纤维丛（第3版）

　　PrefacetotheThirdEdition
　　PrefacetotheSecondEdition
　　PrefacetotheFirstEdition
　　CHAPTER1 PreliminariesonHomotopyTheory
　　1.CategoryTheoryandHomotopyTheory
　　2.Complexes
　　3.TheSpacesMap(X，Y)andMap0(X，Y)
　　4.HomotopyGroupsofSpaces
　　5.FibreMaps
　　PARTITHEGENERALTHEORYOFFIBREBUNDLES
　　CHAPTER2 GeneralitiesonBundles
　　1.DefinitionofBundlesandCrossSections
　　2.ExamplesofBundlesandCrossSections
　　3.MorphismsofBundles
　　4.ProductsandFibreProducts
　　5.RestrictionsofBundlesandInducedBundles
　　6.LocalPropertiesofBundles
　　7.ProlongationofCrossSections
　　Exercises
　　CHAPTER3 VectorBundles
　　1.DefinitionandExamplesofVectorBundles
　　2.MorphismsofVectorBundles
　　3.inducedVectorBundles
　　4.HomotopyPropertiesofVectorBundles
　　5.ConstructionofGaussMaps
　　6.HomotopiesofGaussMaps
　　7.FunctorialDescriptionoftheHomotopyClassificationofVectorBundles
　　8.Kernel，Image，andCokernelofMorphismswithConstantRank
　　9.RiemannianandHermitianMetricsonVectorBundles
　　Exercises
　　CHAPTER4 GeneralFibreBundles
　　1.BundlesDefinedbyTransformationGroups
　　2.DefinitionandExamplesofPrincipalBundles
　　3.CategoriesofPrincipalBundles
　　4.InducedBundlesofPrincipalBundles
　　5.DefinitionofFibreBundles
　　6.FunctorialPropertiesofFibreBundles
　　7.TrivialandLocallyTrivialFibreBundles
　　8.DescriptionofCrossSectionsofaFibreBundle
　　9.NumerablePrincipalBundlesoverBx[0，1]
　　10.TheCofunctork
　　11.TheMilnorConstruction
　　12.HomotopyClassificationofNumerablePrincipalG-Bundles
　　13.HomotopyClassificationofPrincipalG-Bundlesover
　　CW-Complexes
　　Exercises
　　CHAPTER5 LocalCoordinateDescriptionofFibreBundles
　　1.AutomorphismsofTrivialFibreBundles
　　2.ChartsandTransitionFunctions
　　3.ConstructionofBundleswithGivenTransitionFunctions
　　4.TransitionFunctionsandInducedBundles
　　5.LocalRepresentationofVectorBundleMorphisms
　　6.OperationsonVectorBundles
　　7.TransitionFunctionsforBundleswithMetricsExercises
　　CHAPTER6 ChangeofStructureGroupinFibreBundles
　　1.FibreBundleswithHomogeneousSpacesasFibres2.ProlongationandRestrictionofPrincipalBund
　　les
　　3.RestrictionandProlongationofStructureGroupforFibreBundles
　　4.LocalCoordinateDescription.ofChangeofStructureGroup
　　5.ClassifyingSpacesandtheReductionofStructureGroupExercises
　　CHAPTER7 TheGaugeGroupofaPrincipalBundle
　　1.DefinitionoftheGaugeGroup
　　2.TheUniversalStandardPrincipalBundleoftheGaugeGroup
　　3.TheStandardPrincipalBundleasaUniversalBundle
　　4.AbelianGaugeGroupsandtheKiinnethFormula
　　CHPTER8
　　CalculationsInvolvingtheClassicalGroups
　　1.StiefelVarietiesandtheClassicalGroups
　　2.GrassmannManifoldsandtheClassicalGroups
　　3.LocalTrivialityofProjectionsfromStiefelVarieties
　　4.StabilityoftheHomotopyGroupsoftheClassicalGroups
　　5.VanishingofLowerHomotopyGroupsofStiefelVarieties
　　6.UniversalBundlesandClassifyingSpacesfortheClassicalGroups
　　7.UniversalVectorBundles
　　8.DescriptionofallLocallyTrivialFibreBundlesoverSuspensions
　　9.CharacteristicMapoftheTangentBundleoverSn
　　10.HomotopyPropertiesofCharacteristicMaps
　　11.HomotopyGroupsofStiefelVarieties
　　12.SomeoftheHomotopyGroupsoftheClassicalGroups
　　Exercises
　　PARTII
　　ELEMENTSOFK-THEORY
　　CHAPTER9
　　StabilityPropertiesofVectorBundles
　　1.TrivialSummandsofVectorBundles
　　2.HomotopyClassificationandWhitneySums
　　3.TheKCofunctors
　　4.CorepresentationsofKF
　　5.HomotopyGroupsofClassicalGroupsandKF(Si)
　　Exercises
　　CHAPTER10
　　RelativeK-Theory
　　1.CollapsingofTrivializedBundles
　　2.ExactSequencesinRelativeK-Theory
　　3.ProductsinK-Theory
　　4.TheCofunctorL(X，A)
　　5.TheDifferenceMorphism
　　6.ProductsinL(X，A)
　　7.TheClutchingConstruction
　　8.TheCofunctorLn(X.A)
　　9.Half-ExactCofunctors
　　Exercises
　　CHAPTER11
　　BottPeriodicityintheComplexCase
　　1.K-TheoryInterpretationofthePeriodicityResult
　　2.ComplexVectorBundlesoverXxS2
　　3.AnalysisofPolynomialClutchingMaps
　　4.AnalysisofLinearClutchingMaps
　　5.TheInversetothePeriodicityIsomorphism
　　CHAPTER12
　　CliffordAlgebras
　　1.UnitTangentVectorFieldsonSpheres：I
　　2.OrthogonalMultiplications
　　3.GeneralitiesonQuadraticForms
　　4.CliffordAlgebraofaQuadraticForm
　　5.CalculationsofCliffordAlgebras
　　6，CliffordModules
　　7.TensorProductsofCliffordModules
　　8.UnitTangentVectorFieldsonSpheres：II
　　9.TheGroupSpin(k)
　　Exercises
　　CHAPTER13
　　TheAdamsOperationsandRepresentations
　　1.λ-Rings
　　2.TheAdamsψ-Operationsinλ-Ring
　　3.TheγiOperations
　　4.GeneralitiesonG-Modules
　　5.TheRepresentationRingofaGroupGandVectorBundles
　　6.SemisimplicityofG-ModulesoverCompactGroups
　　7.CharactersandtheStructureoftheGroupRF(G)
　　8.MaximalTort
　　9.TheRepresentationRingofaTorus
　　10.TheO-OperationsonK(X)andKO(X)
　　11.TheO-OperationsonK(Sn)
　　CHAPTER14
　　RepresentationRingsofClassicalGroups
　　1.SymmetricFunctions
　　2.MaximalToriinSU(n)andU(n)
　　3.TheRepresentationRingsofSU(n)andU(n)
　　4.MaximalToffinSp(n)
　　5.FormalIdentitiesinPolynomialRings
　　6.TheRepresentationRingofSp(n)
　　7.MaximalToriandtheWeylGroupofSO(n)
　　8.MaximalToriandtheWeylGroupofSpin(n)
　　9.SpecialRepresentationsofSO(n)andSpin(n)
　　10.CalculationofRSO(n)andRSpin(n)
　　11.RelationBetweenRealandComplexRepresentationRings
　　12.ExamplesofRealandQuaternionicRepresentations
　　13.SpinorRepresentationsandtheK-GroupsofSpheres
　　CHAPTER15
　　TheHopflnyariant
　　1.K-TheoryDefinitionoftheHopfInvariant
　　2.AlgebraicPropertiesoftheHopfInvariant
　　3.HopfInvariantandBidegree
　　4.NonexistenceofElementsofHopfInvariant1
　　CHAPTER16
　　VectorFieldsontheSphere
　　1.ThornSpacesofVectorBundles
　　2.S-Category
　　3.S-DualityandtheAtiyahDualityTheorem
　　4.FibreHomotopyType
　　5.StableFibreHomotopyEquivalence
　　6.TheGroupsJ(Sk)andKTop(Sk)
　　7.ThomSpacesandFibreHomotopyType
　　8.S-DualityandS-Reducibility
　　9.NonexistenceofVectorFieldsandReducibility
　　10.NonexistenceofVectorFieldsandCoreducibility
　　11.NonexistenceofVectorFieldsandJ(RPk)
　　12.RealK-GroupsofRealProjectiveSpaces
　　13.RelationBetweenKO(RPn)andJ(RPn)
　　14.RemarksontheAdamsConjecture
　　PARTIII
　　CHARACTERISTICCLASSES
　　CHAPTER17
　　ChernClassesandStiefeI-WhitneyClasses
　　1.TheLeray-HirschTheorem
　　2.DefinitionoftheStiefei-WhitneyClassesandChernClasses
　　3.AxiomaticPropertiesoftheCharacteristicClasses
　　4.StabilityPropertiesandExamplesofCharacteristicClasses
　　5.SplittingMapsandUniquenessofCharacteristicClasses
　　6.ExistenceoftheCharacteristicClasses
　　7.FundamentalClassofSphereBundles.GysinSequence
　　8.MultiplicativePropertyoftheEulerClass
　　9.DefinitionofStiefeI-WhitneyClassesUsingtheSquaring
　　OperationsofSteenrod
　　10.TheThomIsomorphism
　　11.RelationsBetweenRealandComplexVectorBundles
　　12.OrientabilityandStiefeI-WhitneyClasses
　　Exercises
　　CHAPTER18
　　DifferentiableManifolds
　　1.GeneralitiesonManifolds
　　2.TheTangentBundletoaManifold
　　3.OrientationinEuclideanSpaces
　　4.OrientationofManifolds
　　5.DualityinManifolds
　　6.ThornClassoftheTangentBundle
　　7.EulerCharacteristicandClassofaManifold
　　8.WusFormulafortheStiefeI-WhitneyClassofaManifold
　　9.StiefeI-WhitneyNumbersandCobordism
　　10.ImmersionsandEmbeddingsofManifolds
　　Exercises
　　CHAPTER19
　　CharacteristicClassesandConnections
　　1.DifferentialFormsanddeRhamCohomology
　　2.ConnectionsonaVectorBundle
　　3.InvariantPolynomialsintheCurvatureofaConnection
　　4.HomotopyPropertiesofConnectionsandCurvature
　　5.HomotopytotheTrivialConnectionandtheChern-SimonsForm
　　6.TheLevi-CivitaorRiemannianConnection
　　CHAPTER20
　　GeneralTheoryofCharacteristicClasses
　　1.TheYonedaRepresentationTheorem
　　2.GeneralitiesonCharacteristicClasses
　　3.ComplexCharacteristicClassesinDimensionn
　　4.ComplexCharacteristicClasses
　　5.RealCharacteristicClassesMod2
　　6.2-DivisibleRealCharacteristicClassesinDimensionn
　　7.OrientedEven-DimensionalRealCharacteristicClasses
　　8.ExamplesandApplications
　　9.BottPeriodicityandIntegralityTheorems
　　10.ComparisonofK-TheoryandCohomologyDefinitions
　　ofHopfInvariant
　　11.TheBorel-HirzebruchDescriptionofCharacteristicClasses
　　Appendix1
　　DoldsTheoryofLocalPropertiesofBundles
　　Appendix2
　　OntheDoubleSuspension
　　1.H*(ΩS(X))asanAlgebraicFunctorofH(X)
　　2.ConnectivityofthePair(Ω2S2n+1，S2n-1)Localizedatp
　　3.DecompositionofSuspensionsofProductsandliS(X)
　　4.SingleSuspensionSequences
　　5.ModpHopfInvariant
　　6.SpacesWherethepthPowerIsZero
　　7.DoubleSuspensionSequences
　　8.StudyoftheBoundaryMap△：Ω3S2np+1→ΩS2n-1
　　Bibliography
　　Index

　　    The notion of a fibre bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950， the definition of fibre bundle had been clearly formulated， the homotopy classification of fibre bundles achieved， and the theory of characteristic classes of fibre bundles developed by several mathematicians： Chern， Pontrjagin， Stiefel， and Whitney. Steenrods book， which appeared in 1950， gavea coherent treatment of the subject up to that time.
　　    About 1955， Miinor gave a construction ora universal fibre bundle for any topological group. This construction is also included in Part I along with an elementary proof that the bundle is universal.