More Concise Algebraic Topology

542 Pages·2011·2.39 MB·English

by  J. P. May, K. Ponto

More Concise Algebraic Topology
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chicago lectures in mathematics series
Editors:SpencerJ.Bloch,PeterConstantin,BensonFarb,
NormanR.Lebovitz,CarlosKenig,andJ.P.May
OtherChicagoLecturesinMathematicstitlesavailablefromtheUniversityofChicagoPress
SimplicalObjectsinAlgebraicTopology,byJ.PeterMay(1967,1993)
FieldsandRings,SecondEdition,byIrvingKaplansky(1969,1972)
LieAlgebrasandLocallyCompactGroups,byIrvingKaplansky(1971)
SeveralComplexVariables,byRaghavanNarasimhan(1971)
Torsion-FreeModules,byEbenMatlis(1973)
StableHomotopyandGeneralisedHomology,byJ.F.Adams(1974)
RingswithInvolution,byI.N.Herstein(1976)
TheoryofUnitaryGroupRepresentation,byGeorgeV.Mackey(1976)
CommutativeSemigroupRings,byRobertGilmer(1984)
Infinite-DimensionalOptimizationandConvexity,byIvarEkelandandThomasTurnbull(1983)
Navier-StokesEquations,byPeterConstantinandCiprianFoias(1988)
EssentialResultsofFunctionalAnalysis,byRobertJ.Zimmer(1990)
FuchsianGroups,bySvetlanaKatok(1992)
UnstableModulesovertheSteenrodAlgebraandSullivan’sFixedPointSetConjecture,
byLionelSchwartz(1994)
TopologicalClassificationofStratifiedSpaces,byShmuelWeinberger(1994)
LecturesonExceptionalLieGroups,byJ.F.Adams(1996)
GeometryofNonpositivelyCurvedManifolds,byPatrickB.Eberlein(1996)
DimensionTheoryinDynamicalSystems:ContemporaryViews
andApplications,byYakovB.Pesin(1997)
HarmonicAnalysisandPartialDifferentialEquations:EssaysinHonorofAlbertoCaldero´n,
editedbyMichaelChrist,CarlosKenig,andCoraSadosky(1999)
AConciseCourseinAlgebraicTopology,byJ.P.May(1999)
TopicsinGeometricGroupTheory,byPierredelaHarpe(2000)
ExteriorDifferentialSystemsandEuler-LagrangePartialDifferentialEquations,
byRobertBryant,PhillipGriffiths,andDanielGrossman(2003)
Ratner’sTheoremsonUnipotentFlows,byDaveWitteMorris(2005)
Geometry,Rigidity,andGroupActions,editedbyBensonFarbandDavidFisher(2011)
GroupsofCircleDiffeomorphisms,byAndrésNavas(2011)
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More Concise
Algebraic Topology
localization,completion,and
modelcategories
J.P.MayandK.Ponto
theuniversityofchicagopress chicagoandlondon
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J.P.MayisprofessorofmathematicsattheUniversityofChicago;heis
authororcoauthorofmanypapersandbooks,includingSimplicialObjectsin
AlgebraicTopologyandAConciseCourseinAlgebraicTopologyinthisseries.
K.PontoisassistantprofessorofmathematicsattheUniversityofKentucky;
sheisanalgebraictopologistwhoseresearchhasfocusedontopological
fixedpointtheory.
TheUniversityofChicagoPress,Chicago60637
TheUniversityofChicagoPress,Ltd.,London
©2012byTheUniversityofChicago
Allrightsreserved.Published2012.
PrintedintheUnitedStatesofAmerica
201918171615141312 12345
ISBN-13:978-0-226-51178-8(cloth)
ISBN-10:0-226-51178-2(cloth)
LibraryofCongressCataloging-in-PublicationData
May,J.P.
Moreconcisealgebraictopology:localization,completion,andmodel
categories/J.P.MayandK.Ponto.
p. cm.—(Chicagolecturesinmathematicsseries)
ISBN-13:978-0-226-51178-8(hardcover:alk.paper)
ISBN-10:0-226-51178-2(hardcover:alk.paper)
1. Algebraictopology. I. Ponto,Kate. II. Title. III. Series:Chicago
lecturesinmathematics.
QA612.M388 2012
514'.2—dc22 2011012400
(cid:2)∞ThispapermeetstherequirementsofANSI/NISOZ39.48-1992
(PermanenceofPaper).
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Contents
Introduction xi
Someconventionsandnotations xxi
Acknowledgments xxvii
PART 1||Preliminaries:BasicHomotopyTheoryandNilpotentSpaces
1. Cofibrationsandfibrations 3
1.1 Relationsbetweencofibrationsandfibrations 3
1.2 Thefill-inandVerdierlemmas 7
1.3 Basedandfreecofibrationsandfibrations 11
1.4 Actionsoffundamentalgroupsonhomotopyclassesofmaps 15
1.5 Actionsoffundamentalgroupsinfibrationsequences 18
2. Homotopycolimitsandhomotopylimits;lim1 24
2.1 Somebasichomotopycolimits 24
2.2 Somebasichomotopylimits 31
2.3 Algebraicpropertiesoflim1 35
2.4 Anexampleofnonvanishinglim1terms 38
2.5 Thehomologyofcolimitsandlimits 40
2.6 Aprofiniteuniversalcoefficienttheorem 43
3. NilpotentspacesandPostnikovtowers 46
3.1 A-nilpotentgroupsandspaces 46
3.2 NilpotentspacesandPostnikovtowers 49
3.3 CocellularspacesandthedualWhiteheadtheorem 50
3.4 FibrationswithfiberanEilenberg-MacLanespace 56
3.5 PostnikovA-towers 61
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4. Detectingnilpotentgroupsandspaces 69
4.1 Nilpotentactionsandcohomology 69
4.2 Universalcoversofnilpotentspaces 72
4.3 A-mapsofA-nilpotentgroupsandspaces 74
4.4 Nilpotencyandfibrations 76
4.5 Nilpotentspacesandfinitetypeconditions 79
PART 2||LocalizationsofSpacesatSetsofPrimes
5. Localizationsofnilpotentgroupsandspaces 87
5.1 Localizationsofabeliangroups 87
5.2 Thedefinitionoflocalizationsofspaces 91
5.3 Localizationsofnilpotentspaces 94
5.4 Localizationsofnilpotentgroups 97
5.5 Algebraicpropertiesoflocalizationsofnilpotentgroups 101
5.6 FinitelygeneratedT-localgroups 106
6. Characterizationsandpropertiesoflocalizations 111
6.1 Characterizationsoflocalizationsofnilpotentspaces 111
6.2 Localizationsoflimitsandfibersequences 116
6.3 Localizationsoffunctionspaces 119
6.4 Localizationsofcolimitsandcofibersequences 121
6.5 Acellularconstructionoflocalizations 123
6.6 LocalizationsofH-spacesandco-H-spaces 125
6.7 Rationalizationandthefinitenessofhomotopygroups 128
6.8 Thevanishingofrationalphantommaps 130
7. Fracturetheoremsforlocalization:groups 132
7.1 Globaltolocalpullbackdiagrams 133
7.2 Globaltolocal:abelianandnilpotentgroups 137
7.3 Localtoglobalpullbackdiagrams 140
7.4 Localtoglobal:abelianandnilpotentgroups 143
7.5 Thegenusofabelianandnilpotentgroups 145
7.6 Exactsequencesofgroupsandpullbacks 150
8. Fracturetheoremsforlocalization:spaces 154
8.1 Statementsofthemainfracturetheorems 154
8.2 Fracturetheoremsformapsintonilpotentspaces 157
8.3 Globaltolocalfracturetheorems:spaces 161 −1
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8.4 Localtoglobalfracturetheorems:spaces 164
8.5 Thegenusofnilpotentspaces 165
8.6 Alternativeproofsofthefracturetheorems 170
9. RationalH-spacesandfracturetheorems 175
9.1 ThestructureofrationalH-spaces 175
9.2 TheSamelsonproductandH∗(X;Q) 178
9.3 TheWhiteheadproduct 182
9.4 FracturetheoremsforH-spaces 184
PART 3||CompletionsofSpacesatSetsofPrimes
10. Completionsofnilpotentgroupsandspaces 191
10.1 Completionsofabeliangroups 191
10.2 ThedefinitionofcompletionsofspacesatT 199
10.3 Completionsofnilpotentspaces 202
10.4 Completionsofnilpotentgroups 207
11. Characterizationsandpropertiesofcompletions 214
11.1 Characterizationsofcompletionsofnilpotentspaces 214
11.2 Completionsoflimitsandfibersequences 217
11.3 Completionsoffunctionspaces 220
11.4 Completionsofcolimitsandcofibersequences 221
11.5 CompletionsofH-spaces 222
11.6 Thevanishingofp-adicphantommaps 223
12. Fracturetheoremsforcompletion:groups 226
12.1 Preliminariesonpullbacksandisomorphisms 227
12.2 Globaltolocal:abelianandnilpotentgroups 229
12.3 Localtoglobal:abelianandnilpotentgroups 234
12.4 Formalcompletionsandtheadèlicgenus 236
13. Fracturetheoremsforcompletion:spaces 243
13.1 Statementsofthemainfracturetheorems 244
13.2 Globaltolocalfracturetheorems:spaces 246
13.3 Localtoglobalfracturetheorems:spaces 248
13.4 Thetensorproductofaspaceandaring 254
13.5 Sullivan’sformalcompletion 257
13.6 Formalcompletionsandtheadèlicgenus 260
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PART 4||AnIntroductiontoModelCategoryTheory
14. Anintroductiontomodelcategorytheory 267
14.1 Preliminarydefinitionsandweakfactorizationsystems 268
14.2 Thedefinitionandfirstpropertiesofmodelcategories 274
14.3 Thenotionofhomotopyinamodelcategory 280
14.4 Thehomotopycategoryofamodelcategory 286
15. Cofibrantlygeneratedandpropermodelcategories 292
15.1 ThesmallobjectargumentfortheconstructionofWFSs 293
15.2 Compactlyandcofibrantlygeneratedmodelcategories 299
15.3 Overandundermodelstructures 301
15.4 Leftandrightpropermodelcategories 305
15.5 Leftproperness,liftingproperties,andthesets[X,Y] 309
16. Categoricalperspectivesonmodelcategories 314
16.1 Derivedfunctorsandderivednaturaltransformations 314
16.2 QuillenadjunctionsandQuillenequivalences 319
16.3 Symmetricmonoidalcategoriesandenrichedcategories 323
16.4 Symmetricmonoidalandenrichedmodelcategories 329
16.5 Aglimpseathighercategoricalstructures 335
17. Modelstructuresonthecategoryofspaces 339
17.1 TheHurewiczorh-modelstructureonspaces 340
17.2 TheQuillenorq-modelstructureonspaces 344
17.3 Mixedmodelstructuresingeneral 348
17.4 Themixedmodelstructureonspaces 356
17.5 Themodelstructureonsimplicialsets 359
17.6 Theproofofthemodelaxioms 365
18. Modelstructuresoncategoriesofchaincomplexes 372
18.1 Thealgebraicframeworkandtheanalogywithtopology 372
18.2 h-cofibrationsandh-fibrationsinChR 375
18.3 Theh-modelstructureonChR 379
18.4 Theq-modelstructureonChR 383
18.5 Proofsandthecharacterizationofq-cofibrations 387
18.6 Them-modelstructureonChR 392
19. Resolutionandlocalizationmodelstructures 395
19.1 Resolutionandmixedmodelstructures 396
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19.2 ThegeneralcontextofBousfieldlocalization 399
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19.3 Localizationswithrespecttohomologytheories 405
19.4 Bousfieldlocalizationatsetsandclassesofmaps 409
19.5 Bousfieldlocalizationinenrichedmodelcategories 412
PART 5||BialgebrasandHopfAlgebras
20. BialgebrasandHopfalgebras 417
20.1 Preliminaries 417
20.2 Algebras,coalgebras,andbialgebras 421
20.3 AntipodesandHopfalgebras 425
20.4 Modules,comodules,andrelatedconcepts 427
21. ConnectedandcomponentHopfalgebras 432
21.1 Connectedalgebras,coalgebras,andHopfalgebras 432
21.2 Splittingtheorems 435
21.3 Componentcoalgebrasandtheexistenceofantipodes 437
21.4 Self-dualHopfalgebras 440
21.5 ThehomotopygroupsofMOandotherThomspectra 446
21.6 AproofoftheBottperiodicitytheorem 448
22. LiealgebrasandHopfalgebrasincharacteristiczero 453
22.1 GradedLiealgebras 453
22.2 ThePoincaré-Birkhoff-Witttheorem 456
22.3 PrimitivelygeneratedHopfalgebrasincharacteristiczero 460
22.4 CommutativeHopfalgebrasincharacteristiczero 463
23. RestrictedLiealgebrasandHopfalgebrasincharacteristicp 465
23.1 RestrictedLiealgebras 465
23.2 TherestrictedPoincaré-Birkhoff-Witttheorem 467
23.3 PrimitivelygeneratedHopfalgebrasincharacteristicp 469
23.4 CommutativeHopfalgebrasincharacteristicp 471
24. Aprimeronspectralsequences 477
24.1 Definitions 477
24.2 Exactcouples 479
24.3 Filteredcomplexes 482
24.4 Products 484
24.5 TheSerrespectralsequence 486
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24.6 Comparisontheorems 491
24.7 Convergenceproofs 493
Bibliography 497
Index 505
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