代数拓扑的微分形式 英文版
作者： R.Bott，L.W.Tu 著
出版时间： 1999年版
内容简介
The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accordingly， we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients. Although we have in mind an audience with prior exposure to algebraic or differential topology， for the most part a good knowledge of linear algebra， advanced calculus， and point-set topology should suffice. Some acquaintance with manifolds， simplicial complexes， singular homology and cohomology， and homotopy groups is helpful， but not really necessary. Within the text itself we have stated with care the more advanced results that are needed， so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites.本书为英文版。
目录
Introduction
CHAPTERI
DeRhamTheory
1ThedeRhamComplexonRn
ThedeRhamcomplex
Compactsupports
2TheMayer-VietorisSequence
Thefunctor*
TheMayer-Vietorissequence
Thefunctor*andtheMayer-Vietorissequenceforcompactsupports
3OrientationandIntegration
Orientationandtheintegralofadifferentialform
Stokes'theorem
4PoincareLemmas
ThePoincarelemmafordeRhamcohomology
ThePoincarelemmaforcompactlysupportedcohomology
Thedegreeofapropermap
5TheMayer-VietorisArgument
Existenceofagoodcover
FinitedimensionalityofdeRhamcohomology
Poincaredualityonanorientablemanifold
TheKunnethformulaandtheLeray-Hirschtheorem
ThePoincaredualofaclosedorientedsubmanifold
6TheThomIsomorphism
Vectorbundlesandthereductionofstructuregroups
Operationsonvectorbundles
Compactcohomologyofavectorbundle
Compactverticalcohomologyandintegrationalongthefiber
PoincaredualityandtheThomclass
Theglobalangularform.theEulerclass,andtheThomclass
RelativedeRhamtheory
7TheNonorientableCase
ThetwisteddeRhamcomplex
Integrationofdensities,Poincareduality,andtheThomisomorphism
CHAPTERII
TheCech-deRhamComplex
8TheGeneralizedMayer-VietorisPrinciple
ReformulationoftheMayer-Vietorissequence
Generalizationtocountablymanyopensetsandapplications
9MoreExamplesandApplicationsoftheMayer-VietorisPrinciple
Examples:computingthedeRhamcohomologyfromthe
combinatoricsofagoodcover
ExplicitisomorphismsbetweenthedoublecomplexanddeRhamandCech
Thetic-tac-toeproofoftheKfinnethformula
10PresheavesandCechCohomology
Presheaves
Cechcohomology
11SphereBundles
Orientability
TheEulerclassofanorientedspherebundle
Theglobalangularform
Eulernumberandtheisolatedsingularitiesofasection
EulercharacteristicandtheHopfindextheorem
12TheIhomIsomorphismandPoincareDualityRevisited
TheThomisomorphism
Eulerclassandthezerolocusofasection
Atic-tac-toelemma
Poincareduality
13Monodromy
Whenisalocallyconstantpresheafconstant?
Examplesofmonodromy
CHAPTERIII
SpectralSequencesandApplications
14TheSpectralSequenceofaFilteredComplex
Exactcouples
Thespectralsequenceofafilteredcomplex
Thespectralsequenceofadoublecomplex
Thespectralsequenceofafiberbundle
Someapplications
Productstructures
TheGysinsequence
Leray'sconstruction
15CohomologywithIntegerCoefficients
Singularhomology
Theconeconstruction
TheMayer-Vietorissequenceforsingularchains
Singularcohomology
Thehomologyspectralsequence
16ThePathFibration
Thepathfibration
Thecohomologyoftheloopspaceofasphere
17ReviewofHomotopyTheory
Homotopygroups
Therelativehomotopysequence
Somehomotopygroupsofthespheres
Attachingcells
DigressiononMorsetheory
Therelationbetweenhomotopyandhomology
3(S2)andtheHopfinvariant
18ApplicationstoHomotopyTheory
Eilenberg-MacLanespaces
Thetelescopingconstruction
ThecohomologyofK(Z,3)
Thetransgression
Basictricksofthetrade
Postnikovapproximation
Computationof4(S3)
TheWhiteheadtower
Computationof5(S3)
19RationalHomotopyTheory
Minimalmodels
ExamplesofMinimalModels
Themaintheoremandapplications
CHAPTERIV
CharacteristicClasses
20ChernClassesofaComplexVectorBundle
ThefirstChernclassofacomplexlinebundle
Theprojectivizationofavectorbundle
MainpropertiesoftheChernclasses
21TheSplittingPrincipleandFlagManifolds
Thesplittingprinciple
ProofoftheWhitneyproductformulaandtheequality
ofthetopChernclassandtheEulerclass
ComputationofsomeChernclasses
Flagmanifolds
22PontrjaginClasses
Conjugatebundles
Realizationandcomplexification
ThePontrjaginclassesofarealvectorbundle
Applicationtotheembeddingofamanifoldina
Euclideanspace
23TheSearchfortheUniversalBundle
TheGrassmannian
DigressiononthePoincareseriesofagradedalgebra
Theclassificationofvectorbundles
TheinfiniteGrassmannian
Concludingremarks
References
ListofNotations
Index
