Cubical Homotopy Theory

647 Pages·2015·7.993 MB·English

by  Brian A. Munson, Ismar Volić

CubicalHomotopyTheory
Graduate students and researchers alike will benefit from this treatment of various
classical and modern topics in the homotopy theory of topological spaces, with an
emphasisoncubicaldiagrams.Thebookcontainsmorethan300examplesandprovides
detailedexplanationsofmanyfundamentalresults.
PartIfocusesonfoundationalmaterialonhomotopytheory,viewedthroughthelens
ofcubicaldiagrams:fibrationsandcofibrations,homotopypullbacksandpushouts,and
the Blakers–Massey Theorems. Part II includes a brief example-driven introduction
tocategories,limits,andcolimits,andanaccessibleaccount ofhomotopylimitsand
colimits of diagrams of spaces. It also discusses cosimplicial spaces and relates this
topic to the cubical theory of Part I, and provides computational tools via spectral
sequences. The book finishes with applications to some exciting new topics that use
cubicaldiagrams:anoverviewoftwoversionsofcalculusoffunctorsandanaccount
ofrecentdevelopmentsinthestudyofthetopologyofspacesofknots.
briana.munsonisanAssociateProfessorofMathematicsattheUSNavalAcademy.
HehasheldpostdoctoralandvisitingpositionsatStanfordUniversity,HarvardUniver-
sity,andWellesleyCollege,Massachusetts.Hisresearchareaisalgebraictopology,and
hisworkspanstopicssuchasembeddingtheory,knottheory,andhomotopytheory.
ismar volic´ is an Associate Professor of Mathematics at Wellesley College,
Massachusetts. He has held postdoctoral and visiting positions at the University of
Virginia, Massachusetts Institute of Technology, and Louvain-la-Neuve University in
Belgium. His research is in algebraic topology and his articles span a wide variety
of subjects such as knot theory, homotopy theory, and category theory. He is an
award-winningteacherwhoseresearchhasbeenrecognizedbyseveralgrantsfromthe
NationalScienceFoundation.
NEWMATHEMATICALMONOGRAPHS
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Béla Bollobás, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry
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AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
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1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductive
Groups
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.HounieAnIntroductiontoInvolutive
Structures
7. A.ShlapentokhHilbert’sTenthProblem
8. G.MichlerTheoryofFiniteSimpleGroupsI
9. A.BakerandG.WüstholzLogarithmicFormsandDiophantineGeometry
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.S´niatyckiDifferentialGeometryofSingularSpacesandReductionof
Symmetry
24. E.RiehlCategoricalHomotopyTheory
25. B.A.MunsonandI.Volic´CubicalHomotopyTheory
26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition)
27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpaces
onMetricMeasureSpaces
28. Y.-G.OhSymplecticTopologyandFloerHomologyI
29. Y.-G.OhSymplecticTopologyandFloerHomologyII
Cubical Homotopy Theory
BRIAN A. MUNSON
UnitedStatesNavalAcademy,Maryland
ISMAR VOLIC´
WellesleyCollege,Massachusetts
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
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(cid:2)c BrianA.MunsonandIsmarVolic´2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
PrintedintheUnitedKingdombyClays,StIvesplc
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationData
Munson,BrianA.,1976–
Cubicalhomotopytheory/BrianA.Munson,UnitedStatesNavalAcademy,Maryland,
IsmarVolic´,WellesleyCollege,Massachusetts.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-107-03025-1(alk.paper)
1. Homotopytheory. 2. Cube. 3. Algebraictopology. I. Volic´,Ismar,1973–
II. Title.
QA612.7.M86 2015
514′.24–dc23
2015004584
ISBN978-1-107-03025-1Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
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To Marissa and Catherine

Contents
Preface pagexi
PART I CUBICAL DIAGRAMS
1 Preliminaries 3
1.1 Spacesandmaps 3
1.2 Spacesofmaps 12
1.3 Homotopy 14
1.4 Algebra:homotopyandhomology 20
2 1-cubes:Homotopyfibersandcofibers 28
2.1 Fibrations 29
2.2 Homotopyfibers 40
2.3 Cofibrations 47
2.4 Homotopycofibers 56
2.5 Algebraoffibrationsandcofibrations 62
2.6 Connectivityofspacesandmaps 67
2.7 Quasifibrations 80
3 2-cubes:Homotopypullbacksandpushouts 91
3.1 Pullbacks 94
3.2 Homotopypullbacks 100
3.3 Arithmeticofhomotopycartesiansquares 111
3.4 Totalhomotopyfibers 127
3.5 Pushouts 133
3.6 Homotopypushouts 137
3.7 Arithmeticofhomotopycocartesiansquares 149
3.8 Totalhomotopycofibers 168
3.9 Algebraofhomotopycartesianandcocartesiansquares 171
vii
viii Contents
4 2-cubes:TheBlakers–MasseyTheorems 183
4.1 Historicalremarks 183
4.2 Statementsandapplications 188
4.3 Hurewicz,Whitehead,andSerreTheorems 198
4.4 ProofsoftheBlakers–MasseyTheoremsforsquares 203
4.5 ProofsofthedualBlakers–MasseyTheoremsforsquares 213
4.6 Homotopygroupsofsquares 216
5 n-cubes:Generalizedhomotopypullbacksandpushouts 221
5.1 Cubicalandpuncturedcubicaldiagrams 221
5.2 Limitsofpuncturedcubes 226
5.3 Homotopylimitsofpuncturedcubes 230
5.4 Arithmeticofhomotopycartesiancubes 234
5.5 Totalhomotopyfibers 249
5.6 Colimitsofpuncturedcubes 255
5.7 Homotopycolimitsofpuncturedcubes 258
5.8 Arithmeticofhomotopycocartesiancubes 261
5.9 Totalhomotopycofibers 271
5.10 Algebraofhomotopycartesianandcocartesiancubes 275
6 TheBlakers–MasseyTheoremsforn-cubes 288
6.1 Historicalremarks 288
6.2 Statementsandapplications 290
6.3 ProofsoftheBlakers–MasseyTheoremsforn-cubes 299
6.4 ProofsofthedualBlakers–MasseyTheoremsforn-cubes 324
6.5 Homotopygroupsofcubes 329
PART II GENERALIZATIONS, RELATED TOPICS, AND
APPLICATIONS
7 Somecategorytheory 339
7.1 Categories,functors,andnaturaltransformations 339
7.2 Productsandcoproducts 347
7.3 Limitsandcolimits 351
7.4 Modelsforlimitsandcolimits 367
7.5 Algebraoflimitsandcolimits 375
8 Homotopylimitsandcolimitsofdiagramsofspaces 379
8.1 Theclassifyingspaceofacategory 380
8.2 Homotopylimitsandcolimits 388
8.3 Homotopyinvarianceofhomotopylimitsandcolimits 399
8.4 Modelsforhomotopylimitsandcolimits 412
Contents ix
8.5 Algebraofhomotopylimitsandcolimits 423
8.6 Algebraintheindexingvariable 431
9 Cosimplicialspaces 443
9.1 Cosimplicialspacesandtotalization 444
9.2 Examples 458
9.3 Cosimplicialreplacementofadiagram 464
9.4 Cosimplicialspacesandcubicaldiagrams 471
9.5 Multi-cosimplicialspaces 476
9.6 Spectralsequences 479
10 Applications 502
10.1 Homotopycalculusoffunctors 502
10.2 Manifoldcalculusoffunctors 524
10.3 Embeddings,immersions,anddisjunction 542
10.4 Spacesofknots 553
Appendix 570
A.1 Simplicialsets 570
A.2 Smoothmanifoldsandtransversality 578
A.3 Spectra 586
A.4 Operadsandcosimplicialspaces 595
References 600
Index 613
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