Algebraic Topology (EMS Textbooks in Mathematics)

580 Pages·2008·2.6 MB·English

by  Tammo tom Dieck

Dieck_titelei 1.8.2008 13:23 Uhr Seite 1
Dieck_titelei 1.8.2008 13:23 Uhr Seite 2
EMS Textbooks in Mathematics
EMS Textbooks in Mathematics isa book series aimed at students or professional mathematici-
ans seeking an introduction into a particular field. The individual volumes are intended to provide
not only relevant techniques, results and their applications, but afford insight into the motivations
and ideas behind the theory. Suitably designed exercises help to master the subject and prepare
the reader for the study of more advanced and specialized literature.
Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes
Markus Stroppel, Locally Compact Groups
Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations
Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations
Thomas Timmermann, An Invitation to Quantum Groups and Duality
Marek Jarnicki and Peter Pflug, First Steps in Several Complex Variables: Reinhardt Domains
Oleg Bogopolski, Introduction to Group Theory
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Tammo tom Dieck
Algebraic Topology
Dieck_titelei 1.8.2008 13:23 Uhr Seite 4
Author:
Tammo tom Dieck
Mathematisches Institut
Georg-August-Universität Göttingen
Bunsenstrasse 3–5
37073 Göttingen
Germany
E-mail: [email protected]
2000 Mathematics Subject Classification: 55-01, 57-01
Key words: Covering spaces, fibrations, cofibrations, homotopy groups, cell complexes, fibre
bundles, vector bundles, classifying spaces, singular and axiomatic homology and cohomology,
smooth manifolds, duality, characteristic classes, bordism.
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
ISBN 978-3-03719-048-7
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of
use permission of the copyright owner must be obtained.
© 2008 European Mathematical Society
Contact address:
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Seminar for Applied Mathematics
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Switzerland
Phone: [phone]
Email: [email protected]
Homepage: www.ems-ph.org
Typeset using the author’s TEX files: I. Zimmermann, Freiburg
Printed on acid-free paper produced from chlorine-free pulp. TCF°°
Printed in Germany
9 8 7 6 5 4 3 2 1
Preface
Algebraic topology is the interplay between “continuous” and “discrete” mathe-
matics. Continuousmathematicsisformulatedinitsgeneralforminthelanguage
oftopologicalspacesandcontinuousmaps. Discretemathematicsisusedtoexpress
theconceptsofalgebraandcombinatorics. Inmathematicallanguage: weusethe
realnumberstoconceptualizecontinuousformsandwemodeltheseformswiththe
useoftheintegers. Forexample,ourintuitiveideaoftimesupposesacontinuous
process without gaps, an unceasing succession of moments. But in practice we
usediscretemodels,machinesornaturalprocesseswhichwedefinetobeperiodic.
Likewiseweconceiveofaspaceasacontinuumbutwemodelthatspaceasaset
ofdiscreteforms. Thustheessenceoftimeandspaceisofatopologicalnaturebut
algebraictopologyallowstheirrealizationstobeofanalgebraicnature.
Classical algebraic topology consists in the construction and use of functors
fromsomecategoryoftopologicalspacesintoanalgebraiccategory,sayofgroups.
Butonecanalsopostulatethatglobalqualitativegeometryisitselfofanalgebraic
nature. Consequentlytherearetwoimportantviewpointsfromwhichonecanstudy
algebraictopology: homologyandhomotopy.
Homology,inventedbyHenriPoincaré,iswithoutdoubtoneofthemostinge-
niousandinfluentialinventionsinmathematics. Thebasicideaofhomologyisthat
westartwithageometricobject(aspace)whichisgivenbycombinatorialdata(a
simplicialcomplex). Thenthelinearalgebraandboundaryrelationsdeterminedby
thesedataareusedtoproducehomologygroups.
Inthisbook,thechaptersonsingularhomology,homology,homologicalalgebra
andcellularhomologyconstituteanintroductiontohomologytheory(construction,
axiomaticanalysis,classicalapplications). Thechaptersrequireaparallelreading–
thisindicatesthecomplexityofthematerialwhichdoesnothaveasimpleintuitive
explanation. Ifoneknowsoracceptssomeresultsaboutmanifolds,oneshouldread
theconstructionofbordismhomology. Itappearsinthefinalchapterbutoffersa
simpleexplanationoftheideaofhomology.
Thesecondaspectofalgebraictopology,homotopytheory,beginsagainwiththe
constructionoffunctorsfromtopologytoalgebra. Butthisapproachisimportant
fromanotherviewpoint. Homotopytheoryshowsthatthecategoryoftopological
spaceshasitselfakindof(hidden)algebraicstructure. Thisbecomesimmediately
clear in the introductory chapters on the fundamental group and covering space
theory. The study of algebraic topology is often begun with these topics. The
notions of fibration and cofibration, which are at first sight of a technical nature,
are used to indicate that an arbitrary continuous map has something like a kernel
andacokernel–thebeginningoftheinternalalgebraicstructureoftopology. (The
chapteronhomotopygroups,whichisessentialtothisbook,shouldalsobestudied
vi Preface
for its applications beyond our present study.) In the ensuing chapter on duality
theanalogytoalgebrabecomesclearer: Forasuitableclassofspacesthereexists
adualitytheorywhichresemblesformallythedualitybetweenavectorspaceand
itsdualspace.
ThefirstmaintheoremofalgebraictopologyistheBrouwer–Hopfdegreethe-
orem. Weprovethistheorembyelementarymethodsfromhomotopytheory. Itis
afairlydirectconsequenceoftheBlakers–Masseyexcisiontheoremforwhichwe
presenttheelementaryproofofDieterPuppe. Laterweindicateproofsofthede-
greetheorembasedonhomologyandthenondifferentialtopology. Itisabsolutely
essential to understand this theorem from these three view points. The theorem
saysthatthesetofself-mapsofapositivedimensionalsphereunderthehomotopy
relation has the structure of a (homotopically defined) ring – and this ring is the
ringofintegers.
The second part of the book develops further theoretical concepts (like coho-
mology)andpresentsmoreadvancedapplicationstomanifolds,bundles,homotopy
theory, characteristic classes and bordism theory. The reader is strongly urged to
readtheintroductiontoeachofthechaptersinordertoobtainmorecoherentinfor-
mationaboutthecontentsofthebook.
Wordsinboldfaceitalicaredefinedattheplacewheretheyappearevenifthere
isnoindicationofaformaldefinition. Inaddition,thereisalistofstandardorglobal
symbols. Theproblemsectionscontainexercises,examples,counter-examplesand
further results, and also sometimes ask the reader to extend concepts in further
detail. Itisnotassumedthatalloftheproblemswillbecompletelyworkedout,but
it is strongly recommended that they all be read. Also, the reader will find some
familiaritywiththefullbibliography,notjustthereferencescitedinthetext,tobe
crucialforfurtherstudies. Morebackgroundmaterialaboutspacesandmanifolds
may,atleastforawhile,beobtainedfromtheauthor’shomepage.
IwouldliketothankIreneZimmermannandManfredKarbefortheirhelpand
effortinpreparingthemanuscriptforpublication.
Göttingen,September2008 TammotomDieck
Contents
Preface v
1 TopologicalSpaces 1
1.1 BasicNotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Subspaces. QuotientSpaces . . . . . . . . . . . . . . . . . . . . 5
1.3 ProductsandSums . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 CompactSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 ProperMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 ParacompactSpaces . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 TopologicalGroups . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 TransformationGroups. . . . . . . . . . . . . . . . . . . . . . . 17
1.9 ProjectiveSpaces. GrassmannManifolds . . . . . . . . . . . . . 21
2 TheFundamentalGroup 24
2.1 TheNotionofHomotopy . . . . . . . . . . . . . . . . . . . . . 25
2.2 FurtherHomotopyNotions. . . . . . . . . . . . . . . . . . . . . 30
2.3 StandardSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 MappingSpacesandHomotopy . . . . . . . . . . . . . . . . . . 37
2.5 TheFundamentalGroupoid . . . . . . . . . . . . . . . . . . . . 41
2.6 TheTheoremofSeifertandvanKampen . . . . . . . . . . . . . 45
2.7 TheFundamentalGroupoftheCircle . . . . . . . . . . . . . . . 47
2.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.9 HomotopyGroupoids . . . . . . . . . . . . . . . . . . . . . . . 58
3 CoveringSpaces 62
3.1 LocallyTrivialMaps. CoveringSpaces . . . . . . . . . . . . . . 62
3.2 FibreTransport. ExactSequence . . . . . . . . . . . . . . . . . 66
3.3 ClassificationofCoverings . . . . . . . . . . . . . . . . . . . . . 70
3.4 ConnectedGroupoids . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 ExistenceofLiftings . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 TheUniversalCovering . . . . . . . . . . . . . . . . . . . . . . 78
4 ElementaryHomotopyTheory 81
4.1 TheMappingCylinder . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 TheDoubleMappingCylinder . . . . . . . . . . . . . . . . . . . 84
4.3 Suspension. HomotopyGroups . . . . . . . . . . . . . . . . . . 86
4.4 LoopSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
viii Contents
4.5 GroupsandCogroups . . . . . . . . . . . . . . . . . . . . . . . 90
4.6 TheCofibreSequence . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 TheFibreSequence . . . . . . . . . . . . . . . . . . . . . . . . 97
5 CofibrationsandFibrations 101
5.1 TheHomotopyExtensionProperty . . . . . . . . . . . . . . . . 101
5.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 ReplacingaMapbyaCofibration . . . . . . . . . . . . . . . . . 110
5.4 CharacterizationofCofibrations . . . . . . . . . . . . . . . . . . 113
5.5 TheHomotopyLiftingProperty . . . . . . . . . . . . . . . . . . 115
5.6 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.7 ReplacingaMapbyaFibration . . . . . . . . . . . . . . . . . . 120
6 HomotopyGroups 121
6.1 TheExactSequenceofHomotopyGroups . . . . . . . . . . . . 122
6.2 TheRoleoftheBasePoint . . . . . . . . . . . . . . . . . . . . . 126
6.3 SerreFibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 TheExcisionTheorem . . . . . . . . . . . . . . . . . . . . . . . 133
6.5 TheDegree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.6 TheBrouwerFixedPointTheorem . . . . . . . . . . . . . . . . 137
6.7 HigherConnectivity . . . . . . . . . . . . . . . . . . . . . . . . 141
6.8 ClassicalGroups . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.9 ProofoftheExcisionTheorem. . . . . . . . . . . . . . . . . . . 148
6.10 FurtherApplicationsofExcision . . . . . . . . . . . . . . . . . . 152
7 StableHomotopy. Duality 159
7.1 AStableCategory . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2 MappingCones. . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.3 EuclideanComplements . . . . . . . . . . . . . . . . . . . . . . 168
7.4 TheComplementDualityFunctor . . . . . . . . . . . . . . . . . 169
7.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.6 HomologyandCohomologyforPointedSpaces . . . . . . . . . 179
7.7 SpectralHomologyandCohomology . . . . . . . . . . . . . . . 181
7.8 AlexanderDuality . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.9 CompactlyGeneratedSpaces . . . . . . . . . . . . . . . . . . . 186
8 CellComplexes 196
8.1 SimplicialComplexes . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 WhiteheadComplexes . . . . . . . . . . . . . . . . . . . . . . . 199
8.3 CW-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.4 WeakHomotopyEquivalences . . . . . . . . . . . . . . . . . . . 207
8.5 CellularApproximation . . . . . . . . . . . . . . . . . . . . . . 210
8.6 CW-Approximation . . . . . . . . . . . . . . . . . . . . . . . . 211
Contents ix
8.7 HomotopyClassification . . . . . . . . . . . . . . . . . . . . . . 216
8.8 Eilenberg–MacLaneSpaces . . . . . . . . . . . . . . . . . . . . 217
9 SingularHomology 223
9.1 SingularHomologyGroups . . . . . . . . . . . . . . . . . . . . 224
9.2 TheFundamentalGroup . . . . . . . . . . . . . . . . . . . . . . 227
9.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9.4 BarycentricSubdivision. Excision . . . . . . . . . . . . . . . . . 231
9.5 WeakEquivalencesandHomology . . . . . . . . . . . . . . . . 235
9.6 HomologywithCoefficients . . . . . . . . . . . . . . . . . . . . 237
9.7 TheTheoremofEilenbergandZilber . . . . . . . . . . . . . . . 238
9.8 TheHomologyProduct . . . . . . . . . . . . . . . . . . . . . . 241
10 Homology 244
10.1 TheAxiomsofEilenbergandSteenrod . . . . . . . . . . . . . . 244
10.2 ElementaryConsequencesoftheAxioms . . . . . . . . . . . . . 246
10.3 JordanCurves. InvarianceofDomain . . . . . . . . . . . . . . . 249
10.4 ReducedHomologyGroups . . . . . . . . . . . . . . . . . . . . 252
10.5 TheDegree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.6 TheTheoremofBorsukandUlam . . . . . . . . . . . . . . . . . 261
10.7 Mayer–VietorisSequences . . . . . . . . . . . . . . . . . . . . . 265
10.8 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.9 Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11 HomologicalAlgebra 275
11.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.2 ExactSequences . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11.3 ChainComplexes . . . . . . . . . . . . . . . . . . . . . . . . . . 283
11.4 Cochaincomplexes . . . . . . . . . . . . . . . . . . . . . . . . . 285
11.5 NaturalChainMapsandHomotopies . . . . . . . . . . . . . . . 286
11.6 ChainEquivalences . . . . . . . . . . . . . . . . . . . . . . . . 287
11.7 LinearAlgebraofChainComplexes . . . . . . . . . . . . . . . . 289
11.8 TheFunctorsTorandExt . . . . . . . . . . . . . . . . . . . . . 292
11.9 UniversalCoefficients . . . . . . . . . . . . . . . . . . . . . . . 295
11.10 TheKünnethFormula . . . . . . . . . . . . . . . . . . . . . . . 298
12 CellularHomology 300
12.1 CellularChainComplexes . . . . . . . . . . . . . . . . . . . . . 300
12.2 CellularHomologyequalsHomology . . . . . . . . . . . . . . . 304
12.3 SimplicialComplexes . . . . . . . . . . . . . . . . . . . . . . . 306
12.4 TheEulerCharacteristic . . . . . . . . . . . . . . . . . . . . . . 308
12.5 EulerCharacteristicofSurfaces . . . . . . . . . . . . . . . . . . 311