拓扑与几何

Preface

Acknowledgments

CHAPTERⅠGeneralTopology

1.MetricSpaces

2.TopologicalSpaces

3.Subspaces

4.ConnectivityandComponents

5.SeparationAxioms

6.Nets(Moore-SmithConvergence)

7.Compactness

8.Products

9.MetricSpacesAgain

10.ExistenceofRealValuedFunctions

11.LocallyCompactSpaces

12.ParacompactSpaces

13.QuotientSpaces

14.Homotopy

15.TopologicalGroups

16.ConvexBodies

17.TheBaireCategoryTheorem

CHAPTERⅡDifferentiableManifolds

1.TheImplicitFunctionTheorem

2.DifferentiableManifolds

3.LocalCoordinates

4.InducedStructuresandExamples

5.TangentVectorsandDifferentials

6.SardsTheoremandRegularValues

7.LocalPropertiesofImmersionsandSubmersions

8.VectorFieldsandFlows

9.TangentBundles

10.EmbeddinginEuclideanSpace

11.TubularNeighborhoodsandApproximations

12.ClassicalLieGroups

13.FiberBundles

14.InducedBundlesandWhitneySums

15.Transversality

16.Thom-PontryaginTheory

CHAPTERⅢFundamentalGroup

1.HomotopyGroups

2.TheFundamentalGroup

3.CoveringSpaces

4.TheLiftingTheorem

5.TheActionofnlontheFiber

6.DeckTransformations

7.ProperlyDiscontinuousActions

8.ClassificationofCoveringSpaces

9.TheSeifert-VanKampenTheorem

10.RemarksonSO(3)

CHAPTERⅣHomologyTheory

1.HomologyGroups

2.TheZerothHomologyGroup

3.TheFirstHomologyGroup

4.FunctorialProperties

5.HomologicalAlgebra

6.AxiomsforHomology

7.ComputationofDegrees

8.CW-Complexes

9.ConventionsforCW-Complexes

10.CellularHomology

11.CellularMaps

12.ProductsofCW-Complexes

13.EulersFormula

14.HomologyofRealProjectiveSpace

15.SingularHomology

16.TheCrossProduct

17.Subdivision

18.TheMayer-VietorisSequence

19.TheGeneralizedJordanCurveTheorem

20.TheBorsuk-UlamTheorem

21.SimplicialComplexes

22.SimplicialMaps

23.TheLefschetz-HopfFixedPointTheorem

CHAPTERⅤCohomology

1.MultilinearAlgebra

2.DifferentialForms

3.IntegrationofForms

4.StokesTheorem

5.RelationshiptoSingularHomology

6.MoreHomologicatAlgebra

7.UniversalCoefficientTheorems

8.ExcisionandHomotopy

9.deRhamsTheorem

10.ThedeRhamTheoryofCPn

11.HopfsTheoremonMapstoSpheres

12.DifferentialFormsonCompactLieGroups

CHAPTERⅥProductsandDuality

1.TheCrossProductandtheKfinnethTheorem

2.ASignConvention

3.TheCohomologyCrossProduct

4.TheCupProduct

5.TheCapProduct

6.ClassicalOutlookonDuality

7.TheOrientationBundle

8.DualityTheorems

9.DualityonCompactManifoldswithBoundary

10.ApplicationsofDuality

11.IntersectionTheory

12.TheEulerClass,LefschetzNumbers,andVectorFields

13.TheGysinSequence

14.LefschetzCoincidenceTheory

15.SteenrodOperations

16.ConstructionoftheSteenrodSquares

17.Stiefel-WhitneyClasses

18.Plumbing

CHAPTERⅦIHomotopyTheory

1.Cofibrations

2.TheCompact-OpenTopology

3.H-Spaces,H-Groups,andH-Cogroups

4.HomotopyGroups

5.TheHomotopySequenceofaPair

6.FiberSpaces

7.FreeHomotopy

8.ClassicalGroupsandAssociatedManifolds

9.TheHomotopyAdditionTheorem

10.TheHurewiczTheorem

11.TheWhiteheadTheorem

12.Eilenberg-MacLaneSpaces

13.ObstructionTheory

14.ObstructionCochainsandVectorBundles

Appendices

App.A.TheAdditivityAxiom

App.B.BackgroundinSetTheory

App.C.CriticalValues

App.D.DirectLimits

App.E.EuclideanNeighborhoodRetracts

Bibliography

IndexofSymbols

Index

　　Sincethebeginningoftime,oratleasttheeraofArchimedes,smoothmanifolds(curves,surfaces,mechanicalconfigurations,theuniverse)havebeenacentralfocusinmathematics.Theyhavealwaysbeenatthecoreofinterestintopology.AftertheseminalworkofMilnor,Smale,andmanyothers,inthelasthalfofthiscentury,thetopologicalaspectsofsmoothmanifolds,asdistinctfromthedifferentialgeometricaspects,becameasubjectinitsownright.Whilethemajorportionofthisbookisdevotedtoalgebraictopology,Iattempttogivethereadersomeglimpsesintothebeautifulandimportantrealmofsmoothmanifoldsalongtheway,andtoinstillthetenetthatthealgebraictoolsareprimarilyintendedfortheunderstandingofthegeometricworld.