Equivariant stable homotopy theory and the Kervaire invariant problem

880 Pages·2021·6.489 MB·English

by  Michael Anthony Hill,  Michael J. Hopkins, Douglas C. Ravenel

EquivariantStableHomotopyTheoryandtheKervaireInvariantProblem
Thelong-standingKervaireinvariantprobleminhomotopytheoryarosefromgeometricanddiffer-
entialtopologyinthe1960sandwasquicklyrecognizedasoneofthemostimportantproblemsin
thefield.In2009,theauthorsofthisbookannouncedasolutiontotheproblem,whichwaspublished
towideacclaiminalandmarkAnnalsofMathematicspaper.
The proof is long and involved, using many sophisticated tools of modern (equivariant) stable
homotopy theory that are unfamiliar to nonexperts. This book presents the proof together with a
fulldevelopmentofallthebackgroundmaterialtomakeitaccessibletoagraduatestudentwithan
elementaryknowledgeofalgebraictopology.Therearedozensofexplicitexamplesofconstructions
andconceptsusedinsolvingtheproblem,aswellasdetailedaccountsofallthecomputationsneeded
alongtheway.
Also featuring a motivating history of the problem and numerous conceptual and expository
improvementsontheproof,thisisthedefinitiveaccountoftheresolutionoftheKervaireinvariant
problem.
Michael A. Hill isProfessorattheUniversityofCalifornia,LosAngeles.Heistheauthorof
severalpapersonalgebraictopologyandisaneditorforjournalsincludingMathematischeZeitschrift
andTransactionsoftheAmericanMathematicalSociety.
Michael J. Hopkins isProfessoratHarvardUniversity.Hisresearchconcentratesonalge-
braictopology.In2001,hewasawardedtheOswaldVeblenPrizeinGeometryfromtheAMSfor
hisworkinhomotopytheory,followedbytheNASAwardinMathematicsin2012andtheNemmers
PrizeinMathematicsin2014.
Douglas C. Ravenel istheFayerweatherProfessorofMathematicsattheUniversityof
Rochester. He is the author of two influential previous books in homotopy theory and roughly 75
journalarticlesonstablehomotopytheoryandalgebraictopology.
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22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry
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27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetricMeasureSpaces
28. Y.-G.OhSymplecticTopologyandFloerHomologyI
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30. A.BobrowskiConvergenceofOne-ParameterOperatorSemigroups
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32. J.-H.EvertseandK.Gyo˝ryDiscriminantEquationsinDiophantineNumberTheory
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37. A.Defant,D.Garca,M.MaestreandP.Sevilla-PerisDirichletSeriesandHolomorphicFunctions
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38. N.Th.VaropoulosPotentialTheoryandGeometryonLieGroups
39. D.ArnalandB.CurreyRepresentationsofSolvableLieGroups
Equivariant Stable Homotopy Theory and the
Kervaire Invariant Problem
MICHAEL A. HILL
UniversityofCalifornia,LosAngeles
MICHAEL J. HOPKINS
HarvardUniversity
DOUGLAS C. RAVENEL
UniversityofRochester
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DOI:10.1017/9781108917278
©MichaelA.Hill,MichaelJ.HopkinsandDouglasC.Ravenel2021
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Firstpublished2021
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Names:Hill,MichaelA.(MichaelAnthony),author.|
Hopkins,MichaelJ.,1958–author.|Ravenel,DouglasC.,author.
Title:EquivariantstablehomotopytheoryandtheKervaireinvariantproblem/
MichaelA.Hill,UniversityofCalifornia,LosAngeles,MichaelJ.Hopkins,
HarvardUniversity,DouglasC.Ravenel,UniversityofRochester,NewYork.
Description:Cambridge,UK;NewYork,NY:CambridgeUniversityPress,2021.|
Series:Newmathematicalmonographs|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020041986(print)|LCCN2020041987(ebook)|
ISBN9781108831444(hardback)|ISBN9781108932943(paperback)|
ISBN9781108917278(epub)
Subjects:LCSH:Homotopytheory.
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Contents
1 Introduction page1
1.1 TheKervaireInvariantTheoremandtheIngredientsofItsProof 1
1.2 BackgroundandHistory 6
1.3 TheFoundationalMaterialinThisBook 17
1.4 HighlightsofLaterChapters 24
1.5 Acknowledgments 48
PARTONE THECATEGORICALTOOLBOX 49
2 SomeCategoricalTools 51
2.1 BasicDefinitionsandNotationalConventions 53
2.2 NaturalTransformations,AdjointFunctorsandMonads 72
2.3 LimitsandColimitsasAdjointFunctors 91
2.4 EndsandCoends 119
2.5 KanExtensions 129
2.6 MonoidalandSymmetricMonoidalCategories 134
2.7 2-CategoriesandBeyond 156
2.8 GrothendieckFibrationsandOpfibrations 161
2.9 IndexedMonoidalProducts 164
3 EnrichedCategoryTheory 190
3.1 BasicDefinitions 191
3.2 Limits,Colimits,EndsandCoendsinEnrichedCategories 209
3.3 TheDayConvolution 221
3.4 SimplicialSetsandSimplicialSpaces 228
3.5 The Homotopy Extension Property, h-Cofibrations and
NondegenerateBasePoints 234
4 Quillen’sTheoryofModelCategories 244
4.1 BasicDefinitions 246
4.2 ThreeClassicalExamplesofModelCategories 256
4.3 HomotopyinaModelCategory 263
vii
viii Contents
4.4 NonhomotopicalandDerivedFunctors 269
4.5 QuillenFunctorsandQuillenEquivalences 272
4.6 TheSuspensionandLoopFunctors 278
4.7 FiberandCofiberSequences 284
4.8 TheSmallObjectArgument 288
5 ModelCategoryTheorysinceQuillen 294
5.1 HomotopicalCategories 299
5.2 CofibrantlyandCompactlyGeneratedModelCategories 308
5.3 ProperModelCategories 320
5.4 TheCategoryofFunctorsfromaSmallCategorytoaCofibrantly
GeneratedModelCategory 321
5.5 MonoidalModelCategories 335
5.6 EnrichedModelCategories 350
5.7 StableandExactlyStableModelCategories 366
5.8 HomotopyLimitsandColimits 371
6 BousfieldLocalization 391
6.1 It’sAllaboutFibrantReplacement 393
6.2 BousfieldLocalizationinMoreGeneralModelCategories 395
6.3 WhenIsLeftBousfieldLocalizationPossible? 403
PARTTWO SETTINGUPEQUIVARIANTSTABLE
HOMOTOPYTHEORY 409
7 SpectraandStableHomotopyTheory 411
7.1 Hovey’sGeneralizationofSpectra 422
7.2 TheFunctorialApproachtoSpectra 432
7.3 StabilizationandModelStructuresforHoveySpectra 453
7.4 StabilizationandModelStructuresforSmashableSpectra 474
8 EquivariantHomotopyTheory 495
8.1 FiniteG-SetsandtheBurnsideRingofaFiniteGroup 504
8.2 MackeyFunctors 510
8.3 SomeFormalPropertiesofG-Spaces 519
8.4 G-CWComplexes 528
8.5 TheHomologyofaG-CWComplex 531
8.6 ModelStructures 538
8.7 SomeUniversalSpaces 547
8.8 Elmendorf’sTheorem 549
8.9 OrthogonalRepresentationsofGandRelatedStructures 551
9 OrthogonalG-Spectra 566
9.1 CategoricalPropertiesofOrthogonalG-Spectra 568
9.2 ModelStructuresforOrthogonalG-Spectra 584
Contents ix
9.3 NaiveandGenuineG-Spectra 590
9.4 HomotopicalPropertiesofG-Spectra 599
9.5 AHomotopicalApproximationtotheCategoryofG-Spectra 608
9.6 Homotopical Properties of Indexed Wedges and Indexed Smash
Products 615
9.7 TheNormFunctor 617
9.8 ChangeofGroupandSmashProduct 622
9.9 TheRO(G)-GradedHomotopyofHZ 624
9.10 FixedPointSpectra 639
9.11 GeometricFixedPoints 643
10 MultiplicativePropertiesofG-Spectra 663
10.1 EquivariantT-Diagrams 665
10.2 IndexedSmashProductsandCofibrations 666
10.3 TheArrowCategoryandIndexedCornerMaps 670
10.4 IndexedSmashProductsandTrivialCofibrations 671
10.5 IndexedSymmetricPowers 674
10.6 IteratedIndexedSymmetricPowers 684
10.7 CommutativeAlgebrasintheCategoryofG-Spectra 687
10.8 R-ModulesintheCategoryofSpectra 690
10.9 IndexedSmashProductsofCommutativeRings 694
10.10 TwistedMonoidRings 703
PARTTHREE PROVINGTHEKERVAIREINVARIANTTHEOREM 711
11 TheSliceFiltrationandSliceSpectralSequence 713
11.1 TheFiltrationbehindtheSpectralSequence 714
11.2 TheSliceSpectralSequence 731
11.3 SphericalSlices 739
11.4 TheSliceTower,SymmetricPowersandtheNorm 744
12 TheConstructionandPropertiesofMU 750
R
12.1 RealandComplexSpectra 752
12.2 TheRealBordismSpectrum 761
12.3 AlgebraGeneratorsforπuMU((G)) 772
˚
12.4 TheSliceStructureofMU((G)) 777
13 TheProofsoftheGap,PeriodicityandDetectionTheorems 793
13.1 AWarm-Up:TheSliceSpectralSequenceforMU 794
R
13.2 TheGapTheorem 799
13.3 ThePeriodicityTheorem 801
13.4 TheDetectionTheorem 817
References 830
TableofNotations 842
Index 856