-manifold into an orientable 3-manifold; and discuss some of its

properties.

5.1 First Definition

Given a general position mapping f of a 4-manifold M into an orientable

3-manifold, to define the homology obsturction, i.e., the primary obsruction, we

consider the closure of the cusp singular points and take its fundamental homology

class in the group Hi(M; Z2). To define the secondary obstruction, consider the

set of all singular points S(f) of the mapping f. This is a closed 2-dimensional

submanifold of M. We may slightly perturb S(f) so that the new surface S'(f)

intersects the original surface S(f) at finitely many points. With each intersection

point we associate a number 1, the sign of the intersection point; and then define

the secondary obstruction as the sum of the associated numbers. To define the

sign of the intersection point p n p' e S(f) n S'(f) we choose an orientation of

the manifold M and an orientation of a neighborhood of p in S(f). The latter

orientation determines an orientation of a neighborhood of p' in S'(f). Indeed, for

any choice of a Riemannian metric on M the projection of the tangent plane of

S'(f) at p' into the tangent plane of S(f) at p is an isomorphism provided that the

perturbation S'(f) is C1-close to S(f).

The orientation of S(f) at p and the orientation of S'(f) at p' determine an

orientation of M at p = p'. If the latter orientation coincides with one chosen for

M, then the intersection point p is assigned +1, otherwise -1.

Note that to define the secondary obstruction we choose an orientation of M

and after perturbing S(f) choose local orientations of S(f).

The choice of the orientation of M is essential; the change of the orientation

of M leads to the change of signs of each intersection point and thus the secondary

obstruction of the same mapping f corresponding to different orientations on M

differ by sign.

Let us show that the secondary obstruction does not depend on our choice of

the orientations of the singular set in neighborhoods of the intersection points. Let

p n p' be an intersection point, (e1, e2) be two independent vectors tangent to S(f)

and (e3, 4) the corresponding vectors tangent to S'(f). Then the orientation el A e2

of S(f) at p leads to the orientation el A e2 A e3 A e4 of M. The other orientation

e2 A el of S(f) leads to the same orientation e2 A el A e4 A e3 = el A e2 A e3 A e4.

It is easy to show that the secondary obstruction is independent from the

choice of the perturbation S'(f). We omit the proof since it is well-known and will

also follow from the definition of the secondary obstruction which we give below.

In terms of homology groups, the secondary obstruction can be defined as

the homology class of the signed set of intersection points, i.e., as an element of

Ho(M; Z), which under assumption that M is connected can be identified with Z.

5.2 Second Definition

Let K be a 2-dimensional submanifold of an oriented 4-manifold .3', and F

be an orientation system of local coefficients over K. We refer to the paper [21]

for the definition of the normal class or the Euler class of the normal bundle over

K in [1', which is a cohomology class e e H2({K; ). The number (e, [K]), where

[K] E H2(K; .) is the fundamental class of K, is an integer called the normal Euler

number of the embedded manifold K. We denote this number by e(K). Note that

the sign of the normal Euler number depends on the orientation of 31 (see Massey

[21]).

For a general position mapping f from an oriented closed 4-manifold [T into

a 3-manifold N3, we define the secondary obstruction e(f) as the normal Euler

number of the embedded 2-dimensional submanifold S(f).

This definition is consistent with the first definition of the secondary obstruc-

tion.

5.3 Invariance of the Secondary Obstruction Under Homotopy

Let us show that the secondary obstruction e(f) is invariant under homotopy

of the mapping f : 1 -- N3.

Lemma 5.3.1 Suppose that fo and fl are two homotopic general position map-

pings. Then e(fl) = e(f2).

Proof. Let F : 1' x [0, 1] N3 x [0, 1] be a general position homotopy joining

fo : .1T x {0} N3 x {0} and fl : 11 x {1} -- N3 x {1}. The boundary

of the singular set B of F is the union of the singular sets Bo of fo and B1 of fi.

Let it : Bt B, t = 0, 1, denote the inclusion, F the orientation system of local

coefficients on B, and let e E H2(B; F) be the Euler class of the normal bundle of

B in .[1T x [0, 1]. Then

e(fo) e(fi) = (ie, [Bo]) (i*e, [Bi]) (e, io [Bo] i [Bi]) = 0,

since io* [Bo] i*, [B] corresponds to the boundary of B and vanishes in H2(B; T).

*

CHAPTER 6
H-PRINCIPLE

The H-principle applies to differential relations over a fiber bundle and if holds,

it reduces the question of the existence of a solution to the question of the existence

of a section of a certain map.

6.1 Differential Relations

Given a fiber bundle 7 : E -- M, the jet space Jk(w), k = 1, 2,..., oo, associated

with 7 is the space of germs of sections of the bundle 7. We will be interested

in a particular case where the space E is the product M x N and the bundle

map 7 is the projection onto the first factor. The sections of such a bundle are

naturally identified with mappings M -- N into the fiber and the jet space Jk (() is

isomorphic to the jet space Jk(M, N).

A differential relation R of order k over a bundle 7 is an arbitrary subset of

the jet space Jk(r). As in section 2.4, a section f of the bundle 7 leads to a jet

extension jf. We ;?- that f is a solution of the differential relation R if the image

of the jet extension jf is in 7. C Jk(T).

Example. If the manifolds M is R" and the manifolds N is R", then a mapping

M -> N is a set of n smooth functions {fi} and a differential relation R of order

k is an arbitrary relation on the values and the partial derivatives of {fi} of order

order < k. A mapping {fi} is a solution of the differential relation R if its values

and partial derivatives satisfy the relation.

6.2 H-Principle

We ?-v that a mapping f : Y X is a section of a mapping p : X Y if the

composition p o f is the identity mapping of X.

If f is a solution of a differential relation 7, then its jet extension jf is a

section of the composition

7rM : 7Z J(r) -- E M

of the inclusion, projection map of the jet bundle and r.

The H-principle -i-i-, -1- the statement converse to the given one up to

homotopy:

H-Principle. Every section of the mapping 7M : 7 -- M is homotopic through

sections to a section that arises as the jet extension of a section of r.

We refer to [13], [10] and [38] for numerous examples of differential relations

for which the h-principle holds.

Let us conclude the section with illustration of the h-principle by interpreting

the Smale-Hirsh theorem [15] in terms of jet spaces.

The Smale-Hirsch theorem asserts that a smooth mapping f of a manifold

M of dimension m into a manifold N of dimension n, n < m is homotopic to an

immersion if and only if the differential df : TM TN is homotopic to an injective

homomorphism, i.e., to a homomorphism of tangent bundles that is injective on

each fiber.

To interpret the Smale-Hirsch theorem in terms of jet spaces, let us recall that

the space of linear homomorphisms T1M TyN, where x ranges over the points

of M and y ranges over the points of N, is isomorphic to the 1-jet space J1(M, N).

We define the differential relation R as the set of those points in J1(M, N) that

correspond to injective homomorphisms. Then the Smale-Hirsch theorem asserts

that the differential relation R abides by the h-principle.

6.3 Morin Singularities.

Let M be a manifold of dimension m and N a manifold of dimension n < m.

The mildest singularity of a mapping f : M -- N of order 1 is the fold singularity

in some coordinate neighborhood of which the mapping f has the form

P(-,. 11,a X') = (Xi, ...,a' X _-1, *** x ).

We will denote its Thom-Boardman symbol J = (m n + 1,0) by J1. The mildest

singularities of f of order k are called Morin singularities. The normal form of f in

a neighborhood of a Morin singularity of order k is

Ti = ti, i l-1,2,...,n- r,

Li = li, i 2,3,...,r, (6.1)
r
Z = Q+ ltkt+k Q k ...k 2
t=2

Its Thom-Boardman symbol is J = (m n + 1,t1,..., 1, 0) of length r + 1 and will

be denoted by U,. We note that the Morin singularity with symbol J2 is the cusp

singularity and the Morin singularity with symbol J3 is the swallowtail singularity.

Returning to the jet spaces, let 0o = k C Jk(M, N) be the subset corre-

sponding to the set of regular points and for r > 0, Q, r= k denotes the subset

corresponding to the Thom-Boardman singularities with symbols Jt, t < r.

Theorem 6.3.1 (Morin [23]) Suppose that the jet extension of a general position

,'rl. ':,jl f M --- N takes M into the set Q". Then each .:,.i1,.lI, point of the

i,'I; ':, f is Morin.

We emphasize that in the Morin theorem the requirement that the mapping

f is in general position, i.e., that the jet extension jf is transversal to Thom-

Boardman stratum, is essential.

6.4 Ando-Eliashberg Theorem

The Ando-Eliashberg theorem asserts that if dim M > dim N > 2, then the

differential relations Q, abide by the h-principle. The Morin theorem allows to

formulate the Ando-Eliashberg theorem as follows.

Theorem 6.4.1 (Ando-Eliashberg [3]) Let M and N be orientable z,,<,;.: ..],]- Let

dimN > 2. Then for i,;, continuous section s : M --+ Q there exists a Morin map

g : M -- N such that jg : M -- Q becomes a section fiber-wise homotopic to s in

Qr.

The proof of Theorem 6.4.1 in [3] shows that the relative version of Theorem

6.4.1 is valid as well. In other words, suppose that U is an open set in M and

s : M -+ Q, is a section such that the restriction of s to a neighborhood of M \ U

is the jet section jg induced by a Morin mapping g : M \ U -+ N. Then g admits

an extension to a Morin mapping g : M -+ N whose jet section jg is fiber-wise

homotopic in Qr to s by homotopy constant over M \ U.

In particular the Ando-Eliashberg theorem reduces the question of the

existence of a fold mapping of an orientable 4-manifold M into an orientable

3-manifold N to the problem of finding a continuous section of the mapping

7m/r: Q, ) j2 (M, NV) ) M x N M.

CHAPTER 7
UNIVERSAL JET BUNDLES

Let Jok' ,IR) be the vector space of k-jets of germs (-.' ,0) -( (R", 0), i.e.,

the fiber of the jet bundle Jk([.- R") over the point (0, 0) e R" x R".

A universal k-jet bundle J(m, n) is a bundle with fiber Jk(P.' R") over

a universal space BQ such that for any manifolds M and N, the k-jet bundle

Jk(M, N) can be induced from J(m, n) by an appropriate mapping of the base

space M x N -- BQ.

The universal k-jet bundles turns out to be very powerful in study of singulari-

ties of smooth mappings. Using the universal k-jet bundles, Haefliger and Kosinski

proved that the homology obstructions are Poincare dual to polynomials in terms

of Stiefel-Whitney classes. Recently, Kazarian constructed and described spectral

sequences associated with classifications of singularities which encode not only the

fundamental classes of singularities but also the .,li i:ency of classes of different

singularities.

7.1 Construction 1

Let ESOk -> BSOk be a universal vector bundle classifying the orientable

vector bundles of dimension k. For each manifold X of dimension i, let 'x : X -

BSOi denote the mapping classifying the tangent bundle of X.

Let xx y be a point of BSOm x BSO,, F?" be the fiber of ESOm over x, T.

be the fiber of ESO, over y. Then the fiber of the bundle ESOm x ESO, over

xx y is -" x R". We define the set Jf~x(m, n) as the set of k-jets of mappings

(.' 0) (R, 0) and then define a universal k-jet space Jk(m, n) as the union

UJfx(mn, n) that ranges over the pairs x x y of BSOm x BSO,. Note that the

universal k-jet space has a natural structure of a bundle over BSOm x BSO,.

Let us prove that for any m-manifold M and n-manifold N the bundle induced

from the universal k-jet bundle by the mapping

TM X TN : M N -- BSOm x BSO,

is isomorphic to the k-jet bundle Jk(M, N).

Indeed, let Jk(M, N) be the space of k-jets of mappings TzM -- TyN, where

x x y ranges over the points of M x N. By construction, the space Jk(M, N) is

a bundle over M x N isomorphic to the bundle induced from the universal k-jet

bundle by TM x TN.

Lemma 7.1.1 Let M and N be Riemannian ,,,r,:... 4.- Then the k-jet bundle

Jk(M, N) is cano,.:. /ll/; isomorphic to Jk(M, N).

Proof. For each point x of M there is a neighborhood Ux of the origin in TxM

diffeomorphic under the exponential mapping to a neighborhood V, of x in M,

ExpM : UX V,.

Let {Exp } be the set of exponential mappings for the manifold N. Then the

mapping Jk(M, N) Jk(M, N) that takes the k-jet over x x y represented by a

mapping f : TxM TN into the k-jet of the mapping ExpN o f o (ExpfM)1 is a

canonical isomorphism 0.

7.2 Construction 2

In this section we sketch the construction by Kazarjan.

Let k be a positive integer. The group

G JOkDiffR- x JkDiI!. ,

where Dil!F.' and DiffR" stand for the orientation preserving auto-diffeomorphism

groups of (F.' 0) and (R", 0) respectively, is a finite dimensional Lie group with a

natural action on the vector space JOk(.' I R). The natural action of g is defined

so that an element of g represented by a pair of auto-diffeomorphisms dm x dn takes

the k-jet represented by a germ f into the k-jet of the germ d, o f o d,.

Let E -- BQ be the universal g-bundle over BQ. Then the space Jk(m, n) is

defined as the bundle

Jk(mn, E xg ( E xJ ),

over Bg associated with the universal bundle E.

Since the group g is homotopy equivalent to SO, x SO,, the space BQ

is homotopy equivalent to BSO, x BSO,. It is easily verified that the bundle

Jk(m, n) is the universal jet bundle.
7.3 Construction 3

If ( is a vector space, then (or = ( o ( o ... o ( denotes the vector space defined

as the vector space (0r factored by the relation of equivalence: vl 0 v2 & "." r v

w 1 w2 wr if and only if there is a permutation of r elements a such that
I = ,,., for i 1, ..., r. The space (o" is called the symmetric r-tensor product

of (. As in the interpretation of Smale-Hirsch theorem in terms of jet spaces,

for every r, the bundles ( and T give rise to the bundle 'OM/(r, TI). The fiber

of '-OM(0', Tr) over a point x E M is the set of homomorphisms Hom(r'", rx)

between the fibers r" and q, of the bundles (or and r respectively over x. The

space '-OM ((, 7) in the formulation of the Smale-Hirsch Theorem is generalized

by the vector bundle

S' (, T) -= HOM (, T) ED OM.( o Tr) E... E H OM (, r)

over M (see paper [28] of Ronga). As above we define S"(0 Tr) as the inverse limit

lim S(, ).

Let kr(g) denote the rank of the kernel of a linear function g. A point of

S"((, T) over a point x E M is a set g = {g} that consists of homomorphisms

gi E Hom(o', ,i). We set

i, = UE1M{(g S(>z, rI)l kr(gl) ii}. (7.1)

Let Kh and Ch respectively denote the kernel and cokernel of a homomorphism

h E Hom(, r). The composition of natural homomorphisms

Hom( o TI) -) Hom(, Hom(, T7)) Hom(Kgl, Hom(Kgl, Cgl))

takes the homomorphism g2 E Hom( o x, qTp) into some homomorphism g2. We

define

Ei',iz UEM{g E S'(, T,)| Ig E Ei, and kr(g2) 2}. (7.2)

This construction proceeds by induction. We refer the reader to [6], [29].

The bundle J'(M, N) is isomorphic to the bundle S'(, TI), where =- TM and

] = f*TN. Moreover, there is an isomorphism of bundles S"(', rI) and J~(M, N)

that takes each Ej isomorphically onto Ej [29].

7.4 Corollary of Ando-Eliashberg Theorem

As a corollary of the Ando-Eliashberg theorem, we obtain that the existence of

a homotopy eliminating the cusp singular points of a mapping f into an orientable

3-dimensional manifold is independent of f.

Corollary 7.4.1 The homote(.i' class of f : M" -- N3,m > 3, contains a fold

I,'.rl'.:.j if and only if there is a fold in':rl'.:.'j g : Mm R3

Proof. By Ando-Eliashberg theorem, the homotopy class of f contains a fold

mapping if and only if there is a section Mm -- 1 C J2(TMm, f*TN3). The latter

does not depend on f or TN3 since the tangent bundle of an orientable 3-manifold

is trivial. U

CHAPTER 8
SECONDARY OBSTRUCTION

8.1 Definition of the Secondary Obstruction

Let M be a closed connected oriented 4-manifold. The intersection form

defined on the free part of H2(3'; Z) is a quadratic form. Let Q(3[1) denote the

set of integers taken on by this quadratic form. We note that the set Q(.1') and

the normal Euler number of a general position mapping depend on the choice

of orientation of M ['. It is easily verified, however, that for a given mapping

f : [1P -- N3, the condition e(f) E Q([1) does not depend on the orientation of

M\. In sections 8.2 and 8.3 we will prove the main theorem .

Theorem 8.1.1 Let f : [1P -- N3 be a general position inr.,':..j from an orientable

closed connected 4-,,n1.,.:f./.I into an orientable 3-,,,in.;,.: .I. Then the homote'i./; class

of f contains a fold I'"'l'l.:I if and only if e(f) E Q(i1).

8.2 Necessary Condition

The invariant e(f) allows us to give a necessary condition for the existence of a

fold mapping into N3. In the later sections we will prove that this condition is also

sufficient.

With every oriented closed connected 4-dimensional manifold 3.1' we associate

the set Q(.1) of integers each of each is the normal Euler number of an orientable

surface in 1\.

Lemma 8.2.1 If f : 1M' N3 is a fold I'""''r.,:. then e(f) E Q(i[).

1 After the paper was written, O. Saeki informed the author that he obtained
similar results using a different approach [32].

Proof. The singular set of a fold mapping consists of the surfaces S_ (f) of indef-
inite fold singular points and S+(f) of definite fold singular points. Therefore,

e(f) = e(S_(f)) + e(S+(f)). In [30], [1] it is proved that e(S_(f)) = 0. Hence,

e(f) = e(S(f)). Since S+(f) is orientable (see Lemma 4.2.1), we conclude that

e(f) e Q(3'). U
Corollary 8.2.1 Suppose that the homote(i'., class of a general position I''rr'.:"

f : M' N3 contains a fold inqr. Then e(f) E Q9(1'). U

8.3 Sufficient Condition

The objective of this section is to prove that the condition e(f) E Q(01)

is sufficient for the existence of a fold mapping homotopic to a general position

mapping f : [1 -- IR3. In view of Corollaries 4.4.1 and 7.4.1 this completes the

proof of Theorem 8.1.1.

Lemma 8.3.1 Let f be a general position ii,,l'l'.'from a connected closed oriented

4-,,,,r,:f.1,* I[T1 intoR3. Suppose e(f) E Q(3i1). Then .i1 admits a fold in'"r'. :'

into R3.

Proof. The condition e(f) e Q(Pi1) guarantees the existence of an orientable

2-submanifold S of .1 with normal Euler number e(f).

Let us prove that in the complement .1 \ S, there is an orientable possibly

disconnected embedded surface S such that

(P) every orientable surface embedded in 1 1 \ S with non-trivial normal bundle

intersects S.

If l[1 \ S admits no orientable embedded surface with non-trivial normal bundle,

then the property (P) holds for any orientable embedded surface S. Suppose that

in .1 [ \ S there is an orientable embedded surface with non-trivial normal bundle

and that a surface with property (P) does not exist. Then for any positive integer

k there is a family of oriented embedded surfaces {Fi}i, 1...,k such that each of the

surfaces has a non-trivial normal bundle and does not intersect the other surfaces

of the family. Let Tor H2(3 PT \ S; Z) denote the subgroup of H2(3[ P \ S; Z) that

consists of all elements of finite order. The group H2(3 T \ S; Z)/Tor H2(3 \ S; Z)

is finitely generated. Fix a set of generators el, ..., es. Every surface Fi represents

a class [Fi] in H2 (3 \ S; Z)/Tor H2([ \ S; ), which is not trivial since F, has a

non-trivial normal bundle. Moreover, [Fi] [Fi] / 0 and [Fi] [Fj] = 0 for i / j. If

the number k of the surfaces is greater than the number s of the generators, then

there is a combination

al[FI] + a2[F] + + ak[Fk] = 0

with ac + + ac / 0. Multiplication of both sides by [Fi], i = 1,..., k, gives

aci[Fi] [F] = 0. Therefore, ai = 0 for every i = 1, ..., k. Contradiction. Thus a

surface S with property (P) exists.

Let us construct a mapping for which the set S U S is the part of the singular

set. We recall that we identify the base of a vector bundle with the zero section.

Lemma 8.3.2 There is a general position Ini'l'.:.j h : NS R3 from the normal

bundle NS of S in 1' such that the set S is the set of /. I;,'.:/. fold .:,,i.,lr, points

of h and h has no other : ..:l.,, points.

Proof. The fiber of the bundle NS is diffeomorphic to the standard disc D2

{ (x, y) E R2 x2 + y2 < 1}. Let m : D2 [-2, 2] be the mapping defined by the
formula m(x, y) = x2 + 2. Then m is a Morse function on D2 with one singular

point.

For every open disc U, in S, the restriction of the normal bundle NS to U, is

a trivial bundle U, x D2 -+ U,. Let 13 denote the segment (-3, 3). We define the

mapping g_ : U x D2 U x I3 by ga(u, z) = (u, m(z)), where u E Ua, and z E D2.

Note that rotations of D2 do not change the function m. We may assume that

the fiber bundle NS -- S is an SO2-bundle. Then the mappings g9 give rise to a

mapping g : NS -- S x I3. The open oriented manifold S x I3 admits an immersion
into IR3. We define h : NS -- IR as the composition of g and this immersion. U

Lemma 8.3.3 Let NS be the normal bundle of S in M1'. Then there is a general

position i,.r'l'':', h : NS --- R3 such that the set S is the :u.gqlar set of h and every

component of S has at least one cusp .:,u.;,1.i, point of h.

Proof. For a closed disc D C S, the restriction of the bundle NS S to S \ D

is a trivial bundle (S \ D) x D2 (S \ D), where D2 is the disc as in Lemma
8.3.2. Let 13 denote the segment (-3, 3). The function mi : D2 -- 3 defined

by mi (x, y) = x2 y2 is a Morse function with one singular point at the origin.
Let idl : S \ D -, S \ D be the identity mapping. Put El = (S \ D) x D2 and

BI = (S \ D) x Is. Then idi x mi : El B is a fold mapping.

Set E2 = S1 x (-1, 1) x D2 and B2 = S1 x (-1, 1) x I3. There is a mapping

g : E2 B2 such that
1) the set S1 x (-1,0) x {(0, 0)} is the set of all indefinite fold singular points of g,

2) the set S1 x (0, 1) x {(0, 0)} is the set of all definite fold singular points of g,

3) the curve S1 x {0} x {(0, 0)} is the set of all cusp singular points of g.

Let U denote the intersection of S \ D and a collar neighborhood of 9(S \ D)
in S \ D. Then U is diffeomorphic to S1 x (-1, -1/2). We can identify the subset

U x D2 of E1 with the subset S1 x (-1, -1/2) x D2 of E2 and the subset U x I3 of

B1 with the subset S1 x (-1, -1/2) x I3 of B2 so that the obtained sets El U E2
and B1 U B2 are manifolds and the mapping idl x mi coincides with the mapping

g on the common part of the domains El n E2 C El U E2. Thus, idl x m1 and g
define a cusp mapping c : E1 U E2 B1 U B2. Note that E1 U E2 is diffeomorphic

to (S \ D) x D2 and B1 U- B2 is diffeomorphic to (S \ D) x 13.

Let m3 : D2 -- I be the Morse function, defined by m3(x, y) = 2 + y2,
0 0 0
and id3 : D be the identity mapping of the open 2-disc D D \ OD. Then

id3 x m3 : D x D2 D x I3 is a fold mapping. Let V be the intersection of D

and a tubular neighborhood of OD in S. Then V is diffeomorphic to S1 x (1/2, 1).

We identify the part V x D2 of E3 = Dx D2 with the part S' x (1/2, 1) x D2 of

E2 C El U- E2 and the part V x I3 of B3 = Dx Is with the part S1 x (1/2, 1) x Is

of B2 C B1 U- B2 so that

1) the obtained sets E = El U E2 U- E3 and B = B1 U B2 U- B3 are manifolds,

2) the mapping id3 x m3 coincides with c on the common part of the domains,

3) the manifold E is diffeomorphic to NS.

The condition (3) can be achieved since the mapping m3 does not change

under rotations of the fiber D2

Then id3 x m3 and c define a cusp mapping NS -- B. Note that B M S x 3I

is an open orientable 3-manifold. Thus, it admits an immersion into R The

composition of NS -> B and the immersion B -> R3 is a cusp mapping satisfying

the conditions of the lemma. U

We identify NS and NS with open tubular neighborhoods of S and S in PT

respectively. There is a general position mapping g : 1P -- IR3 which extends

h : NS IR3 and h : NS -- R. In general the extension g has some swallowtail

singular points. Let us prove that we may choose g to be a cusp mapping.

Ando (see Section 5 in [3]) showed that the obstruction to the existence of

a section of the bundle Q2(Tl 1 TR3) over the orientable closed 4-manifold .1

coincides with the number of the swallowtail singular points of a general position

mapping .1 -- R3 modulo 2. Also Ando calculated that this obstruction is trivial.

Since the mapping h U h does not have swallowtail singular points, the obstruction

to the existence of an extension of the section j3(h U h), defined over NS U NS,

to a section of Q2 over 1 P is trivial. Therefore, the relative version of the Ando

Eliashberg theorem (see chapter 7) implies the existence of an extension to a cusp

mapping g : 3[1 -- R3.

The singular set S(g) consists of S U S and probably of some other connected

submanifolds A, ..., Ak of 1\. We have

e(f) = e(S(g)) = e(S) + e(S) + e(Ai) + e(A2) + + e(Ak). (8.1)

The normal Euler number of the submanifold S equals e(f). Hence the sum of the

normal Euler numbers e(Ai) + + e(Ak) equals -e(S).

Let At be a component of UAj. Suppose At is a surface of definite fold singular

points. The surface At is orientable (see Lemma 4.2.1) and does not intersect

S U S. By definition of S, this implies e(At) = 0. Suppose At is a surface of

indefinite fold singular points. Then again e(At) = 0 [30], [1]. Therefore, e(At)

is nontrivial only if the surface At contains cusp singular points. Let us recall

that the union of those components of the singular submanifold S(g) that contain

cusp singular points is denoted by C = C(g). The equation (8.1) implies that

e(C) e(S) + e(Ai) +... + e(Ak) 0.

It remains to prove the following lemma.

Lemma 8.3.4 If g : [1V -- R3 is a cusp ina'l.':hg and e(C) = 0, then there exists a

homote'l,, of g eliminating all cusp /.:,tuli r points.

Proof. If the curve of cusp singular points is not connected, then there exists a

homotopy of g to a mapping with one component of the curve of cusp singular

points. We may require that the homotopy preserves the number e(C). We omit

the proof of these facts since the reasoning are similar to those in section 4.5.

We will assume that the curve of cusp singular points 7(g) is connected and

hence so is C(g).

Lemma 8.3.5 Let v(x) be a characteristic vector field on 7(g). If e(C) = 0, then

v(x) can be extended on C(g) as a normal vector field.

Proof. For a general position mapping g : 3i1 R3, the set

F =f- (f(C (g))) 9 1/'

is an immersed 3-manifold. The self-intersection points of F correspond to the
points of the surface C_(g).
We v- that two vectors vl and v2 of a vector space have the same direction

if vl = Av2 for some scalar A / 0. There is an unordered pair of directions

(I1(p), 12(P)) over C_(g) [1] with the following property. For every point p of
C_(g) there are a neighborhood U about p with coordinates (xl, X2, x3, x4) and
a coordinate neighborhood about g(p) such that the restriction glu has the form

(xi, x2, x3 x4) and the directions of the vectors 8/8x3 and 8/8x4 coincide with
l1(p) and 12(p) respectively. An L-pair is a pair (/i(p),12(p)) that satisfies this
property.

Let FI C C_(g) denote the complement of a regular neighborhood of the curve

7(g) in C_(g). The proof of Lemma 3 in [1] shows that there is a vector field v(p)
in the normal bundle over Fi with directions li(p) + 12(p) or 11(p) 1(p) over the

boundary OF, for some -pair (li(p), 2(p)).
We v- that a direction at a cusp singular point is an x-direction if it is

tangent to the surface S(f) and transversal to the curve 7(g). Note that for a

special coordinate neighborhood about a cusp singular point the direction of the
vector 8/8x has an x-direction.

It is easily verified that for an -pair (II(p),12(p)), the directions lI(p) 2(p)

are tangent to F at every point p in C_(f). Furthermore the directions li(p) + 2(p)
and 11(p) 2(p) approach the same x-direction as p approaches 7(g). It implies

that the vector field v(p) over Fi has an extension to C_(g) such that v(p) is
transversal to C (g) at every point of C (g) and has an x-direction at every point

of 7(g). If necessary, we multiply the vector field v(p) by -1 to get a vector field
which points toward C (g) over 7(g). Now the vector field v(p) can be modified
in a neighborhood of 7(g) so that a new v(p) is normal to C_ (g) at every point of

C_(g) and the restriction of v(p) to 7(g) is the characteristic vector field v(p).

47

The obstruction to the existence of an extension of v(p) to a vector field over

C(g) is the normal Euler number e(C). Since e(C) = 0, such an extension exists.E

The ends of the vectors v(p), p c C+(g), define an embedding of an orientable

surface H diffeomorphic to C+(g) into M '. We modify the embedding in a neigh-

borhood of the boundary OH so that the new embedding defines a basis of surgery.

Now Lemma 8.3.4 follows from Lemma 4.5.1. U

The proof of Lemma 8.3.1 is complete. U

CHAPTER 9
COMPUTATION OF THE SECONDARY OBSTRUCTION

In this chapter we will express the secondary obstruction in terms of the

Pontrjagin class pi(C1V) of the tangent bundle of Mi'. Namely, we will prove the

formula1 e(f) = (pi(i/C), [M4]), where [M4] is the fundamental class of the

manifold 1'.

Lemma 9.0.6 Let f be a general position i'.ri.':.j from a closed oriented connected

4- i,n,,:f.4.-1 11' into an orientable 3-if,,,.',.....1N3. Let pi(M ) denote the first

Pontrjagin class of 1 and [M4] the fundamental class of I 1I. Then e(f)

(pi( ), [M4]).
It allows us to formulate the theorem 8.1.1 in terms of the cohomology ring of

m\.

Theorem 9.0.1 Let f : [1P -- N3 be a continuous i'n'r'.:',j from an orientable

closed connected 4-,,,,/,..:.. into an orientable 3-n,,.n,..'.11 Then the homotel,'q

class of f contains a fold il.ni1'. i if and only if there is a 4'l..1,. /.i./; class x E

H2(l ;Z) such that pi(C) =x2.

A smooth mapping f : P1 N3 induces a section j2f of the 2-jet bundle

J2(T 1, f*TN3) over .M. To calculate the invariant e(f) we consider sections
Ai1 [ J2( l), where ( is an arbitrary orientable 4-vector bundle over [11 and Tr is

an arbitrary orientable 3-vector bundle over 11.

1 After the paper was written, the author learned that this equality is a special
case of a result obtained in [24].

The singular set E in the bundle J2( rT) over .31 is a manifold with sin-
gularities. By dimensional reasoning, the image of a general position section
j : ll -- J2((, T) does not contain singular points of the manifold with singular-
ities E. Consequently, the -:,i.l;,jri set j-1(E) of the section j is a submanifold of
i[1. We define the normal Euler number e(j) of the section j as the normal Euler
number e(j-1(E)).
A regular neighborhood E of E in J = J2(, r( ) is an open manifold. There
is a system of local coefficients F over E, the restriction FlT of which gives a
Z-orientation of E. The Poincar6 homomorphism for cohomologies and homologies
with twisted coefficients takes the fundamental class [E] onto some class Tr
H2(E, E \ E; ). Note that T-T is in H4(E, E \ E; Z). Let i be the composition

H4(E, E \ E; Z) H4(J, J \ E; Z) H4(J, 0; Z)

of the excision isomorphism and the homomorphism induced by the inclusion. We
define h((, Tr) = i(r-r). Then we claim that

e(j) (j*h((, f ), 1']). (9.1)

Lemma 9.0.7 For every general position section j : ll J2 (, I), the normal
Euler class of the -;n, i'.,. j-'() is given by (9.1). In particular, for every i,,j'~r.,:,
f : N3, we have e(f) (( 2f)*h(TM, f*TN3), [M4]).
Proof. Let A C [ll denote the singular set j-1(E) and B denote a tubular
neighborhood of A. The tubular neighborhoods E of E and B of A may be viewed
as vector bundles. Since j : B -> E is transversal to E, there is a commutative
diagram of vector bundles
B- E

A 1
A S .

The Thorn class 7 possesses the property of naturality. Hence the Thorn class of

the bundle B -- A is j*(r). We have a commutative diagram

i: H4(E, E \ ; Z) H (JJ E;Z) H4(J;Z)

H4(B,B \ A;Z) H4( F1', [1 \ A;Z) H4 (I-;Z),

which completes the proof. U

Lemma 9.0.7 shows that the number e(j) depends only on the bundles ( and T1.

That is why we will denote this number by e( T).

In the following, for an arbitrary manifold V, we denote the trivial line bundle

over V by r(V) or simply by 7.

Lemma 9.0.8 There is an integer k / 0 such that for i,:, orientable 4-vector

bundle ( over wiq:; closed oriented 4-il., .f.'41./ 1\', the c(,;,;/I (pi(0, [M41)

ke(, 3r) holds.
Proof. We recall (see chapter 7) that the bundle J2( 3r) over P1I is induced

by an appropriate mapping p : 3[1 BS04 x BSO3 from some bun-

dle J2(E4, E3) over BS04 x BSOs. As above we define a cohomology class

h(E4, E3) E H4(j2(E4, E3); Z). Let a be an arbitrary section of the bundle

J2(E4,E3). Together with p, the section a defines a section j : Pi -- J2(, 3r)
such that the diagram

j2 2 (E4, E3

31 BSO4 x BSO3
commutes. We have

j*h(, 3r) = j*p*h(E4, E3) = p*a*h(E4, E3),

where p denotes the upper horizontal homomorphism of the diagram. Con-

sequently, the class j*h(3, 37) is induced by p from some class a*h(E4, E3) in

H4(BS04 x BSO3; Z). Moreover, since 3, is a trivial bundle, the mapping p is
homotopic to a mapping .1 P- BS04 x pt c BS04 x BSO3 and therefore

j*h(, 37) is induced from some class in H4(BSO4; Z). Modulo torsion the group
H4(BSO4; Z) is isomorphic to Z E Z and is generated by the first Pontrjagin class

pi and the Euler class W4. Since H4(3F; Z) is torsion free, for some integers k and
1, we have

j*h(r, 37) =kpi( ) + lW4(). (9.2)

Let us apply (9.2) to the 4-sphere S4 with = TS4. The singular set of the

standard projection f : S4 R4 -+ R3 is a 2-sphere with trivial normal bundle in
S4. Hence (j*h((, 37), [S4]) e(f) = 0. Since pi(TS4) = 0 and W4(TS4) = 2, we
conclude that 1 0.

Finally, k / 0 follows from pi(0) / 0 for some U
To find the number k of Lemma 9.0.8 we need another description of the

invariant e(j). Let Tr and ( be vector bundles over a manifold .1'. There are
natural projections pr : ( E ()or _+ or and inclusion i : TI -- TI (. A point
of J"'(, rl) is a set of n homomorphisms {gi}i 1,...,, (see chapter 7). Define the

embedding

by

sn(gl,---,gk) =(gl id, io g2 op2, ... io gn opn).

The homomorphism s,, n > 1, is called the stabilization homomorphism

afforded by (.
Lemma 9.0.9 (Ronga 7'])

2. E2 1(( TI E D )) i(,) and
2. s21(Yi,j( E(,TI(D)) -Ei^j(,TI), and

3. the embedding s2 is transversal to the ,D',l,,,i.:l..1 j (, () and

Let ( be an orientable 4-vector bundle, T the trivial 3-vector bundle over
. ', and S2 the stabilization homomorphism afforded by the trivial line bundle
7 over .i. Lemma 9.0.9 allows us to give a definition of e(f) in terms of some
cohomology class of H4(J2( T Z; Z).
Lemma 9.0.10 There is a /. ,./.,i/; class h E H4(J2( T, T); Z) such that
for a section j : 31 J2(, l), we have e(j) ((2 o j)*(h), [M4]).
The proof of Lemma 9.0.10 is the same as that of Lemma 9.0.7.
Let ( and rl be the vector bundles of dimensions 4 and 3 respectively over
the standard 4-disc D4. The set of regular points in J2( E Tr, rl E r) is homotopy
equivalent to SO5. Therefore, each section j : S3 j 2( Tr TI r) that sends
S3 into the set of regular points defines an element j in the set of homotopy classes

[S3, Os]. The space SOs is an H-space; hence j is an element of w3(S05) =Z.
Since J2( TE, 'E ) is contractible, the section j admits the extension to a section
over D4. We obtain a mapping e : 73(S05) Z that sends the homotopy class ) of
a section j to the normal Euler number of the singular set of the section j extended
over D4

Lemma 9.0.11 The i''r!.:.! e : 73(SO5) Z is a well /. J,... homomorphism.

Proof. Let jl and j2 be two sections of the bundle J2 ( T', T E r) over D4 whose
restrictions ji aD4 and j2 aD4 map S3 = OD4 into the set of regular points of

J2( T~ T E r) and represent the same homotopy class j e [S3, SOs]. The
arguments similar to those in the proof of Lemma 5.3.1 show that the normal Euler
numbers of the submanifolds ji l() and j2 1(), where Z is the singular set of

J2( T-, T r-), are equal. Therefore, the number e(j) does not depend on the
choice of representative of the homotopy class j.

We need to verify that the equality

e(j + j2) = e(I) +e(2) (9.3)

holds for every pair of elements 1, j2 of r3(S05).
For i = 1, 2, let ji : OD' J2( D, T E r) be a section that leads to Ji. We can
modify jl by homotopy that does not intersect the singular set of J2( E T, rl D T) so
that the sections ji and j2 agree on some open subset of OD = OD. Then jl and

j2 determine a section j3 : aOD#OD/ J2( T, y T), which leads to an element
ji + J2 of 73(S05). Extensions of ji and j2 to D' and D' respectively give rise to
an extension of j3 to DI#D' the singular set of which is the union of the singular
sets of the extensions of jl and j2. Therefore, (9.3) holds. U
We have defined the normal Euler number of a general position section of
the bundle J2((, r) in the case where ( is an orientable 4-vector bundle and Tl is
an orientable 3-vector bundle. If ( is an orientable 5-vector bundle and Tl is an
orientable 4-vector bundle, then again the singular set of a general position section

j of J2( q) is a 2-submanifold of .31 and therefore we can define the normal Euler
number e(j) and the number e(, Tr) in the same way as above.
Let g : S3 -- SOs be a generator of 73(S0O) and d the number e(g). Let
us calculate e(6 E T, 4r), where 6 is the 4-vector bundle associated with the Hopf
fibration S7 S4.
Lemma 9.0.12 e(6, 37) = e(6 E T, 47) = d, up to sign.

Proof. The sphere S4 is a union of two discs D1 and D2 with OD1 = OD2. A choice
of trivializations of 6 over D1 and D2 defines a gluing homomorphism

a: 6 E rlaDi 6 T aD2, (9.4)

which being identified with a mapping S3 SOs represents a generator [a] E

73(SO5).

Let J1 and J0 respectively denote the space J1(6 E r, 4r) and the complement
to the singular set E in J1. To prove Lemma 9.0.12, it suffices to determine the
normal Euler number of j-1(E) for a particular section j : S4 -- J1. We regard a

section of J1 as a bundle homomorphism 6 T -- 4r. If j is given over D1, then
the diagram

t JD1 ido(jlD1)o0 ) 1
4lT I idD2
shows that in the trivialization of 6 over D2 the section jlo D is id o (jl aD) o a-1,
where id is the identity mapping. If we choose j to be constant over D1, then in
the trivialization of 6 over D2 the section jlo D induces a mapping S3 J0 M S05
representing the homotopy class -[a] E 73(SOs). Thus the normal Euler number of
j extended over D2 is 0 up to sign. U
Lemma 9.0.13 There is an integer q such that e(TCP2, 3) 1 + Id.
Proof. There is a mapping f of a regular neighborhood E of CP1 C CP2 into
R3 such that the singular set of f is CP1 (see Lemma 8.3.2). Let f be a general
position extension of f on CP2. The number e(f) is the sum of the normal Euler
number of CP1 and the normal Euler number of the surface of singular points that
lies in the disc D4 = P2 \ E. The latter number is a multiple of 0. Hence for some

q, e(f) 1 + -'. U
To calculate the exact value of e(TCP2, 3r) we use the notion of the connected
sum of two bundles.
For i = 1, 2, let 31 T be a closed oriented 4-manifold and (i an orientable

4-vector bundle over 31T'. Identifying the fiber of (1 over some point in 3 [f with
the fiber of 12 over some point in 3i\ we obtain a bundle over 3 / V i\1 which is
transferred to a bundle over .3 #3 1 IT by a natural mapping .3 T # -3 J 3 [ V 3 f.

We denote the resulting bundle over .3[/ #3 IT by (1#2. It follows that the

(Pi(I #62), :- (pi(I), [M4]) + (pi (2), -!L

and

e(1#2, 3wT/: V) e(I, 3,T +)) +e(2, 3T(1j))

take place.

Lemma 9.0.14 e(TCP2, 3-) 3.

Proof. Let 6 be the 4-vector bundle over S4 with pl(6) = 2. Lemma 9.0.12 implies

that e(6, 37) 0. For K = 2CP2, we have pi(TK#6#3) = 0, where 6#3 stands for

6#6#6. Lemma 9.0.8 shows that e(TK#6#3, 3T) is also zero. By additivity,

0 = e(TK#6#3, 3) = -2(1 + qd) + 30.

Since q and 0 are integers, we conclude that e(TK, 3r) = 6. This implies that

e(TCP2, 3r) = 3. On the other hand it is known (see Sakuma [36] and [37]) that

e(TCP2, 3) 3 (mod 4). Therefore, e(TCP2,23T) 3. U

Lemma 9.0.14 shows that the integer k in Lemma 9.0.8 equals 1. Thus, for

every oriented 4-manifold .1' and a general position mapping f : I' -- R3, we

have e(f) = (pi(3 ), [M4]). This completes the proof of Lemma 9.0.6. U

Theorem 9.0.1 is proved. U

CHAPTER 10
SIMPLY CONNECTED CASE

This chapter is devoted to the case where the manifold .31 is simply con-

nected. We examine the equation pi(3 [) = x2 and determine when it has a

solution.

Theorem 10.0.2 Suppose that .1' is an orientable closed connected -.:i,,'lp

connected 4-,m,,. ,: f. /.I and N3 is an orientable 3- n,,',:. f./I Then a homote('i.; class

of a f.'' ':j f 3: I -- N3 has no fold ii l'l!"''! if and only if 3i1 is homot(.ji'/

equivalent to CP2 or CP2# CP2. Here homote(.c' equivalence is not supposed to be

orientation preserving.

As has been shown, a homotopy class of a general position mapping f from a

connected closed oriented 4-manifold .3 into an orientable 3-manifold N3 has a

fold mapping if and only if e(f) = (pi(.1), [M4]) E Q(1). That is the number

(pi(,3), [M4]) is a value of the intersection form of .31. First, let us consider the
case where the intersection form of .31 is indefinite. If pi( ) = 0, then for every

f, e(f) = 0 E Q(3P). Suppose pi(3 ) / 0.

Lemma 10.0.15 If the intersection form of a closed -.:,,1m; connected m,,;,,...f..1.1 3

with pl(M ) / 0 is .,/. I,./: odd, then Q(3 ) = Z. In particular (pi(1), [41) e

Proof. Since the intersection form of .1I is odd, in H2(3P; Z) there exists a basis

g,9g2, ...,g s *..., k such that the value of the intersection form at ale + + aes +
2 2 2 2
S+ akek is C+ +1 -... -2. Therefore, the number e(f) is in Q(.1)

if and only if e(f) can be represented in the form ca + + ca ac2+ a for

some integers ca, i 1, ..., k. Since the intersection form is indefinite, this sum has

at least one positive square and at least one negative square. Since the signature

a(o-(1) = pi(P\') / 0, the number k of squares is at least 3. Suppose that the
number e(f) is odd. Then it can be represented as the difference of two squares.
Suppose that e(f) is even. Then the odd number e(f) 1 can be represented as the
difference of two squares and the third square of the sum can be used to add T1 to
the difference to get e(f). Hence Q(lP) Z. U
Suppose that the intersection form of .3 [ is indefinite even. Being even, it is
isomorphic to a direct sum of some copies of Es-form and some copies of the form
0 1
with matrix Consequently, the number (pi(\l ), [M4]) = 3o(31) is
1 0)
even. Every even indefinite intersection form contains a subform isomorphic to

Since this subform takes every even value, we have (pi(\l ), [M4])
1 0

Thus, every closed simply connected 4-manifold with indefinite or trivial
intersection form admits a fold mapping into R3.
To treat the case where the intersection form of .3 1 is definite, we need the
Donaldson Theorem. Let kJ, k / 0, denote the form of rank Ikl given by the
diagonal matrix with eigenvalues 1 if k > 0 and -1 if k < 0.
Theorem 10.0.3 (Donaldson [71) Every 1 /7i,,'/. intersection form of a closed
oriented smooth 4-11,,:ni f./. is isomorphic to the form kJ for some integer k / 0.
Lemma 10.0.16 Suppose that the intersection form of a connected closed -.: ,, '1
connected innf.-./,1 i3 with pji(f) / 0 is 1, [f,,/, Then (pi(p'), [14]) E Q(i[1)
if and only if the intersection form is isomorphic to kJ, Ik > 3.
Proof. It suffices to consider only the case where k > 0. If k = 1, then the inter-
section form is isomorphic to that of CP2 and (pi(i ), [M41) = 3 1(1) is not in
Q(.1l). For k = 2, the set Q(3.1) consists only of integers that can be represented
as the sum of at most two squares. Hence the number (pi(i. ), [M4) = 6 is not
in Q(lM'). If k = 3, then (pi(fl1), [M4]) = 9 E Q([1). Finally, by the Lagrange

58

theorem, every positive integer can be represented as a sum of four squares. Thus

for k > 4, we have (pl(3 i), [M4]) C Q(i1). *

In view of Theorem 8.1.1, Lemmas 10.0.15 and 10.0.16 imply that .3l1 admits a

fold mapping into R3 if and only if the intersection form of 31 T is different from J

and 2J. By the J. H. C. Whitehead Theorem about the oriented homotopy type

of a simply connected 4-manifold [22], [41], this completes the proof of Theorem

10.0.2. U

CHAPTER 11
FINAL REMARKS

Remark 1. If two manifolds 3M/ and .3j admit a fold mapping into R3, then the

connected sum M,#M2 also admits a fold mapping into R3. In [34] the authors

conjectured that the obstruction to the existence of a fold mapping into R3 is

additive with respect to connected sum, and the manifold kCP2#lP2 admits

a fold mapping into R3 if and only if k + 1 is odd. Theorem 10.0.2 solves the

conjecture in the negative.

Remark 2. Sakuma conjectured (see Remark 2.3 in [19]) that a closed orientable

manifold with odd Euler characteristic does not admit a fold mapping into R"

for n = 3, 7. Saeki [33] presented an explicit counterexample to this conjecture.

Theorem 10.0.2 shows that there are many manifolds with odd Euler characteristic

admitting fold mappings into R3. However, it should be mentioned that Theorem

10.0.2 does not -i--.- -1 a method of an explicit construction and the question of an

explicit construction of a fold mapping for a given manifold seems difficult.

Remark 3. If the manifolds M" and N" are orientable and m n is odd, then

every Morin mapping f : M" -> N" is homotopic to a mapping with at most cusp

singular points [29]. Theorem 10.0.2 gives a restriction on further simplification of

Morin mappings by homotopy.

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BIOGRAPHICAL SKETCH

I was born in 1978 in Russia. In 1996-2001 I studied at Moscow State Univer-

sity. In 2001 I become a graduate student at the University of Florida.