纤维丛

　　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.

暂缺《纤维丛》作者简介

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