代数拓扑中微分形式

图书标准信息

• 作者 Raoul、Loring W.Tu 著
• 出版社 世界图书出版公司
• 出版时间 2009-03
• 版次 1
• ISBN 9787506291903
• 定价 55.00元
• 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

【内容简介】

　　Theguidingprincipleinthisbookistousedifferentialformsasanaidinexploringsomeofthelessdigestibleaspectsofalgebraictopology.Accord-ingly，wemoveprimarilyintherealmofsmoothmanifoldsandusethedeRhamtheoryasaprototypeofallofcohomology.Forapplicationstohomotopytheorywealsodiscussbywayofanalogycohomoiogywitharbitrarycoefficients.Althoughwehaveinmindanaudiencewithpriorexposuretoalgebraicordifferentialtopology，forthemostpartagoodknowledgeoflinearalgebra，advancedcalculus，andpoint-settopologyshouldsuffice.Someacquaintancewithmanifolds，simplicialcomplexes，singularhomologyandcohomology，andhomotopygroupsishelpful，butnotreallynecessary.Withinthetextitselfwehavestatedwithcarethemoreadvancedresultsthatareneeded，sothatamathematicallymaturereaderwhoacceptsthesebackgroundmaterialsonfaithshouldbeabletoreadtheentirebookwiththeminimalprerequisites.

【作者简介】

Raoul Bott ，美国哈佛大学剑桥分校（Harvard University Cambridge）数学系教授。

内容摘要

《代数拓扑中的微分形式》以微分形式为主要手段，简洁明快地介绍代数拓扑中的许多比较深刻的概念和定理。全书不拘泥于叙述格式，而是强调有关问题的具体背景，从而使读者开阔思路和加深对概念的理解。本书可供拓扑工作者参考，亦可作代数拓扑课研究生教材。 目次：de Rham理论；Cech-de Rham复形；谱序列和应用；示性类。

主编推荐

Raoul Bott ，美国哈佛大学剑桥分校（Harvard University Cambridge）数学系教授。

【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