代数拓扑中微分形式

代数拓扑中微分形式
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作者: ,
2009-03
版次: 1
ISBN: 9787506291903
定价: 55.00
装帧: 平装
开本: 32开
纸张: 其他
页数: 331页
正文语种: 英语
原版书名: Differential Forms in Algebraic Topology
分类: 自然科学
109人买过
  •   Theguidingprincipleinthisbookistousedifferentialformsasanaidinexploringsomeofthelessdigestibleaspectsofalgebraictopology.Accord-ingly,wemoveprimarilyintherealmofsmoothmanifoldsandusethedeRhamtheoryasaprototypeofallofcohomology.Forapplicationstohomotopytheorywealsodiscussbywayofanalogycohomoiogywitharbitrarycoefficients.Althoughwehaveinmindanaudiencewithpriorexposuretoalgebraicordifferentialtopology,forthemostpartagoodknowledgeoflinearalgebra,advancedcalculus,andpoint-settopologyshouldsuffice.Someacquaintancewithmanifolds,simplicialcomplexes,singularhomologyandcohomology,andhomotopygroupsishelpful,butnotreallynecessary.Withinthetextitselfwehavestatedwithcarethemoreadvancedresultsthatareneeded,sothatamathematicallymaturereaderwhoacceptsthesebackgroundmaterialsonfaithshouldbeabletoreadtheentirebookwiththeminimalprerequisites. Introduction
    CHAPTERⅠ
    DeRhamTheory
    §1ThedeRhamComplexonR
    ThedeRhamcomplex
    Compactsupports
    §2TheMayer-VietorisSequence
    ThefunctorQ
    TheMayer-Vietorissequence
    ThefunctorandtheMayer—Vietorissequenceforcompactsupports
    §3OrientationandIntegration
    Orientationandtheintegralofadifferentialform
    Stokes’theorem
    §4Poincar6Lemmas
    ThePoincarelemmafordeRham~ohomoiogy
    ThePoincarelemmaforcompactlysupportedcohomology
    Thedegreeofapropermap
    §5TheMayer-VietorisArgument
    Existenceofagoodcover
    FinitedimensionalityofdeRhamcohomology
    Poincar6dualityonanorientablemanifold
    TheKiinnethformulaandtheLeray-Hirschtheorem
    ThePoincar6dualofaclosedorientedsubmanifold
    §6TheThornIsomorphism
    Vectorbundlesandthereductionofstructuregroups
    Operationsonvectorbundles
    Compactcohomologyofavectorbundle
    Compactverticalcohomologyandintegrationalongthefiber
    Poincar6dualityandtheThornclass
    Theglobalangularform,theEulerclass,andtheThornclass
    RelativedeRhamtheory
    §7TheNonorientableCase
    ThetwisteddeRhamCODrplex
    Integrationofdensities,Poincardduality,andtheThomisomorphism

    CHAPTERⅡ
    TheCech——deRhamComplex
    §8TheGeneralizedMayer-VietorisPrinciple
    ReformulationoftheMayer-Vietorissequence
    Generalizationtocountablymanyopensetsandapplications
    §9MoreExamplesandApplicationsoftheMayer—VietorisPrinciple
    Examples:computingthedeRhamcohomologyfromthe
    combinatoricsofagoodcover
    ExplicitisomorphismsbetweenthedoublecomplexanddeRhamandeach
    Thetic—tac-toeproofoftheKfinnethformula
    §10PresheavesandCechCohomology
    Presheaves
    Cechcohomology
    §11SphereBundles
    Orientability
    TheEulerclassofanorientedspherebundle
    Theglobalangularform
    Eulernumberandtheisolatedsingularitiesofasection
    EulercharacteristicandtheHopfindextheorem
    §12TheThornIsomorphismandPoincar6DualityRevisited
    TheThornisomorphism
    Eulerclassandthezcr0locusofasection
    Atic—tac-toelemma
    Poincar6duality
    §13Monodromy
    Whenisalocallyconstantpresheafconstant?
    Examplesofmonodromy

    CHAPTERⅢ
    SpectralSequencesandApplications
    §14TheSpectralSequenceofaFilteredComplex
    ExactCouples
    Thespectralsequenceofafilteredcomplex
    Thespectralsequenceofadoublecomplex
    Thespectralsequenceofafiberbundle
    Someapplications
    PfodUctstructures
    TheGysinsequence
    Leray’Sconstruction
    §15CohomologywithIntegerCoefficients
    Singularhomology
    Theconeconstruction
    TheMayer-Vietorissequenceforsingularchains
    Singularcohomology
    Thehomologyspectralsequence
    §16ThePathFibration
    Thepathfibration
    Thecohomologyoftheloopspaceofasphere
    §17ReviewofHomotopyTheory
    Homotopygroups
    Therelativehomotopysequence
    Somehomotopygroupsofthespheres
    Attachingcells
    DigressiononMorsetheory
    Therelationbetweenhomotopyandhomology
    π3(S2)andtheHopfinvariant
    §18ApplicationstoHomotopyTheory
    Eilenberg-MacLanespaces
    Thetelescopingconstruction
    ThecohomologyofK(Z,3)
    Thetransgression
    Basictricksofthetrade
    Postnikovapproximation
    Computationofπ4(S3)
    TheWhiteheadtower
    Computationofπ5(S3)
    §19RationalHomotopyTheory
    Minimalmodds
    ExamplesofMinimalModels
    Themaintheoremandapplications

    CHAPTERⅣ
    CharacteristicClasses
    §20ChernClassesofaComplexVectorBundle
    ThefirstChernclassofacomplexlinebundle
    Theprojectivizationofavectorbundle
    MainpropertiesoftheChernclasses
    §21TheSplittingPrincipleandFlagManifolds
    Thesplittingprinciple
    ProofoftheWhitneyproductformulaandtheequality
    ofthetopChernclassandtheEulerclass
    ComputationofsomeChernclasses
    Flagmanifolds
    §22PontrjaginClasses
    Conjugatebundl
    Realizationandcomplexification
    ThePontrjaginclassesofarealvectorbundle
    Applicationtotheembeddingofamanifoldina
    Euclideanspace
    §23TheSearchfortheUniversalBund
    TheGrassmannian
    DigressiononthePoincar6seriesofagradedalgebra
    Theclassificationofvectorbundles
    TheinfiniteGrassmannian
    Concludingremarks
    References
    ListofNotations
    Index
  • 内容简介:
      Theguidingprincipleinthisbookistousedifferentialformsasanaidinexploringsomeofthelessdigestibleaspectsofalgebraictopology.Accord-ingly,wemoveprimarilyintherealmofsmoothmanifoldsandusethedeRhamtheoryasaprototypeofallofcohomology.Forapplicationstohomotopytheorywealsodiscussbywayofanalogycohomoiogywitharbitrarycoefficients.Althoughwehaveinmindanaudiencewithpriorexposuretoalgebraicordifferentialtopology,forthemostpartagoodknowledgeoflinearalgebra,advancedcalculus,andpoint-settopologyshouldsuffice.Someacquaintancewithmanifolds,simplicialcomplexes,singularhomologyandcohomology,andhomotopygroupsishelpful,butnotreallynecessary.Withinthetextitselfwehavestatedwithcarethemoreadvancedresultsthatareneeded,sothatamathematicallymaturereaderwhoacceptsthesebackgroundmaterialsonfaithshouldbeabletoreadtheentirebookwiththeminimalprerequisites.
  • 目录:
    Introduction
    CHAPTERⅠ
    DeRhamTheory
    §1ThedeRhamComplexonR
    ThedeRhamcomplex
    Compactsupports
    §2TheMayer-VietorisSequence
    ThefunctorQ
    TheMayer-Vietorissequence
    ThefunctorandtheMayer—Vietorissequenceforcompactsupports
    §3OrientationandIntegration
    Orientationandtheintegralofadifferentialform
    Stokes’theorem
    §4Poincar6Lemmas
    ThePoincarelemmafordeRham~ohomoiogy
    ThePoincarelemmaforcompactlysupportedcohomology
    Thedegreeofapropermap
    §5TheMayer-VietorisArgument
    Existenceofagoodcover
    FinitedimensionalityofdeRhamcohomology
    Poincar6dualityonanorientablemanifold
    TheKiinnethformulaandtheLeray-Hirschtheorem
    ThePoincar6dualofaclosedorientedsubmanifold
    §6TheThornIsomorphism
    Vectorbundlesandthereductionofstructuregroups
    Operationsonvectorbundles
    Compactcohomologyofavectorbundle
    Compactverticalcohomologyandintegrationalongthefiber
    Poincar6dualityandtheThornclass
    Theglobalangularform,theEulerclass,andtheThornclass
    RelativedeRhamtheory
    §7TheNonorientableCase
    ThetwisteddeRhamCODrplex
    Integrationofdensities,Poincardduality,andtheThomisomorphism

    CHAPTERⅡ
    TheCech——deRhamComplex
    §8TheGeneralizedMayer-VietorisPrinciple
    ReformulationoftheMayer-Vietorissequence
    Generalizationtocountablymanyopensetsandapplications
    §9MoreExamplesandApplicationsoftheMayer—VietorisPrinciple
    Examples:computingthedeRhamcohomologyfromthe
    combinatoricsofagoodcover
    ExplicitisomorphismsbetweenthedoublecomplexanddeRhamandeach
    Thetic—tac-toeproofoftheKfinnethformula
    §10PresheavesandCechCohomology
    Presheaves
    Cechcohomology
    §11SphereBundles
    Orientability
    TheEulerclassofanorientedspherebundle
    Theglobalangularform
    Eulernumberandtheisolatedsingularitiesofasection
    EulercharacteristicandtheHopfindextheorem
    §12TheThornIsomorphismandPoincar6DualityRevisited
    TheThornisomorphism
    Eulerclassandthezcr0locusofasection
    Atic—tac-toelemma
    Poincar6duality
    §13Monodromy
    Whenisalocallyconstantpresheafconstant?
    Examplesofmonodromy

    CHAPTERⅢ
    SpectralSequencesandApplications
    §14TheSpectralSequenceofaFilteredComplex
    ExactCouples
    Thespectralsequenceofafilteredcomplex
    Thespectralsequenceofadoublecomplex
    Thespectralsequenceofafiberbundle
    Someapplications
    PfodUctstructures
    TheGysinsequence
    Leray’Sconstruction
    §15CohomologywithIntegerCoefficients
    Singularhomology
    Theconeconstruction
    TheMayer-Vietorissequenceforsingularchains
    Singularcohomology
    Thehomologyspectralsequence
    §16ThePathFibration
    Thepathfibration
    Thecohomologyoftheloopspaceofasphere
    §17ReviewofHomotopyTheory
    Homotopygroups
    Therelativehomotopysequence
    Somehomotopygroupsofthespheres
    Attachingcells
    DigressiononMorsetheory
    Therelationbetweenhomotopyandhomology
    π3(S2)andtheHopfinvariant
    §18ApplicationstoHomotopyTheory
    Eilenberg-MacLanespaces
    Thetelescopingconstruction
    ThecohomologyofK(Z,3)
    Thetransgression
    Basictricksofthetrade
    Postnikovapproximation
    Computationofπ4(S3)
    TheWhiteheadtower
    Computationofπ5(S3)
    §19RationalHomotopyTheory
    Minimalmodds
    ExamplesofMinimalModels
    Themaintheoremandapplications

    CHAPTERⅣ
    CharacteristicClasses
    §20ChernClassesofaComplexVectorBundle
    ThefirstChernclassofacomplexlinebundle
    Theprojectivizationofavectorbundle
    MainpropertiesoftheChernclasses
    §21TheSplittingPrincipleandFlagManifolds
    Thesplittingprinciple
    ProofoftheWhitneyproductformulaandtheequality
    ofthetopChernclassandtheEulerclass
    ComputationofsomeChernclasses
    Flagmanifolds
    §22PontrjaginClasses
    Conjugatebundl
    Realizationandcomplexification
    ThePontrjaginclassesofarealvectorbundle
    Applicationtotheembeddingofamanifoldina
    Euclideanspace
    §23TheSearchfortheUniversalBund
    TheGrassmannian
    DigressiononthePoincar6seriesofagradedalgebra
    Theclassificationofvectorbundles
    TheinfiniteGrassmannian
    Concludingremarks
    References
    ListofNotations
    Index
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