PREFACE
In1982,MichaelFreedman,buildingupontheideasandconstructionsofAndrew Casson,provedthe h-cobordismtheoremandtheexactnessofthesimplyconnected surgerysequenceindimensionfour,deducingaclassificationtheoremfortopological 4manifolds,aspecialcaseofwhichwasthe 4-dimensionaltopologicalPoincaréconjecture.
Thekeyingredientinhisproofisthe discembeddingtheorem.Inmanifoldsofdimension fiveandhigher,genericmapsofdiscsareembeddings,whereasindimensionfoursuch mapshaveisolateddoublepoints,preventingthehigh-dimensionalproofsfromapplying. Freedmanshowedhowtoembeddiscsinsimplyconnected 4-manifolds,revealingthat incertainsituationstopological 4-manifoldsbehavelikehigher-dimensionalmanifolds. ContemporaneousresultsofSimonDonaldsonshowedthatsmooth 4-manifoldsdonot. Indeed,dimensionfourexhibitsaremarkabledisparitybetweenthesmoothandtopological categories,asdemonstratedbytheexistenceofexoticsmoothstructuresonR4,forexample.
FreedmanandDonaldsonbothreceivedtheFieldsMedalin1986fortheircontributions totheunderstandingof 4-manifolds.SoonafterFreedman’sworkappeared,FrankQuinn expandedonthetechniquesofFreedman,provingfoundationalresultsfortopological 4-manifolds,suchastransversalityandtheexistenceofnormalbundlesforlocallyflat submanifolds.TheworkofFreedmanandQuinnwascollectedinthebook[FQ90],which becamethecanonicalsourcefortopological 4-manifoldsinthedecadesthatfollowed.
TheOriginofThisBook
InJanuaryandFebruaryof2013,Freedmangaveaseriesof12lecturesattheUniversity ofCaliforniaSantaBarbara(UCSB)intheUSAwiththegoalofexplaininghisproofof thediscembeddingtheorem.ThelectureswerebroadcastlivetotheMax-Planck-Institut fürMathematik(MPIM)inBonn,Germanyaspartofthe Semesteron 4-manifoldsand theircombinatorialinvariants organizedbyMatthiasKreckandPeterTeichner,where QuinnandTeichnerransupplementarydiscussionsessions.RobertEdwards,intheUCSB audience,notonlycontributedvariousremarksbutalsosteppedinasaguestlecturer andpresentedhisperspectiveonakeystepoftheproof,namelytheconstructionof ‘thedesign’.Thelectureswererecordedandarecurrentlyavailableonlineatwww.mpimbonn.mpg.de/FreedmanLectures.
ThisbookbeganasannotatedtranscriptsofFreedman’slecturestypedbyStefanBehrens. InMayandJuneof2013,theroughdraftofthenoteswasrevisedandaugmentedinacollaborativeeffortoftheMPIMaudience,coordinatedbyBehrensandTeichner.Thefollowing peoplewereinvolvedinthisprocess:XiaoyiCui,MatthewHogancamp,DanielKasprowski,
JuA.Lee,WojciechPolitarczyk,MarkPowell,HenrikRüping,NathanSunukjian,and DanieleZuddas.
Threeyearslater,inNovemberandDecemberof2016,PeterFellerandMarkPowellorganizedaseminaronthediscembeddingtheoremattheHausdorffInstituteforMathematics (HIM)inBonn.ThisincludedscreeningsofFreedman’sUCSBlecturesondecomposition spacetheoryandaseriesoftalksbyPowell,alongwithguestlecturesbyStefanBehrens, PeterFeller,BoldizsárKalmár,AllisonN.Miller,andDanielKasprowskiontheconstructive partoftheprooffollowingtheapproachinthebookbyFreedmanandQuinn[FQ90]. TheHIMaudienceincludedmanyoftheparticipantsintheJuniorTrimesterProgramin Topology:ChristopherW.Davis,PeterFeller,DuncanMcCoy,JeffreyMeier,AllisonN. Miller,MatthiasNagel,PatrickOrson,JungHwanPark,MarkPowell,andArunimaRay. Together,thespeakersandtheaudiencerevisedthestructureofthe2013notes,fleshing outmanydetailsandrewritingcertainpartsfromscratch.From2017to2020,Kalmár,Kim, Powell,andRaysynthesizedtheindividualcontributionsoftheauthorsintotheartefact youpresentlybehold.Newchaptersongoodgroups,theapplicationsofthediscembedding theoremtosurgeryandthePoincaréconjecture,thedevelopmentoftopological 4-manifold theory,andremainingopenproblemswerewritten.Duringthisperiod,KimandMiller,in particular,createdthemanycomputerizedfiguresappearingthroughoutthebook.
ThistextfollowsFreedman’sintroductiontodecompositionspacetheoryinhis2013 lecturesinPartI,beforegivingacompleteproofofthediscembeddingtheoreminPartsII andIV.Thelatterpartsfollowthe2016lecturesbasedon[FQ90],althoughtheyare naturallybasedontheideaslearntfromFreedman’soriginallecturesandtheconcurrent explanationsandguestlecturesbyEdwards,Quinn,andTeichner.Inparticular,wegivea detailednewdescriptionoftowerembeddingandthedesign.PartIIIcontainsadiscussion ofmajorapplicationsandconjecturesrelatedtothediscembeddingtheorem.Itdescribes howtousethediscembeddingtheoremtoprovethe s-cobordismtheorem,thePoincaré conjecture,theexactnessofthesurgerysequenceindimensionfourforgoodgroups,and thetopologicalclassificationofsimplyconnectedclosed 4-manifolds.
Sincesomuchof 4-dimensionaltopologicalmanifoldtheoryrestsontheseminalwork ofFreedman,ithasbeenfeltbythecommunitythatanotherindependentandrigorous accountoughttoexist.Wehopethatthismanuscriptwillmakethishighpointin 4-manifold topologyaccessibletoawideraudience.
CassonTowers
Wechoosetofollowtheprooffrom[FQ90],usinggropes,whichdiffersinmanyrespects fromFreedman’soriginalproofusingCassontowers[Fre82a].Theinfiniteconstruction usinggropes,whichwecalla skyscraper,simplifiesseveralkeystepsoftheproof,andthe knownextensionsofthetheorytothenon-simplyconnectedcaserelyonthisapproach. ReadersinterestedinCassontowersshouldrefertotheMPIMvideosofFreedman’s2013 lectures,whereheexplainedmuchaboutCassontowersandtheiruseintheoriginalproof. Apartfrom[Fre82a],furtherliteratureonCassontowersincludes[GS84,Biž94,Sie82,
CP16].Moreover,thecombinationof[Sie82]andtheCassontowerembeddingtheorem from[GS84]givesanotheraccountoftheoriginalCassontowersprooffrom[Fre82a].
Differences
Webrieflyindicate,fortheexperts,thesalientdifferencesbetweentheproofgiveninthis bookandthatgivenin[FQ90].First,thereisaslightchangeinthedefinitionoftowers(and thereforeofskyscrapers),whichwepointoutpreciselyinRemark12.8.Withourdefinition, itisclearthatthecorrespondingdecompositionspacesaremixedramifiedBing–Whitehead decompositions.Thispossibilitywasmentionedin[FQ90,p.238].
Additionally,thestatementofthediscembeddingtheoremin[FQ90]assertsthat immerseddiscs,undercertainconditionsincludingtheexistenceofframed,algebraically transversespheres,maybereplacedbyflatembeddeddiscswiththesameboundaryandgeometricallytransversespheres.Theproofsgivenin[Fre82a,FQ90]producetheembedded discsbutnotthegeometricallytransversespheres.Weremedythisomissionbymodifying thestartoftheproofgivenin[FQ90],asin[PRT20].Thegeometricallytransversespheres areessentialforthesphereembeddingtheorem,whichisthekeyresultusedintheapplicationofthediscembeddingtheoremtosurgeryfortopological 4-manifoldswithgoodfundamentalgroupandtheclassificationofsimplyconnected,closed,topological 4-manifolds,as wedescribeinChapter22.Wealsoobservethatthegeometricallytransversespheresinthe outputarehomotopictothealgebraicallytransversespheresintheinput[PRT20].Besides thesepoints,theproofofthediscembeddingtheoremgiveninthisbookonlydiffersfrom thatin[FQ90]intheincreasedamountofdetailandnumberofillustrations.
Welargelyfocusonthefirstfewchaptersof[FQ90].Inparticular,weassumethatthe ambient 4-manifoldissmooth.WedonotdelveintotheworkofQuinnonthesmoothing theoryofnoncompact 4-manifolds,theannulustheorem,transversality,ornormalbundles forlocallyflatsubmanifolds,insteaddescribingthesedevelopmentsbroadlyinChapter1, andinmoredetailinChapter21.
SeminarOrganization
Themajorityofthechaptersinthisbookmaybecoveredinasingleseminartalkeach. WeexpectthatPartsIIandIV,evenwithoutgoingthroughallthedetailsinPartIV, willrequireasemester.Wethereforesuggestthefollowingalternativetothestandard approach.AfterusingChapters1and2toprovidecontext,workthroughPartsIIandIV alongsidegroupviewingsofthevideosofFreedman’sUCSBlectures2–5,whichdiscussed thedecompositionspacetheoryofPartI.TheexpositioninPartIofthisbookshouldsupply enoughadditionaldetailtosupportthelectures,anditaddstothecharmoflearningthis mathematicstowatchthemanhimselfexplainit.ThisalsoallowsPartsIandIItobecovered simultaneously.Inthelatterpartoftheseminar,resultsfrombothcanbecombinedforthe proofthatskyscrapersarestandardinPartIV.PartIIIisnotdirectlyapplicabletotheproof ofthediscembeddingtheoremandmaybesafelyskippedinthefirstreading.
Credit
Thismanuscriptistheoutcomeofacollaborativeprojectofmanymathematicians,as describedearlier.AfterFreedman,whoofcoursegavetheoriginallecturesandprovedthe discembeddingtheoreminthefirstplace,andStefanBehrens,whotypeduptheinitialdraft, manypeoplecontributedtoimprovingindividualchapters,orinsomecasesdeveloping themfromscratch.Wethereforeattributeeachchaptertothosewhocontributedthebulkof theworktowardsit,whetherthroughanewlecturethattheywroteanddelivered,polishing theexposition,creatingoriginalpictures,addingnewmaterialtofillindetailsthatcouldnot becoveredinthelectures,orwritingachapterontheirownbycombininginformationfrom varioussourcesintheliterature.
Apartfromtheauthors,theprojectbenefittedfromtheinputofBobEdwardsandFrank Quinn,aswellasJaeChoonCha,DiarmuidCrowley,JimDavis,StefanFriedl,BobGompf, ChuckLivingston,MichaelKlug,MatthiasKreck,ChristianKremer,SlavaKrushkal,Andy Putman,BenRuppik,andAndrásStipsicz.
1ContextfortheDiscEmbeddingTheorem
StefanBehrens,MarkPowell,andArunimaRay
PARTI DECOMPOSITIONSPACETHEORY
3TheSchoenfliesTheoremafterMazur,Morse,andBrown
StefanBehrens,AllisonN.Miller,MatthiasNagel,andPeterTeichner
4DecompositionSpaceTheoryandtheBingShrinkingCriterion
ChristopherW.Davis,BoldizsárKalmár,MinHoonKim,andHenrikRüping
5TheAlexanderGoredBallandtheBingDecomposition
StefanBehrensandMinHoonKim
6ADecompositionThatDoesNotShrink
StefanBehrens,ChristopherW.Davis,andMarkPowell
7TheWhiteheadDecomposition
XiaoyiCui,BoldizsárKalmár,PatrickOrson,andNathanSunukjian
8MixedBing–WhiteheadDecompositions
JeffreyMeier,PatrickOrson,andArunimaRay
StefanBehrens,BoldizsárKalmár,andDanieleZuddas
PARTII BUILDINGSKYSCRAPERS
11IntersectionNumbersandtheStatementoftheDiscEmbedding
MarkPowellandArunimaRay 12Gropes,Towers,andSkyscrapers
MarkPowellandArunimaRay
DuncanMcCoy,JungHwanPark,andArunimaRay
14ArchitectureofInfiniteTowersandSkyscrapers ...................
StefanBehrensandMarkPowell
15BasicGeometricConstructions ...............................
MarkPowellandArunimaRay
16FromImmersedDiscstoCappedGropes
WojciechPolitarczyk,MarkPowell,andArunimaRay
17GropeHeightRaisingand 1-storeyCappedTowers ................
PeterFellerandMarkPowell
18TowerHeightRaisingandEmbedding ..........................
AllisonN.MillerandMarkPowell
PARTIII INTERLUDE
19GoodGroups .............................................
MinHoonKim,PatrickOrson,JungHwanPark,andArunimaRay
20The s-cobordismTheorem,theSphereEmbeddingTheorem,andthe PoincaréConjecture ........................................
PatrickOrson,MarkPowell,andArunimaRay
21TheDevelopmentofTopological4-manifoldTheory
MarkPowellandArunimaRay
22SurgeryTheoryandtheClassificationofClosed,SimplyConnected 4-manifolds 331
PatrickOrson,MarkPowell,andArunimaRay
23OpenProblems ............................................
MinHoonKim,PatrickOrson,JungHwanPark,andArunimaRay
PARTIV SKYSCRAPERSARESTANDARD
24ReplicableRoomsandBoundaryShrinkableSkyscrapers ...........
StefanBehrensandMarkPowell
25TheCollarAddingLemma
DanielKasprowskiandMarkPowell
26KeyFactsaboutSkyscrapersandDecompositionSpaceTheory ......
MarkPowellandArunimaRay
1.1Tryingtosurgerahyperbolicpair..............................4
1.2Tryingtocancelalgebraicallydual 2-and 3-handlesinan s-cobordism....6
1.3Adjustingthealgebraicself-intersectionnumber....................7 1.4TheWhitneymove........................................7
1.5TheHopflinkatatransverseintersection.........................8
1.6ABingdoublealongaWhitneycircle...........................9
1.8Tradingintersectionsforself-intersections........................12 1.9Adjustingtheintersectionnumberofspheresintheproofofthe s-cobordism
1.10ObtainingaWhitneydiscintheproofofthe s-cobordismtheorem.......14 1.11Whitneymovetoresolveaself-intersection.......................15
2.1Transformingacappedsurfaceintoasphere.......................30
2.3Schematicpictureofthe 2-dimensionalspineofaheightthreegrope......32 2.4Schematicpictureofthe 2-dimensionalspineofaheighttwocappedgrope.33
2.5Schematicpictureofthe 2-dimensionalspineofa 2-storeytower........35
2.6TheBingdoubleandaWhiteheaddoubleofthecoreofasolidtorus......37 2.7Aniterated,ramifiedBing-Whiteheadlinkinasolidtorus.............38
5.4Twothickenedarcsembeddedin
5.5Twothickenedarcsembeddedin D2 × [ 1, 1] andatorus...........81
5.6AdecompositionassociatedwiththeAlexandergoredball.............82
5.7ThedefiningpatternoftheBingdecompositionanditssecondstage......84
5.8Bing’sproofshowingthattheBingdecompositionshrinks.............85
6.1Thedefiningpatternforthedecomposition B2.....................88
6.2Asubstantialintersection...................................89
6.3Liftsofthecomponentsofthedefiningpatternfor B2................90
7.1ThedefiningpatternfortheWhiteheaddecomposition...............96
7.2Ahomeomorphismfollowedbyarotation/shear...................101
8.1Ameridional3-interlacing...................................106
8.2An (n,m)-link..........................................111
8.3A (3, 2)-linkintersectingan 8-interlacingoptimally..................111
9.1AlternativedefiningpatternsfortheBingdecompositionandthedecomposition B2...............................................116
9.2Examplesofstarlike,starlike-equivalent,andrecursivelystarlike-equivalent sets..................................................119
9.3Aredbloodcellwithonedimensioninthe S1 × D3 piecesuppressed....121
9.4Shrinkingaredbloodcelldisc................................121
9.5Proofofthestarlikeshrinkinglemma...........................123
9.6ProofofLemma9.12......................................124
9.7ConstructingtheternaryCantorset............................128
10.1AmodificationoftheCantorfunction...........................133
10.2Anadmissiblediagram.....................................139
11.1AmodelWhitneymove....................................158
11.2Amodelfingermove.......................................159
11.3Computationoftheintersectionnumberforspheresina 4-manifold......161
11.4Aregularhomotopyacrosstheboundaryofanimmerseddiscwhichdoesnot preservetheintersectionnumber..............................162
11.5Computationoftheself-intersectionnumberforanimmersedsphereina 4-manifold.............................................164
11.6FindingWhitneydiscspairingself-intersectionpointsofanimmersedsphere withtrivialself-intersection..................................168
12.1Astandardsurfaceblockwithgenusone.........................173
12.2Tipregionsforastandarddiscblock............................174
12.3Schematicpictureofthe 2-dimensionalspineofaheightthreegrope......176
12.4Schematicpictureofthe 2-dimensionalspineofaheighttwocappedgrope.176
12.5Schematicpictureofthe 2-dimensionalspineofa 2-storeytower........177
12.6Splittingthehigherstagesofa 2-storeytowerinto (+)-and ( )-sides.....179
13.1Schematicofa 1-anda 2-handle...............................186
13.2Notationfor 1-handles.....................................188
13.3BasicKirbydiagrams......................................189
13.4Sliding 2-handles.........................................191
13.5Cancellinga 1-/2-handlepairinthreedimensions..................191
13.6HandlecancellationinaKirbydiagram..........................192
13.7Decomposingadiscintohandlesrelativetoitsboundary..............193
13.8Plumbingtwo 2-handles....................................193
13.9Diagramsforself-plumbingsof 2-handles........................194
13.10Kirbydiagramforasurfaceblockwithgenusone...................195
13.11Kirbydiagramforasurfaceblockwithgenustwo...................196
13.12Kirbydiagramforadiscblock................................196
13.13Identifyingtwosolidtoriusinga 1-handleanda 2-handle.............198
13.14Kirbydiagramforaheighttwogrope...........................198
13.15Thespineofaheightonecappedgrope..........................199
13.16AKirbydiagramofa 1-storeycappedtower.......................199
13.17PatternsforBingandWhiteheaddoubling........................200
13.18SimplificationforBingdoubles...............................202
13.19SimplificationforWhiteheaddoubles...........................203
13.20Atreeassociatedwithaheighttwogrope.........................205
13.21Agraphassociatedwithaheightonecappedgrope..................205
13.22AsimplifiedKirbydiagramofaheightonecappedgrope..............207
13.23AsimplifiedKirbydiagramforaheighttwocappedgrope.............208
15.1ACliffordtorusatadoublepoint..............................218
15.2Tubingintoatransversesphere...............................219
15.3Tubingmultiplepointsofintersection...........................219
15.4Boundarytwisting........................................220
15.5Boundarypush-offtoensureWhitneycirclesaredisjoint..............220
15.6Pushingdownintersections..................................221
15.7(Symmetric)contractionofacappedsurface......................222
15.8Contractionfollowedbypushingoffintersectionswiththecaps.........223
15.9ProofofthegeometricCassonlemma...........................224
15.10AnimmersedWhitneymoveisaregularhomotopy..................225
16.1SummaryofProposition16.1................................229
16.2Modifyinganimmerseddiscwithatransversespheretohavezero self-intersection..........................................230
16.3ArrangingforWhitneydiscsinthecomplement....................230
16.4ObtainingageometricallytransversecappedsurfacefromaCliffordtorus..231
16.5RemovingintersectionsbetweentheWhitneydiscsandthecapsofthe transversecappedsurfaces...................................232
16.6SummaryofProposition16.2................................233
16.7Constructingaheightonecappedgrope.........................234
16.8Asinglecappedsurface.....................................234
16.9Thefirstgeometricallytransversesphere.........................234
16.10SeparatingcapsintheproofofProposition16.2....................235
16.11AfterseparatingcapsintheproofofProposition16.2................236
16.12Managingcapintersections..................................236
16.13Thesecondgeometricallytransversesphere.......................237
17.1Schematicpictureofthe 2-dimensionalspineofaheighttwocappedgrope.240
17.2Constructingatransversecappedgrope.........................241
17.3Separating (+)-sideand ( )-sidecaps..........................242
17.4Raisingtheheightofthe ( )-sideofagrope......................243
17.5Obtainingdiscsfromnullhomotopiesofthedoublepointloops........246
17.6Tubingintothegeometricallytransversesphere....................246
17.7A 1-storeycappedtowerwithageometricallytransversesphere.........248
17.8Tubingtoremoveintersections...............................251
18.1Aschematicpictureofthe 2-dimensionalspineofa 2-storeytower.......254
18.2AccessorytoWhitneylemma.................................256
18.3Splittingthesecondandhigherstagesofa 2-storeytowerinto (+)-and ( )-sides..............................................257
18.4Asinglepairofself-intersectionsgivesrisetoeightnewpairsofintersections.258
18.5Thegrope-Whitneymove...................................258
18.6Atowerwithtwostoreysonthe (+)-sideandonestoreyonthe ( )-side..261
18.7A 1-complexinagropeandinadiscblock........................263
18.8Embeddingtheattachingcircletimesanintervalinasurfaceandadiscblock.264
18.9Embeddingtheattachingcircletimesanintervalawayfromintersections...264
18.10Embeddingtheattachingcircletimesanintervalneartheattachingcircles closetointersectionpoints..................................265
18.11Fingermovetomakethetrackofahomotopyembedded..............265
18.12Movingthesecondstoreyandtowercapsofatowerintosmallballs......266
19.1Newdoublepointloopsafteracontractionfollowedbyapush-off........275
19.2Newdoublepointloopsinthecapseparationlemma................276
19.3Newdoublepointloopsingropeheightraising....................277
20.1ObtainingaWhitneydiscintheproofofthe s-cobordismtheorem.......286
20.2ObtainingatransversespherefromaCliffordtorus.................287
20.3Proofofanalternativeversionofthediscembeddingtheorem..........290
20.4FindinganembeddedWhitneydisc............................291
20.5Summaryofthesphereembeddingtheorem......................292
21.1Thedevelopmentoftopological 4-manifoldtheory..................296
23.1Thediscembeddingconjectureinrelationtootherconjectures..........354
23.2Anelement L ofthefamily L1................................357
23.3Thesurgeryconjectureinrelationtootherconjectures...............361
23.4TubingalongWhitneycirclestomodifyanimmersedsphereintoanembeddedclosedsurface........................................374
23.5Resolvingintersectionsbetweenaspheretransversetoasurfaceanditscaps.374
23.6Managingcapintersections..................................375
23.7ResolvingintersectionsbetweenasurfaceandWhitneydiscspairingintersectionsbetweenitscapsandatransversesphere.....................375
23.8AKirbydiagramforasinglepairofgeometricallytransversesphere-like cappedgropesofheightone..................................378
25.1Addingacollartoadefiningsequence...........................393
27.1Thedesigninsidetheskyscraperandthedesigninsidethestandardhandle..402
27.2AspanningdiscforaWhiteheadcurve..........................403
27.3Aschematicpictureofahole+ showingaredbloodcelldisc...........403
28.1ConstructingtheternaryCantorset............................408
28.2Anexplicitparametrizationofthestandardhandle..................411
28.3Skyscrapersandcollarscorrespondingtofinitebinarywordsoflengthat mosttwo...............................................415
28.4Skyscrapersandcollarscorrespondingtofinitebinarywords oflengththree...........................................416
28.5Skyscrapersandcollarscorrespondingtofinitebinarywordsoflengthat mostthree..............................................416
28.6Thedesignintheskyscraper..................................420
28.7Thedesigninthestandardhandle..............................427
28.8AspanningdiscforaWhiteheadcurveandparametrizingits self-intersections.........................................431
28.9Agraphofthefunction b restrictedtotheabscissa..................432
28.10Theproofthatthesingularimageof f isnowheredense...............443
A.1Cassonhandles..........................................451
A.2Theproduct
A.3TheAndrews-Rubinshrink..................................453
A.4Themap α crushes {holes+} and β crushes {gaps+} ..............454 A.5Thegraphof β ..........................................455
ContextfortheDiscEmbeddingTheorem
stefanbehrens,markpowell,andarunimaray
1.1BeforetheDiscEmbeddingTheorem
1.1.1High-dimensionalSurgeryTheory
By1975,classificationproblemsformanifoldsofdimension n atleastfive,betheysmooth, piecewiselinear(PL),ortopological,hadbeentranslatedintoquestionsinhomotopy theoryandalgebra.Foreachofthesecategories,classificationproblemsaretypicallyoftwo types:the existenceproblem concernstheexistenceofamanifoldwithinagivenhomotopy type,whilethe uniquenessproblem concernsthenumberofsuchmanifoldsuptoisomorphism.Theinputforsuchquestionsisa Poincarécomplex—roughlyspeakingafinitecell complexthatsatisfies n-dimensionalPoincarédualityforsome n
Fixthecategory CAT tobeeithersmooth, PL,ortopological.Twoclosed n-manifolds aresaidtobe h-cobordant iftheycoboundan (n +1)-manifoldsuchthattheinclusionof eachboundarycomponentisahomotopyequivalence.The structureset ofagivenPoincaré complex X,denotedby S(X),isthesetof n-dimensionalclosedmanifolds M along withahomotopyequivalence M → X,upto h-cobordism,wherethecobordismhasa compatiblemapto X.ForPoincarécomplexesofdimensionatleastfive, surgerytheory can decideif S(X) isnonempty,andifso,cancomputeitexplicitlyusingalgebraictopology,at leastinfavourablecircumstances[Bro72,Nov64,Sul96,Wal99,KS77].Moreprecisely,the structuresetofaPoincarécomplex X withdimensionatleastfiveisnonemptyifandonly if(i)acertainsphericalfibrationover X,calledthe Spivaknormalfibration,liftstoa CAT bundle,and(ii)an L-theoreticsurgeryobstructionvanishes.Thiscompletelyanswers,at leastinprinciple,thequestionofwhether X ishomotopyequivalenttoa CAT manifold. Moreover,ifthestructuresetforaPoincarécomplex X ofdimension n atleastfive isnonempty,itfitsinthefollowingexactsequenceofpointedsets,calledtheBrowder–Novikov–Sullivan–Wall surgeryexactsequence
Here N (X) denotesthesetof normalinvariants of X,namelybordismclassesofdegree onemapsfromsome n-manifoldto X,togetherwithnormalbundledata.Viatransversality, thiscanbecomputedusinghomotopytheory.The L-groupsarepurelyalgebraicanddepend onlyonthegroup π1(X),theorientationcharacter,andtheresidueof n modulo 4. Letusdescribetheexistenceprogrammeinmoredetail.AssumingthattheSpivak normalfibrationon X liftstoa CAT-bundle,achoiceofliftgivesrisetoanelementof N (X),namelyaclosedmanifold N togetherwithadegreeonemapto X thatrespects thenormaldatacorrespondingtothechosenlift.Wewishtoimprovesuchanelementtoa manifold M equippedwithahomotopyequivalenceto X,attheexpenseofmodifying N bytheprocessof surgery.An elementarysurgery consistsoffindinganembedded Sp × Dq withina (p + q)-dimensionalmanifold,cuttingoutitsinterior,andgluingin Dp+1 × Sq 1 alongitsboundaryinstead.Thisprocesskillsthehomotopyclassrepresentedby Sp ×{0} andthereforecanassistinachievingagivenhomotopytype.Themaintheoremofsurgery theorysaysthatsuchasequenceofelementarysurgerieson N canproduceamanifold homotopyequivalentto X ifandonlyiftheobstructionin Ln(Z[π1(X)]) associatedwith N vanishes.Thisisencodedbythemap σ inthesequenceabove.Inotherwords,every elementof σ 1(0) canbemodifiedbysurgerytoproduceanelementofthestructureset S(X),namelyaclosedmanifold M equippedwithahomotopyequivalencetothePoincaré complex X.Thisargumentshowsthat,forPoincarécomplexesofdimensionatleastfive,we haveaprocedurefordecidingwhetherthestructuresetisnonempty—thatis,whetherthe existenceproblemhasapositiveresolution.
Exactnessofthesurgerysequencecanbeusedtocalculatethesizeofthestructureset, whichaddressespartoftheuniquenessproblem.Inordertofullysolvetheuniqueness problem,wealsoneedtounderstandwhen h-cobordantmanifoldsareisomorphicinthe category CAT.The s-cobordismtheorem [Sma62,Bar63,Maz63,Sta67,KS77](seealso [Mil65,RS72])statesthatan h-cobordismbetweenclosedmanifoldsofdimensionatleast fiveisaproductifandonlyifitsWhiteheadtorsionvanishes.Thetheorem,whichholdsfor allsmooth, PL,andtopologicalmanifolds,allowsonetoobtainuniquenessresults.
Itsprecursor,the h-cobordismtheorem,statesthatevery simplyconnected h-cobordism betweenclosedmanifoldsofdimensionatleastfiveisaproduct.Thisisastraightforward corollaryofthe s-cobordismtheorem,sinceasimplyconnected h-cobordismhasWhiteheadtorsionvaluedintheWhiteheadgroupofthetrivialgroup,whichitselfvanishes.
Summarizing,bytheearly1970s,armedwiththepowerfultoolsofthesurgeryexact sequenceandthe s-cobordismtheorem,topologistshadadeepunderstandingofboththe existenceanduniquenessproblemsformanifoldsofdimensionatleastfive.
1.1.2Attempting4-dimensionalSurgery
Bycontrast,intheearly1970sverylittlewasknownabout 4-manifolds.Whitehead[Whi49] andMilnor[Mil58]hadshownthatthehomotopytypeofasimplyconnected 4dimensionalPoincarécomplexisdeterminedbyitsintersectionform.Moreprecisely, thehomotopytypes,togetherwithachoiceoffundamentalclass,areinonetoone correspondencewithisometryclassesofunimodularsymmetricintegralbilinearforms, or,equivalently,congruenceclasses A ∼ PAP T ofsymmetricintegralmatriceswith
determinant ±1.So 4-manifoldtopologistswereinterestedindeterminingwhichof theseformsarerealizedby,smooth(equivalently, PL [HM74;FQ90,Theorem8.3B]) ortopological,closed 4-manifolds;whetherhomotopyequivalent 4-manifoldsare scobordant;andwhether s-cobordant 4-manifoldsare CAT-isomorphic.Duetoits remarkablesuccessinaddressinghigh-dimensionalmanifolds,surgerytheoryseemed likeapromisingtool.However,themaintheoremsofsurgerywerenotknowntohold indimensionfour.Similarly,the h-and s-cobordismtheoremsfor 4-manifoldsremained openinallthreecategories.
Let E8 denotetheeven 8 × 8 integerCartanmatrixoftheeponymousexceptionalLie algebra;thatis,
Thisisasymmetricintegralmatrixwithdeterminantone,andsobytheMilnor–Whitehead classificationthereisasimplyconnected, 4-dimensionalPoincarécomplexwithintersection formrepresentedbythismatrix.Isthereaclosed 4-manifoldhomotopyequivalenttothis Poincarécomplex?
ByRochlin’stheorem[Roc52,Kir89],theintersectionformofasmooth,closed,spin 4-manifoldmusthavesignaturedivisibleby 16.Since E8 correspondstoanevenintersectionform,hassignature 8,andanysimplyconnected 4-manifoldwithevenintersectionform isspin,therecannotbeanysmooth,closed,simplyconnected 4-manifoldwith E8 asits intersectionform.Nevertheless,thequestionremained:isthereatopological,closed,simply connected 4-manifoldwith E8 asitsintersectionform?Thiswasanintractablequestionin the1970s(refertoSection1.6fortheanswer).
InordertobypasstheobstructionfromRochlin’stheorem,letusconsiderthematrix E8 ⊕ E8,whichhassignature 16.Thefollowingisastrategyforconstructingasmooth, closed,simplyconnected 4-manifoldwith E8 ⊕ E8 asitsintersectionform.Startwiththe simplyconnected 4-manifold K knownasthe K3surface,givenbythesolutionsetforthe quartic x4 + y4 + z4 + w4 =0 in CP3.Itsintersectionformisrepresentedbythematrix
8 ⊕ E8 ⊕ H ⊕ H ⊕ H,
where H = ( 01 10 ) isthehyperbolicmatrixcorrespondingtotheintersectionformof S2 × S2 and ⊕ denotesthejuxtapositionofblocksdownthediagonal.
Wehavetheobviousalgebraicprojection
Wewouldsucceedinconstructingthedesiredmanifoldifthisalgebraicprojectionwere realizedgeometrically.Thatis,wewishtoperformsurgeryon K withtheeffectofremoving thethreehyperbolicpairsfromtheintersectionform,resultinginaclosed 4-manifoldwith intersectionform E8 ⊕ E8.Letusattempttodothisinthesmoothcategory,andseewhere andwhywefail.
Since K issmoothandsimplyconnected,weknowbytheHurewicztheoremthat theelementsof H2(K; Z) correspondingtothehyperbolicpairsintheintersectionform canberepresentedbymaps S2 → K,whichwecantaketobesmoothimmersionsin generalposition.Henceforth,immersionswillbeassumedwithoutfurthercommenttobe ingeneralposition.AsinglehyperbolicpairisshownschematicallyontheleftofFigure1.1. Accordingtothematrix H,thetwospheresintersecteachotheralgebraicallyonce,but ingeneraltherewillbeexcessintersectionpointsgeometrically.Additionally,thespheres mayonlybeassumedtobeimmersed,withalgebraicallyzeroself-intersections.Ofcourse, thespherescorrespondingtodifferenthyperbolicpairsmighthavealgebraicallytrivialbut geometricallynontrivialintersectionsaswell,butweignorethosefornow.Ifthehyperbolic paircouldberepresentedbyframed,embeddedsphereswhichintersectexactlyonce,such asontherightofFigure1.1,wecoulddosurgeryoneitherofthetwospheresbycutting outaregularneighbourhood(diffeomorphicto S2 × D2)andreplacingitwith D3 × S1 , withtheeffectofremovingthecorrespondinghyperbolicmatrixfromtheintersectionform. Wesaythattwospheresinanambient 4-manifoldare geometricallydual iftheyintersectata singlepoint.Theexistenceofthesecondsphere,geometricallydualtothefirst,ensuresthat thissurgerywouldnotchangethefundamentalgroupoftheambientmanifold.Forthisthe secondspheredoesnotneedtobeembedded.Thesituationisentirelysymmetric:wecould dothesurgeryonanembeddinghomotopictothesecondsphere,withthesameeffecton homologyandthefundamentalgroup.
Thisstrategyisanalogoustotheideabehindtheclassificationofclosed,orientable 2-manifolds,inwhichwereducethegenusofanygivensurfacebyidentifyingadualpair ofsimpleclosedcurvesingivenhomologyclasses,cuttingoutanannularneighbourhood ofoneofthem,andfillinginthetworesultingboundarycomponentswithdiscs;the classificationcountsthenumberofsuchmovesneededtoproduceasphere.Theobstruction
Figure1.1 Tryingtosurgerahyperbolicpair.Left:Immersedspheres,depictedschematically, whichintersecteachotheralgebraicallyoncebutgeometricallythrice.Right:Thedesiredsituation, wherewehaveembeddedsphereswhichintersectgeometricallyonce.
tocarryingoutthisstrategyindimensionfourliesingeometricallyrealizingthealgebraic intersectionnumber,passing,asitwere,fromthelefttotherightofFigure1.1.Inthe smoothcategory,Donaldson’sdiagonalizationtheorem[Don83](Section21.2.2)implies thatthisisarealobstruction,sinceitshowsthereisnosmooth,closed,simplyconnected 4-manifoldwithintersectionform E8 ⊕ E8.Sowehaveseenwhyanaïveattempttodo surgeryfails.
Forsurgeryonnon-simplyconnectedmanifolds,oneseekstoremovehyperbolicsummandsintheequivariantintersectionformon H2(M ),thesecondhomologyoftheuniversalcoverofaclosedmanifold M ,thoughtofasamoduleoverthegroupringZ[π1(M )]. Inthiscontext,intersectioncountsarealgebraicallytrivialiftheyaretrivialoverZ[π1(M )]. Theprincipleinsuchasituationisstillthesame,namelywewishtorepresentthisalgebraic situationgeometrically.
1.1.3AttemptingtoProvethe s-cobordismTheorem
Asimilarproblemwithdisjointlyembedding2-spheresoccurswhenwetrytoprovethe s-cobordismtheoremfor 5-dimensionalcobordismsbetween 4-manifolds.Letustrytoimitatetheproofofthehigh-dimensionalsmooth s-cobordismtheorem,andseewhatobstructs thestrategyfromsucceeding.Let N beasmooth,compact s-cobordismbetweentwo closed 4-manifolds M0 and M1;thatis, ∂N = M0 ⊔ M1,eachinclusion Mi → N isa homotopyequivalence,andtheWhiteheadtorsion τ (N,M0) istrivial.Considerarelative handledecompositionof N builton M0 × [0, 1].SincetheWhiteheadtorsionvanishes,the relativechaincomplexoffinitelygenerated,free Z[π1(N )]-modulesforthepair (N,M0) canbesimplifiedalgebraicallysothatthereareonly 2-chainsand 3-chainsandtheboundary mapbetweenthemisanisomorphismrepresentedbytheidentitymatrixinsuitablebases (thismightalsorequiresomepreliminarystabilizationinthecaseofnontrivialfundamental groups).Asbefore,wewouldliketorepresentthisalgebraicsituationgeometrically.
Wefindsomeinitialsuccess:since N isconnected,wemayassumethereareno 0-handlesor 5-handles,andsince N hasdimensionfiveandisan h-cobordism,astandard procedurecalled handletrading allowsustotrade 1-handlesfor 3-handles,and 4-handles for 2-handles(seetheproofofTheorem20.1).Thusweseethat N isbuiltfrom M0 × [0, 1] byattachingonly 2-handlesand 3-handles,inthatorder.Since N isan s-cobordism,we arrangebyhandleslides—possiblyafterstabilizationbyaddingcancelling 2-and 3-handle pairs—thatthe 2-handlesand 3-handlesoccurinalgebraicallycancellingpairs.Let M1/2 denotethe 4-manifoldobtainedbyattachingthe 2-handlesto M0 ×{1}⊆ M0 × [0, 1].By turningthe 3-handlesof N upsidedown,weseethat M1/2 isalsoobtainedbyattaching 2handlesto M1 ×{1}⊆ M1 × [0, 1].Inotherwords, M1/2 canbeobtainedfromeither M0 or M1 byasequenceofsurgeriesonembeddedcircles.Sincetheinclusionof M0 in N inducesanisomorphismonfundamentalgroups,theattachingcirclesforthe 2-handles arenull-homotopicin M0.Similarly,theattachingcirclesin M1 arealsonull-homotopic in M1.Indimensionfour,homotopyimpliesisotopyforloops,andsothesurgeriesare performedonstandardtrivialcircles.Thisproduceseither S2 × S2 or S2×S2 summands in M1/2 [Wal99,Lemma5.5].
Thebeltspheres {0}× S2 ⊆ D2 × D3 ofthe 2-handlesformapairwisedisjointcollectionofframed,embedded 2-spheresin M1/2.Eachofthesesphereshasanembedded, geometricallydualspherecomingfrompushingthecoreofthecorresponding 2-handle unionanullhomotopyoftheattachingcircleinto M1/2.Thelatternullhomotopyprovides anembeddeddisc,sincetheattachingcircleistrivial.Iftheframingoftheattachmentissuch thatwegetan S2×S2 summand,thenthisdualsphereneednotbeframed.Similarly,when weturnthehandlesupsidedown,theattachingcirclesofthe 3-handlesattachedto M1/2 becomethebeltspheresfor 2-handlesattachedto M1.Bythesamereasoningasabove, theattachingspheresfor 3-handlesin M1/2 formapairwisedisjointcollectionofframed, embeddedspheresin M1/2 equippedwithembedded,geometricallydualspheres,which againneednotbeframed.
Recallthatwehavearrangedthateachbeltsphereofa 2-handleintersectstheattaching sphereofthecorresponding 3-handlealgebraicallyonce.However,theymayintersect multipletimesgeometrically.Aschematicpictureforasinglepairofa 2-handlebeltsphere and 3-handleattachingsphereisshownontheleftofFigure1.2,where,asbefore,we ignorepossibleinteractionswithotherpairs.Ifthe 3-handleattachingspherescouldbe isotopedin M1/2 toachievethesituationontherightofthefigureforeachpair,then thecorresponding 2-and 3-handlescouldbecancelled.Sincecancellingalltherelative handlesofthecobordism (N,M0) yieldstheproduct M0 × [0, 1],theproofwouldbe complete.Howeversuchanisotopyisingeneralnotpossibleinthesmoothcategory: Donaldson[Don87a](Section21.2.2)showedthereare h-cobordant,smooth,closed, simplyconnected 4-manifoldsthatarenotdiffeomorphic.Sowehaveseenwhyimitating theproofofthehigh-dimensional s-cobordismtheoremdoesnotsucceed.
Insummary,akeyinputneededinsurgeryaswellasintheproofofthe s-cobordism theoremistheabilitytoremovepairsofalgebraicallycancellingintersectionpointsbetween spheres,andthencegeometricallyrealizealgebraicintersectionnumbers.Asmentioned above,thisisingeneralnotpossiblesmoothly,butfortopological 4-manifoldshope remains.WediscussthesurgeryproblemfurtherinSection1.3.1,andwereturntoa discussionofthe s-cobordismtheoreminSection1.3.2.
Figure1.2 Analgebraicallydualpairconsistingofa 2-handlebeltsphere(red)anda 3-handle attachingsphere(blue)isshown.Thelightcurvesdenotethecorrespondinggeometricallydual spheres.Left:Thebeltsphereandtheattachingsphereintersectalgebraicallyoncebutgeometrically thrice.Right:Thedesiredsituationwherethebeltsphereandattachingsphereintersectgeometricallyonce.
1.2TheWhitneyMoveinDimensionFour
Consideramapofsmooth,orientedmanifolds Xd → Y 2d.Ingeneralposition,theonly singularpointsareisolated,signed,transversedoublepoints.Byinsertinglocalkinks(see Figure1.3forasketch),wecanarrangethatthesumofthesignsoftheself-intersection pointsiszero.Inthecaseofexactlytwoself-intersectionpointsofoppositesign,the situationislikeintheleftofFigure1.4,withtwoarcsintheimageof X joiningthetwo self-intersectionpointsondifferentsheets.Thecirclevisibleinthepicture,consistingof twoarcsjoiningthetwointersectionpoints,iscalleda Whitneycircle.Adiscbounded byaWhitneycircleiscalleda Whitneydisc.SupposethattheWhitneycircleboundsan embeddedWhitneydisc, W ,whoseinteriorliesintheexterioroftheimageof X in Y Underaconditiononthenormalbundleof W in Y describedinthenextparagraph,wecan pushonesheetof X along W andovertheothersheet,asindicatedinFigure1.4,which geometricallycancelsthetwoalgebraicallycancellingintersectionpoints.Thisprocessis calledthe Whitneytrick orthe Whitneymove [Whi44].
For dim X = d ≥ 3,theWhitneymoveturnsouttobesurprisinglysimple.Ifthe Whitneycircleisnull-homotopicin Y ,thenbygeneralpositionwecanassumeitbounds anembeddedWhitneydisc W whoseinteriorisdisjointfromtheimageof X.Anydisc D withboundaryacircle C pairingself-intersectionpointsintheimageof X determines a (d 1)-dimensionalsub-bundleofthenormalbundle νD⊆Y |C of D restrictedto C, byrequiringthatthesub-bundlebenormaltoonesheetoftheimageof X andtangent totheothersheet.InordertoperformtheWhitneymove,weneedthissub-bundleover thecircle C toextendovertheentiredisc D.Standardbundletheoryimpliesthatthe
Figure1.3 Adjustingthealgebraicself-intersectionnumberofanimmersedsubmanifoldby addinglocalkinks. + W
Figure1.4 TheWhitneymove.Left:AWhitneydisc W isshowninblue.Right:TheWhitney moveacross W removestwointersectionpoints.
