Foundations of stable homotopy theory PDF
Preview Foundations of stable homotopy theory
CAMBRIDGESTUDIESINADVANCEDMATHEMATICS185 EditorialBoard B.BOLLOBA´S,W.FULTON,F.KIRWAN, P.SARNAK,B.SIMON,B.TOTARO FOUNDATIONSOFSTABLEHOMOTOPYTHEORY Thebeginninggraduatestudentinhomotopytheoryisconfrontedwithavastliterature onspectrathatisscatteredacrossbooks,articlesanddecades.Thereismuchfolklore, but very few easy entry points. This comprehensive introduction to stable homotopy theorychangesthat.Itpresentsthefoundationsofthesubjecttogetherinoneplacefor thefirsttime,fromthemotivatingphenomenatothemoderntheory,atalevelsuitable forthosewithonlyafirstcourseinalgebraictopology. Startingfromstablehomotopygroupsand(co)homologytheories,theauthorsstudy the most important categories of spectra and the stable homotopy category, before movingontocomputationalaspectsandmoreadvancedtopicssuchasmonoidalstruc- tures,localisationsandchromatichomotopytheory.Theappendixcontainingessential factsonmodelcategories,thenumerousexamplesandthesuggestionsforfurtherread- ingmakethisafriendlyintroductiontoanoftendauntingsubject. DavidBarnesisaSeniorLecturerinMathematicsatQueen’sUniversityBelfast.His workcentresonstablehomotopytheory,usuallywitheitheramonoidalorequivariant focus,oftenusingalgebratodescribethestructuresinquestion. Constanze Roitzheim is a Senior Lecturer in Mathematics at the University of Kent.Herworkfocusesonlocalisationsofthestablehomotopycategoryandrelated questionsinalgebra. CAMBRIDGESTUDIESINADVANCEDMATHEMATICS EditorialBoard B.Bolloba´s,W.Fulton,F.Kirwan,P.Sarnak,B.Simon,B.Totaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress. Foracompleteserieslisting,visitwww.cambridge.org/mathematics. AlreadyPublished 148 S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontoModelSpacesandTheirOperators 149 C.Godsil&K.MeagherErdo˝s–Ko–RadoTheorems:AlgebraicApproaches 150 P.MattilaFourierAnalysisandHausdorffDimension 151 M.Viana&K.OliveiraFoundationsofErgodicTheory 152 V.I.Paulsen&M.RaghupathiAnIntroductiontotheTheoryofReproducingKernelHilbertSpaces 153 R.Beals&R.WongSpecialFunctionsandOrthogonalPolynomials 154 V.JurdjevicOptimalControlandGeometry:IntegrableSystems 155 G.PisierMartingalesinBanachSpaces 156 C.T.C.WallDifferentialTopology 157 J.C.Robinson,J.L.Rodrigo&W.SadowskiTheThree-DimensionalNavier–StokesEquations 158 D.HuybrechtsLecturesonK3Surfaces 159 H.Matsumoto&S.TaniguchiStochasticAnalysis 160 A.Borodin&G.OlshanskiRepresentationsoftheInfiniteSymmetricGroup 161 P.WebbFiniteGroupRepresentationsforthePureMathematician 162 C.J.Bishop&Y.PeresFractalsinProbabilityandAnalysis 163 A.BovierGaussianProcessesonTrees 164 P.SchneiderGaloisRepresentationsand(ϕ,Γ)-Modules 165 P.Gille&T.SzamuelyCentralSimpleAlgebrasandGaloisCohomology(2ndEdition) 166 D.Li&H.QueffelecIntroductiontoBanachSpaces,I 167 D.Li&H.QueffelecIntroductiontoBanachSpaces,II 168 J.Carlson,S.Mu¨ller-Stach&C.PetersPeriodMappingsandPeriodDomains(2ndEdition) 169 J.M.LandsbergGeometryandComplexityTheory 170 J.S.MilneAlgebraicGroups 171 J.Gough&J.KupschQuantumFieldsandProcesses 172 T.Ceccherini-Silberstein,F.Scarabotti&F.TolliDiscreteHarmonicAnalysis 173 P.GarrettModernAnalysisofAutomorphicFormsbyExample,I 174 P.GarrettModernAnalysisofAutomorphicFormsbyExample,II 175 G.NavarroCharacterTheoryandtheMcKayConjecture 176 P.Fleig,H.P.A.Gustafsson,A.Kleinschmidt&D.PerssonEisensteinSeriesandAutomorphic Representations 177 E.PetersonFormalGeometryandBordismOperators 178 A.OgusLecturesonLogarithmicAlgebraicGeometry 179 N.NikolskiHardySpaces 180 D.-C.CisinskiHigherCategoriesandHomotopicalAlgebra 181 A.Agrachev,D.Barilari&U.BoscainAComprehensiveIntroductiontoSub-RiemannianGeometry 182 N.NikolskiToeplitzMatricesandOperators 183 A.YekutieliDerivedCategories 184 C.DemeterFourierRestriction,DecouplingandApplications 185 D.Barnes&C.RoitzheimFoundationsofStableHomotopyTheory 186 V.Vasyunin&A.VolbergTheBellmanFunctionTechniqueinHarmonicAnalysis 187 M.Geck&G.MalleTheCharacterTheoryofFiniteGroupsofLieType Foundations of Stable Homotopy Theory DAVIDBARNES Queen’sUniversityBelfast CONSTANZEROITZHEIM UniversityofKent UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 79AnsonRoad,#06-04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108482783 DOI:10.1017/9781108636575 (cid:2)c DavidBarnesandConstanzeRoitzheim2020 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2020 PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:Barnes,David,1981-author. Title:Foundationsofstablehomotopytheory/DavidBarnes,Queen’s UniversityBelfast,ConstanzeRoitzheim,UniversityofKent,Canterbury. Description:Cambridge,UnitedKingdom;NewYork,NY:Cambridge UniversityPress,2020.|Series:Cambridgestudiesinadvanced mathematics|Includesbibliographicalreferencesandindex. Identifiers:LCCN2019035713|ISBN9781108482783(hardback)|ISBN 9781108636575(ebook) Subjects:LCSH:Homotopytheory. Classification:LCCQA612.7.B3752020|DDC514/.24--dc23 LCrecordavailableathttps://lccn.loc.gov/2019035713 ISBN978-1-108-48278-3Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Introduction page1 1 BasicsofStableHomotopyTheory 7 1.1 StablePhenomena 7 1.2 TheSpanier–WhiteheadCategory 28 1.3 AFirstAttemptatSpectra 32 2 SequentialSpectraandtheStableHomotopyCategory 37 2.1 TheLevelwiseModelStructure 38 2.2 HomotopyGroupsofSpectra 47 2.3 TheStableModelStructure 57 2.4 ExplicitFibrantReplacement 67 2.5 TheSteenrodAlgebra 71 2.6 TheAdamsSpectralSequence 86 3 TheSuspensionandLoopFunctors 93 3.1 DefinitionoftheFunctors 94 3.2 StableModelCategories:AFirstLook 101 3.3 TheCoactionofaCofibre 105 3.4 DefinitionofFibreandCofibreSequences 112 3.5 ShiftingFibreandCofibreSequences 115 3.6 TheLongExactPuppeSequence 125 4 TriangulatedCategories 128 4.1 DefinitionandBasicProperties 128 4.2 TheHomotopyCategoryofaStableModelCategory 138 4.3 ComparisonofFibreandCofibreSequences 146 4.4 ConsequencesofStability 149 v vi Contents 4.5 ExactFunctorsandQuillenFunctors 152 4.6 TodaBrackets 155 4.7 Muro’sExoticTriangulatedCategory 161 5 ModernCategoriesofSpectra 166 5.1 TheStableHomotopyCategory–Revisited 167 5.2 OrthogonalSpectra 174 5.3 SymmetricSpectra 186 5.4 PropernessofSpectra 210 5.5 OtherCategoriesofSpectra 212 5.6 CompactObjects 222 5.7 RigidityofSpectra 231 6 MonoidalStructures 234 6.1 MonoidalModelCategories 234 6.2 ASmashProductontheStableHomotopyCategory 251 6.3 ClosedMonoidalStructuresonSpectra 253 6.4 MonoidalModelCategoriesofSpectra 269 6.5 Spanier–WhiteheadDuality 280 6.6 RingSpectraandModules 283 6.7 CommutativeRingSpectra 292 6.8 ApplicationsofMonoidality 296 6.9 HomotopyMappingObjectsandFramings 298 7 LeftBousfieldLocalisation 317 7.1 GeneralLocalisationTechniques 317 7.2 LocalisationofStableModelCategories 325 7.3 Localisation of Spectra with Respect to Homology Theories 336 7.4 ExamplesofLeftLocalisation 343 Appendix ModelCategories 376 References 411 Index 418 Introduction AimsandIntendedReadership The aim of this book is to be an accessible introduction to stable homotopy theory that novices, particularly graduate students, can use to learn the fun- damentals of the subject. For the experts, we hope to have provided a useful compendiumofresultsacrossthemainareasofstablehomotopytheory. Thisbookisnotintendedtoreplaceanyspecificpartoftheexistinglitera- ture, but instead to give a smoother, more coherent introduction to stable ho- motopytheory.Weusemoderntechniquestogiveastreamlineddevelopment that avoids a number of outdated, and often over-complicated, constructions ofasuitablestablehomotopycategory.Wecoverthemostpressingtopicsfor a novice and give a narrative to motivate the development. This narrative is missing from much of the current literature, which often assumes the reader alreadyknowsstablehomotopytheoryandhenceunderstandswhyanygiven definitionorresultisimportant. Themajorityofsectionshavebeenwrittento(hopefully)containalldetails requiredforagraduatestudent.Theremainingsectionsareintendedtogivean overviewofmorespecialisedoradvancedtopics,withreferencestothecentral texts for those areas. It is hoped that once the reader has read the chapters relevanttotheirresearch,theywillbewellpreparedtodiveintotherestofthe literatureandtoknowwhattoreadnext. Prerequisites Ratherthanrewritemanypagesonmodelcategories,categorytheoryandun- stablehomotopytheory,wedependuponseveralexcellent,andquitestandard, references.Assuch,thereadershouldknowafairamountofpoint-settopology andalgebraictopology.ThestandardtextsareGray[Gra75],Hatcher[Hat02], 1 2 Introduction May[May99a]andMayandPonto[MP12].Thereadershouldalsoknowthe basics of model categories. The best introductions are Dwyer and Spalin´ski [DS95]andthefirstchaptersofHovey[Hov99]. A certain amount of category theory is used throughout, the standard text is Mac Lane [Mac71]. For the chapters on the monoidal smash product, the reader will need some enriched category theory, easily obtained from Kelly [Kel05]. They may also like to have access to Borceaux [Bor94]. The chap- teronlocalisationsreferstoHirschhorn[Hir03]forsomeproofsandtechnical results,butthereaderwillnotneedtohavereadthebooktofollowthedevel- opment. AHistoricalNarrative Thebook[Jam99]givesatreatmentofthehistoryoftopology,whilethechap- terofMay[May99b](50pages)coversstablehomotopytheoryfrom1945to 1966.Sincethen,thepaceofdevelopmentandpublicationhasonlyquickened, athoroughhistoryofstablehomotopytheorywouldbeabookbyitself. A basic problem in homotopy theory is the calculation of the homotopy groups of spheres. This problem is well known to be hopelessly difficult, but certain patterns in the homotopy groups were noticed. The Freudenthal Sus- pensionTheoremgivesaclearstatementofamajorpattern:thegroupπ (Sk+n) n is independent of n for n > k+1. This and the suspension isomorphisms of homologyandcohomologywereastartingpointofstablehomotopytheory. CalculationscontinuedandtheSpanier–Whiteheadcategorywasdeveloped to study duality statements. It was also a useful category for the study of spacesunderequivalencesofstablehomotopygroups.However,theSpanier– Whiteheadcategoryhassomesubstantialdrawbacks,inparticular,itdoesnot containrepresentativesforallreducedcohomologytheories. Severalsolutionstothiswereconstructed,includingBoardman’sstableho- motopy category, Lima’s notion of spectra, Kan’s semisimplicial spectra and Whitehead’sdevelopmentsofthenotionsofspectra.Noneofthesecategories were entirely satisfactory, so we jump ahead to Adams’s construction of the stable homotopy category [Ada74], which was based on ideas of Boardman. This category contained the Spanier–Whitehead category, represented all co- homologytheoriesandhadacommutativesmashproduct. Havingagoodconstructionwithsensibleaxiomsallowedforfurtherdevel- opmentofstablehomotopytheory.Agoodnotionof“categoriesoffractions”, nowknownasBousfieldlocalisationsathomologytheories,greatlyimproved the ability to calculate stable homotopy groups via the Adams spectral se- quence.ThroughworkofBousfield,theselocalisationswerefurtherdeveloped Introduction 3 ([Bou75],[Bou79]),andvastamountsofcalculationswerenowpossible.This ledtotheintroductionofchromatichomotopytheory,whichgivesaframework for major structural results about the stable homotopy category (see [Rav84] and [Rav92a]), as well as techniques for even further calculations of stable homotopygroupsofspheres[Rav86]. The lack of a good commutative monoidal point-set model for the stable homotopy category still held the subject area back. Brown representability posited the existence of function spectra and allowed for some homotopical calculations,butdirectconstructionswereoftenimpossibletogive.Thestudy of(commutative)ringspectrauptohomotopywasdifficult–keepingtrackof the homotopies and their coherence was particularly burdensome. Moreover, constructionsuptohomotopypreventedgeometricconstructionssuchasbun- dlesofspectraordiagramsofspectra. Thedevelopmentofcoordinate–freespectrabyMayandothersofferedsev- eral improvements to the area. The use of operads to manage commutative multiplications up to homotopy allowed for serious study of derived algebra inspectra,theso-called“bravenewalgebra”,seetheworkofMay,Quinnand Ray[May77]. Coordinate–freespectraalsoledtothedevelopmentof“spectrification”func- tors,whichsimplifiedtheconstructionofmapsbetweenspectra.Thesefunc- torsplayedacentralroleinworkofLewis,MayandSteinberger[LMSM86], whichgaveaconstructionofG-equivariantspectraandtheG-equivariantsta- blehomotopycategoryforGacompactLiegroup. Whilethistechnologydidallowforausefuldefinitionofaninternalfunction objectforspectra,thesmashproductwasstillonlycommutativeandassocia- tive up to homotopy. The work of Lewis [Lew91] even suggested that there may be no commutative monoidal point-set model for the stable homotopy category,butthispessimismturnedouttobeunfounded. Twoindependentsolutionstotheproblemofcommutativesmashproducts came about in surprisingly quick succession: the S–modules of Elmendorff, Kriz,MandellandMay[EKMM97]andthesymmetricspectraofHovey,Ship- ley and Smith [HSS00]. These references gave closed symmetric monoidal modelcategoriesofspectraandmodelcategoriesof(commutative)ringspec- tra. By using model categories, one had point-set level smash products and function objects which would have the correct homotopical properties after passingtothehomotopycategories. Thisreinvigoratedtheareaandallowedforagreatdealoffurtherdevelop- mentin“bravenewalgebra”,namely,theimportingofstatementsfromalge- braintostablehomotopytheory.Forexample,Hovey,PalmieriandStrickland [HPS97]wereabletogiveanaxiomatisationofstablehomotopytheories.
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