代数拓扑导论

　　This textbook is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible. The principal topics treated are 2-dimensional manifolds， the fundamental group， and covering spaces， plus the group theory needed in these topics. The only prerequisites are some group theory， such as that normally contained in an undergraduate algebra course on the junior-senior level， and a one-semester undergraduate course in general topology.The topics discussed in this book are "standard" in the sense that several well-known textbooks and treatises devote a few sections or a chapter to them. This， I believe， is the first textbook giving a straightforward treatment of these topics， stripped of all unnecessary definitions， terminology， etc.， and with numerous examples and exercises， thus making them intelligible to advanced undergraduate students.

暂缺《代数拓扑导论》作者简介

CHAPTERONETwo-DimensionalManifolds
1 Introduction
2 Definitionandexamplesofn-manifolds
3 Orientablevs.nonorientablemanifolds
4 Examplesofcompact，connected2-manifolds
5 Statementoftheclassificationtheoremforcompactsurfaces
6 Triangulationsofcompactsurfaces
7 ProofofTheorem5.1
8 TheEulercharacteristicofasurface
9 Manifoldswithboundary
10 Theclassificationofcompact，connected2-manifoldswithboundary
11 TheEulercharacteristicofaborderedsurface
12 ModelsofcompactborderedsurfacesinEuclidean3-space
13 Remarksonnoncompactsurfaces
CHAPTERTWOTheFundamentalGroup
1 Introduction
2 Basicnotationandterminology
3 Definitionofthefundamentalgroupofaspace
4 Theeffectofacontinuousmai)pingonthefundamentalgroup
5 Thefundamentalgroupofacircleisinfinitecyclic
6 Application：TheBrouwerfixed-pointtheoremilldimension2
7 Thefundamentalgroupofaproductspace
8 Homotopytypeandhomotopyequivalenceofspaces
CHAPTERTHREEFreeGroupsandFreeProductsofGroups
1 Introduction
2 Theweakproductofabeliangroups
3 Freeabeliangroups
4 Freeproductsofgroups
5 Freegroups
6 Thepresentationofgroupsbygeneratorsandrelations
7 Universalmappingproblems
CHAPTERFOURScifertandVanKampenTheoremontheFundamentalGroupoftheUnionofTwoSpaces.Applic
ations
1 Introduction
2 StatementandproofofthetheoremofSeifertandVanKampen
3 FirstapplicationofTheorem2.1
4 SecondapplicationofTheorem2.1
5 Structureofthefundamentalgroupofacompactsurface
6 Applicationtoknottheory
CHAPTERFIVECoveringSpaces
1 Introduction
2 Definitionandsomeexamplesofcoveringspaces
3 Liftingofpathstoacoveringspace
4 Thefundamentalgroupofacoveringspace
5 Liftingofarbitrarymapstoacoveringspace
6 Homomorphismsandautomorphismsofcoveringspaces
7 Theactionofthegroupπ(X，x)onthesetp-(x)
8 Regularcoveringspacesandquotientspaces
9 Application：TheBorsuk-Ulamtheoremforthe2-sphere
10 Theexistencetheoremforcoveringspaces
11 Theinducedcoveringspaceoverasubspace
12 Pointsettopologyofcoveringspaces
CHAPTERSIXTheFundamentalGroupandCoveringSpacesofaGraph.ApplicationstoGroupTheory
1 Introduction
2 Definitionandexamples
3 Basicpropertiesofgraphs
4 Trees
5 Thefundamentalgroupofagraph
6 TheEulercharacteristicofafinitegraph
7 Coveringspacesofagraph
8 Generatorsforasubgroupoffreegroup
CHAPTERSEVENTheFundamentalGroupofHigherDimensionalSpaces
1 Introduction
2 Adjunctionof2-cellstoaspace
3 Adjunctionofhigherdimensionalcellstoaspace
4 CW-complexes
5 TheKuroshsubgrouptheorem
6 GrushkosTheorem
CHAPTEREIGHTEpilogue
APPENDIXATheQuotientSpaceorIdentificationSpaceTopology
1 Definitionsandbasicproperties
2 Ageneralizationofthequotientspacetopology
3 Quotientspacesandproductspaces
4 Subspaceofaquotientspacevs.quotientspaceofasubspace
5 ConditionsforaquotientspacetobeaHausdorffspace
APPENDIXBPermutationGroupsorTransformationGroups
1 Basicdefinitions
2 HomogeneousG-spaces
Index