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GlobalHomotopyTheory Equivariant homotopy theory started from geometrically motivated questions about symmetriesofmanifolds.Severalimportantequivariantphenomenaoccurnotjustfor a particular group, but in a uniform way for all groups. Prominent examples include stable homotopy, K-theory or bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., universal symmetries encoded by simultaneous and compatibleactionsofallcompactLiegroups. Thisbookintroducesgraduatestudentsandresearcherstoglobalequivarianthomo- topy theory. The framework is based on the new notion of global equivalences for orthogonalspectra,amuchfinernotionofequivalencethanistraditionallyconsidered. Thetreatmentislargelyself-containedandcontainsmanyexamples,makingitsuitable as a textbook for an advanced graduate class. At the same time, the book is a comprehensiveresearchmonographwithdetailedcalculationsthatrevealtheintrinsic beautyofglobalequivariantphenomena. Stefan Schwede is Professor in the Mathematical Institute at the University of Bonn. His main area of expertise is algebraic topology, specifically stable homotopy theory. NEW MATHEMATICAL MONOGRAPHS EditorialBoard Be´laBolloba´s,WilliamFulton,FrancesKirwan, PeterSarnak,BarrySimon,BurtTotaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity Press.Foracompleteserieslistingvisitwww.cambridge.org/mathematics. 1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups 2. J.B.GarnettandD.E.MarshallHarmonicMeasure 3. P.CohnFreeIdealRingsandLocalizationinGeneralRings 4. E.BombieriandW.GublerHeightsinDiophantineGeometry 5. Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns 6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures 7. A.ShlapentokhHilbert’sTenthProblem 8. G.MichlerTheoryofFiniteSimpleGroupsI 9. A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry 10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds 11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T) 12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory 13. M.GrandisDirectedAlgebraicTopology 14. G.MichlerTheoryofFiniteSimpleGroupsII 15. R.SchertzComplexMultiplication 16. S.BlochLecturesonAlgebraicCycles(2ndEdition) 17. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups 18. T.DownarowiczEntropyinDynamicalSystems 19. C.SimpsonHomotopyTheoryofHigherCategories 20. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI 21. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII 22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory 23. J.SńiatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry 24. E.RiehlCategoricalHomotopyTheory 25. B.A.MunsonandI.VolicĆubicalHomotopyTheory 26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition) 27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetric MeasureSpaces 28. Y.-G.OhSymplecticTopologyandFloerHomologyI 29. Y.-G.OhSymplecticTopologyandFloerHomologyII 30. A.BobrowskiConvergenceofOne-ParameterOperatorSemigroups 31. K.CostelloandO.GwilliamFactorizationAlgebrasinQuantumFieldTheoryI 32. J.-H.EvertseandK.Gyo¨ryDiscriminantEquationsinDiophantineNumberTheory Global Homotopy Theory STEFAN SCHWEDE RheinischeFriedrich-Wilhelms-Universita¨tBonn 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/9781108425810 DOI:10.1017/9781108349161 ©StefanSchwede2018 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2018 PrintedandboundinGreatBritainbyClaysLtd,ElcografS.p.A. AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-1-108-42581-0Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Preface pagevii 1 Unstableglobalhomotopytheory 1 1.1 Orthogonalspacesandglobalequivalences 2 1.2 Globalmodelstructurefororthogonalspaces 31 1.3 Monoidalstructures 54 1.4 Globalfamilies 64 1.5 Equivarianthomotopysets 77 2 Ultra-commutativemonoids 92 2.1 Globalmodelstructure 94 2.2 Globalpowermonoids 110 2.3 Examples 133 2.4 GlobalformsofBO 156 2.5 Globalgroupcompletionandunits 186 3 Equivariantstablehomotopytheory 227 3.1 Equivariantorthogonalspectra 228 3.2 TheWirthmu¨llerisomorphismandtransfers 261 3.3 Geometricfixedpoints 288 3.4 Thedoublecosetformula 305 3.5 Products 332 4 Globalstablehomotopytheory 348 4.1 Orthogonalspectraasglobalhomotopytypes 349 4.2 Globalfunctors 367 4.3 Globalmodelstructuresfororthogonalspectra 385 4.4 Triangulatedglobalstablehomotopycategories 409 4.5 Changeoffamilies 430 5 Ultra-commutativeringspectra 461 5.1 Poweroperations 462 v vi Contents 5.2 Comonadicdescriptionofglobalpowerfunctors 481 5.3 Examples 506 5.4 Globalmodelstructure 524 6 GlobalThomandK-theoryspectra 543 6.1 GlobalThomspectra 544 6.2 Equivariantbordism 584 6.3 ConnectiveglobalK-theory 626 6.4 PeriodicglobalK-theory 661 AppendixA Compactlygeneratedspaces 687 AppendixB Equivariantspaces 735 AppendixC Enrichedfunctorcategories 793 References 809 Indexofsymbols 819 Index 822 Preface Equivariant stable homotopy theory has a long tradition, starting from geometrically motivated questions about symmetries of manifolds. The homotopy-theoretic foundations of the subject were laid by tom Dieck, Se- gal and May and their students and collaborators in the 1970s, and over the intervening decades equivariant stable homotopy theory has been very use- fulforsolvingcomputationalandconceptualproblemsinalgebraictopology, geometrictopologyandalgebraicK-theory.Variousimportantequivariantthe- ories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples of this are equivariant sta- blehomotopy,equivariantK-theoryorequivariantbordism.Globalequivariant homotopytheorystudiessuchuniformphenomena,i.e.,theadjective‘global’ referstosimultaneousandcompatibleactionsofallcompactLiegroups. This book introduces a new context for global homotopy theory. Various ways of providing a home for global stable homotopy types have previously beenexploredin[100,Ch.II],[68,Sec.5],[18]and[19].Weuseadifferentap- proach:weworkwiththewell-knowncategoryoforthogonalspectra,butuse anotionofequivalence,theglobalequivalence,whichismuchfinerthanwhat istraditionallyconsidered.Thebasicunderlyingobservationisthatanorthog- onal spectrum gives rise to an orthogonalG-spectrum for every compact Lie groupG, and the fact that all these individual equivariant objects come from oneorthogonalspectrumimplicitlyencodesstrongcompatibilityconditionsas thegroupGvaries.AnorthogonalspectrumthushasG-equivarianthomotopy groupsforeverycompactLiegroup,andaglobalequivalenceisamorphism oforthogonalspectrathatinducesisomorphismsforallequivarianthomotopy groupsforallcompactLiegroups(basedon‘completeG-universes’,compare Definition4.1.3). The structure of the equivariant homotopy groups of an orthogonal spectrum gives an idea of the information encoded in a global homotopy type in our sense: the equivariant homotopy groups πG(X) are contravariantly func- k vii viii Preface torialforcontinuousgrouphomomorphisms(‘restrictionmaps’),andtheyare covariantlyfunctorialforinclusionsofclosedsubgroups(‘transfermaps’).The restrictionandtransfermapsenjoyvarioustransitivitypropertiesandinteract via a double coset formula. This kind of algebraic structure has been studied before under different names, e.g., ‘global Mackey functor’, ‘inflation functor’, .... From a purely algebraic perspective, there are various parameters herethatonecanvary,namelytheclassofgroupstowhichavalueisassigned andtheclassesofhomomorphismstowhichrestrictionmapsortransfermaps areassigned,andlotsofvariationshavebeenexplored.However,thedecision toworkwithorthogonalspectraandequivarianthomotopygroupsoncomplete universesdictatesacanonicalchoice:weproveinTheorem4.2.6thatthealge- braofnaturaloperationsbetweentheequivarianthomotopygroupsoforthog- onal spectra is freely generated by restriction maps along continuous group homomorphisms and transfer maps along closed subgroup inclusion, subject toexplicitlyunderstoodrelations. WedefinetheglobalstablehomotopycategoryGH bylocalizingthecate- gory of orthogonal spectra at the class of global equivalences. Every global equivalenceisinparticularanon-equivariantstableequivalence,sothereisa ‘forgetful’ functor U: GH −→ SH on localizations, where SH denotes the traditional non-equivariant stable homotopy category. By Theorem 4.5.1 this forgetfulfunctorhasaleftadjoint LandarightadjointR,bothfullyfaithful, whichparticipateinarecollementoftriangulatedcategories: (cid:4)(cid:4) i∗ (cid:4)(cid:4) L GH+ i∗ (cid:2)(cid:2)GH U (cid:2)(cid:2)SH . (cid:3)(cid:3) (cid:3)(cid:3) i! R HereGH+denotesthefullsubcategoryoftheglobalstablehomotopycategory spannedbytheorthogonalspectrathatarestablycontractibleinthetraditional, non-equivariantsense. Theglobalspherespectrumandsuspensionspectraareintheimageofthe left adjoint (Example 4.5.11). Global Borel cohomology theories are the im- age of the right adjoint (Example 4.5.19). The ‘natural’ global versions of Eilenberg–MacLanespectra(Construction5.3.8),Thomspectra(Section6.1), ortopologicalK-theoryspectra(Sections6.3and6.4)arenotintheimageof eitherofthetwoadjoints.PeriodicglobalK-theory,however,isrightinduced fromfinitecyclicgroups,i.e.,intheimageoftheanalogousrightadjointfrom anintermediateglobalhomotopycategoryGH basedonfinitecyclicgroups cyc (Example6.4.27). Looking at orthogonal spectra through the eyes of global equivalences is like using a prism: the latter breaks up white light into a spectrum of colors, Preface ix andglobalequivalencessplitatraditional,non-equivarianthomotopytypeinto manydifferentglobalhomotopytypes.Thefirstexampleofthisphenomenon that we will encounter refines the classifying space of a compact Lie group G.Ontheonehand,thereistheconstantorthogonalspacewithvalueanon- equivariantmodelfor BG;andthereistheglobalclassifyingspace B G (see gl Definition1.1.27).Theglobalclassifyingspaceisanalogoustothegeometric classifyingspaceofalinearalgebraicgroupinmotivichomotopytheory[123, 4.2],anditisthecounterparttothestackofG-principalbundlesintheworld ofstacks. Anothergoodexample isthesplittingupofthenon-equivariant homotopy type of the classifying space of the infinite orthogonal group. Again there is the constant orthogonal space with value BO, the Grassmannian model BO (Example 2.4.1), a different Grassmannian model bO (Example 2.4.18), the bar construction model B◦O (Example 2.4.14), and finally a certain ‘cofree’ orthogonalspaceR(BO).TheorthogonalspacebOisalsoahomotopycolimit, asngoestoinfinity,oftheglobalclassifyingspacesB O(n).Wediscussthese gl differentglobalformsof BOinsomedetailinSection2.4,andtheassociated ThomspectrainSection6.1. In the stable global world, every non-equivariant homotopy type has two extremeglobalrefinements,the‘leftinduced’(theglobalanalogofaconstant orthogonalspace,seeExample4.5.10)andthe‘rightinduced’globalhomotopy type (representing Borel cohomology theories, see Example 4.5.19). Many important stable homotopy types have other natural global forms. The non- equivariant Eilenberg–MacLane spectrum of the integers has a ‘free abelian groupfunctor’model(Construction5.3.8),andanotherincarnationistheEilen- berg–MacLanespectrumoftheconstantglobalfunctorwithvalueZ(Remark 4.4.12).ThesetwoglobalrefinementsoftheintegralEilenberg–MacLanespec- trum agree on finite groups, but differ for compact Lie groups of positive di- mensions. Asalreadyindicated,thereisagreatvarietyoforthogonalThomspectra,in real(orunoriented)flavorsasmOandMO,ascomplex(orunitary)versions mU and MU, and there are periodic versions mOP, MOP, mUP and MUP ofthese;wediscussthesespectrainSection6.1.Thetheoriesrepresentedby mO and mU have the closest ties to geometry; for example, the equivariant homotopy groups of mO receive Thom–Pontryagin maps from equivariant bordismrings,andtheseareisomorphismsforproductsoffinitegroupsandtori (compareTheorem6.2.33).ThetheoriesrepresentedbyMOaretomDieck’s homotopicalequivariantbordism,isomorphicto‘stableequivariantbordism’. Connective topological K-theory also has two fairly natural global refine- ments, in addition to the left and right induced ones. The ‘orthogonal sub- space’ model ku (Construction 6.3.9) represents connective equivariant