代数拓扑基础教程 （英文版）

Thisbookisintendedtoserveasatextbookforacourseinalgebraictopologyatthebeginninggraduatelevel.Themaintopicscoveredaretheclassificationofcompact2-manifolds，thefundamentalgroup，coveringspaces，singularhomologytheory，andsingularcohomologytheory(includingcupproductsandthedualitytheoremsofPoincareandAlexander).ItconsistsofmaterialfromthefirstfivechaptersoftheauthorsearlierbookAlgebraicTopology：AnIntroduction(GTM56)togetherwithalmostallofhisbookSingularHomologyTheory(GTM70).Thismaterialfromthetwoearlierbookshasbeenrevised，corrected，andbroughtuptodate.Thereisenoughhereforafull-yearcourse.

Theauthorhastriedtogiveastraightforwardtreatmentofthesubjectmatter，strippedofallunnecessarydefinitions，terminology，andtechnicalmachinery.Hehasalsotried，whereverfeasible，toemphasizethegeometricmotivationbehindthevariousconcepts.Severalapplicationsofthemethodsofalgebraictopologytoconcretegeometrical-topologicalproblemsaregiven(e.g.，Brouwerfixedpointtheorem，Brouwer-Jordanseparationtheorem，lnvarianceofDomain.Borsuk-UlamtheoremS.

本书的主要内容包括：紧2-流形的分类、基本群、覆叠空间、奇异同调论以及奇异上同调理论。这些主题得到系统地展开，并且摒弃了所有不必要的定义、术语及技术工具。作者力求突出各种概念的几何背景。本书纳入了作者以前的著作：《代数拓扑学引论》（GTM 56）中的前五章内容，以及现已绝版的《奇异同调论》（GTM 70）著作中的几乎全部内容，这些内容材料现都已经过了细致地修改、校正与翻新。目次：2维流形；基本群； 自由群及群的自由积  两空间并集的基本群上的Seifert及Van Kampen定理及其应用；覆叠空间； 同调理论的背景及动机；同调理论的定义及基本性质；一些空间的同调群的计算: 同调理论的应用及更多性质；CW复形的同调；任意系数群的同调； 积空间的同调；上同调理论；同调及上同调的积；流形上的同调对偶定理；射影空间的上积及其应用。            

Preface
NotationandTerminology
CHAPTERⅠ
Two-DimensionalManifolds
1.Introduction
2.DefinitionandExamplesofn-Manifolds
3.Orientablevs.NonorientableManifolds
4.ExamplesofCompact，Connected2-Manifolds
5.StatementoftheClassificationTheoremforCompactSurfaces
6.TriangulationsofCompactSurfaces
7.ProofofTheorem5.1
8.TheEulerCharacteristicofaSurface
References

CHAPTERⅡTheFundamentalGroup
1.Introduction
2.BasicNotationandTerminology
3.DefinitionoftheFundamentalGroupofaSpace
4.TheEffectofaContinuousMappingontheFundamentalGroup
5.TheFundamentalGroupofaCircleISInfiniteCyclic
6.Application：TheBrouwerFixed-PointTheoreminDimension2
7.TheFundamentalGroupofaProductSpace
8.HomotopyTypeandHomotopyEquivalenceofSpaces
References

CHAPTERⅢFreeGroupsandFreeProductsofGroups
1.Introduction
2.TheWeakProductofAbelianGroups
3.FreeAbelianGroups
4.FreeProductsofGroups
5.FreeGroups
6.ThePresentationofGroupsbyGeneratorsandRelations
7.UniversalMappingProblems
References

CHAPTERⅣSeifertandVanKampenTheoremontheFundamentalGroupoftheUnionofTwoSpaces.Applications
1.Introduction
2.StatementandProofoftheTheoremofSeifertandVanKampen
3.FirstApplicationofTheorem2.1
4.SecondApplicationofTheorem2.1
5.StructureoftheFundamentalGroupofaCompactSurface
6.ApplicationtoKnotTheory
7.ProofofLemma2.4
References

CHAPTERⅤCoveringSpaces
1.Introduction
2.DefinitionandSomeExamplesofCoveringSpaces
3.LiftingofPathstoaCoveringSpace
4.TheFundamentalGroupofaCoveringSpace
5.LiftingofArbitraryMapstoaCoveringSpace
6.HomomorphismsandAutomorphismsofCoveringSpaces
10.TheExistenceTheoremforCoveringSpacesReferences

CHAPTERⅥ
BackgroundandMotivationforHomologyTheory
1.Introduction
2.SummaryofSomeoftheBasicPropertiesofHomologyTheory
3.SomeExamplesofProblemswhichMotivatedtheDevelopmentofHomologyTheoryintheNineteenthCenturyReferences

CHAPTERⅦ
DefinitionsandBasicPropertiesofHomologyTheory
1.Introduction
2.DefinitionofCubicalSingularHomologyGroups
3.TheHomomorphismInducedbyaContinuousMap
4.TheHomotopyPropertyoftheInducedHomomorphisms
5.TheExactHomologySequenceofaPair
6.TheMainPropertiesofRelativeHomologyGroups
7.TheSubdivisionofSingularCubesandtheProofofTheorem6.4

CHAPTERⅧ
DeterminationoftheHomologyGroupsofCertainSpaces：
ApplicationsandFurtherPropertiesofHomologyTheory
1.Introduction
2.HomologyGroupsofCellsandSpheres——Applications
3.HomologyofFiniteGraphs
4.HomologyofCompactSurfaces
5.TheMayer-VietorisExactSequence
6.TheJordan-BrouwerSeparationTheoremandlnvarianceofDomain
7.TheRelationbetweentheFundamentalGroupandtheFirstHomologyGroup
References

CHAPTERⅨ
HomologyofCW-Complexes
1.Introduction
2.AdjoiningCellstoaSpace
3.CW-Complexes
4.TheHomologyGroupsofaCW-Complex
5.IncidenceNumbersandOrientationsofCells
6.RegularCW-Complexes
7.DeterminationofIncidenceNumbersforaRegularCellComplex
8.HomologyGroupsofaPseudomanifoldReferences

CHAPTERⅩ
HomologywithArbitraryCoefficientGroups
1.Introduction
2.ChainComplexes
3.DefinitionandBasicPropertiesofHomologywithArbitraryCoefficients
4.IntuitiveGeometricPictureofaCyclewithCoefficientsinG
5.CoefficientHomomorphismsandCoefficientExactSequences
6.TheUniversalCoefficientTheorem
7.FurtherPropertiesofHomologywithArbitraryCoefficientsReferences

CHAPTERⅪ
TheHomologyofProductSpaces
1.Introduction
2.TheProductofCW-ComplexesandtheTensorProductofChainComplexes
3.TheSingularChainComplexofaProductSpace
4.TheHomologyoftheTensorProductofChainComplexes(TheKiinnethTheorem)
5.ProofoftheEilenberg-ZilberTheorem
6.FormulasfortheHomologyGroupsofProductSpacesReferences