【正版书籍】Differentialformsinalgebraictopology
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作者Raoul,Loring W.Tu
出版社世界图书出版公司
ISBN9787506291903
出版时间2009-03
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定价79元
货号J9787506291903
上书时间2024-07-03
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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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