代数拓扑

　　To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the relations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology （homotopy theory，simplicial complexes， singular theory， axiomatic homology， differential topology， etc.）， we concentrate our attention on concrete problems in low dimensions， introducing only as much algebraic machinery as necessary for the problems we meet. This makes it possible to see a.wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists：without， we hope， discouraging budding topologists. We also feel that this approach is in better harmony with the historical development of the subject.本书为英文版。

暂缺《代数拓扑》作者简介

Preface
PARTI
CALCULUSINTHEPLANE
CHAPTER1
PathIntegrals
1a.DifferentialFormsandPathIntegrals
1b.WhenArePathIntegralsIndependentofPath?
1c.ACriterionforExactness
CHAPTER2
AnglesandDeformations
2a.AngleFunctionsandWindingNumbers
2b.ReparametrizingandDeformingPaths
2e.VectorFieldsandFluidFlow
PARTII
WINDINGNUMBERS
CHAPTER3
TheWindingNumber
3a.DefinitionoftheWindingNumber
3b.HomotopyandReparametrization
3c.VaryingthePoint
3d.DegreesandLocalDegrees
CHAPTER4
ApplicationsofWindingNumbers
4a.TheFundamentalTheoremofAlgebra
4b.FixedPointsandRetractions
4c.Antipodes
4d.Sandwiches
PARTIII
COHOMOLOGYANDHOMOLOGY,I
CHAPTER5
DeRhamCohomologyandtheJordanCurveTheorem
5a.DefinitionsoftheDeRhamGroups
5b.TheCoboundaryMap
5c.TheJordanCurveTheorem
5d.ApplicationsandVariations
CHAPTER6
Homology
6a.Chains,Cycles,andHoU
6b.Boundaries,H1U,andWindingNumbers
6c.ChainsonGrids
6d.MapsandHomology
6e.TheFirstHomologyGroupforGeneralSpaces
PARTIV
VECTORFIELDS
CHAPTER7
IndicesofVectorFields
7a.VectorFieldsinthePlane
7b.ChangingCoordinates
7c.VectorFieldsonaSphere
CHAPTER8
VectorFieldsonSurfaces
8a.VectorFieldsonaTorusandOtherSurfaces
8b.TheEulerCharacteristic
PARTV
COHOMOLOGYANDHOMOLOGY,II
CHAPTER9
HolesandIntegrals
9a.MultiplyConnectedRegions
9b.IntegrationoverContinuousPathsandChains
9c.PeriodsofIntegrals
9d.ComplexIntegration
CHAPTER10
Mayer-Vietoris
10a.TheBoundaryMap
10b.Mayer-VietorisforHomology
10c.VariationsandApplications
10d.Mayer-VietorisforCohomology
PARTVI
COVERINGSPACESANDFUNDAMENTALGROUPS,I
CHAPTER11
CovetingSpaces
11a.Definitions
11b.LiftingPathsandHomotopies
11c.G-Coverings
11d.CoveringTransformations
CHAPTER12
TheFundamentalGroup
12a.DefinitionsandBasicProperties
12b.Homotopy
12c.FundamentalGroupandHomology
PARTVII
COVERINGSPACESANDFUNDAMENTALGROUPS,II
CHAPTER13
TheFundamentalGroupandCoveringSpaces
13a.FundamentalGroupandCoverings
13b.AutomorphismsofCoverings
13c.TheUniversalCovering
13d.CoveringsandSubgroupsoftheFundamentalGroup
CHAPTER14
TheVanKampenTheorem
14a.G-CoveringsfromtheUniversalCovering
14b.PatchingCoveringsTogether
14c.TheVanKampenTheorem
14d.Applications:GraphsandFreeGroups
PARTVIII
COHOMOLOGYANDHOMOLOGY,III
CHAPTER15
Cohomology
15a.PatchingCoveringsandCechCohomology
15b.CechCohomologyandHomology
15c.DeRhamCohomologyandHomology
15d.ProofofMayer-VietorisforDeRhamCohomology
CHAPTER16
Variations
16a.TheOrientationCovering
16b.Coveringsfrom1-Forms
16c.AnotherCohomologyGroup
16d.G-SetsandCoverings
16e.CoveringsandGroupHomomorphisms
16f.G-CoveringsandCocycles
PARTIX
TOPOLOGYOFSURFACES
CHAPTER17
TheTopologyofSurfaces
17a.TriangulationandPolygonswithSidesIdentified
17b.ClassificationofCompactOrientedSurfaces
17c.TheFundamentalGroupofaSurface
CHAPTER18
CohomologyonSurfaces
18a.1-FormsandHomology
18b.Integralsof2-Forms
18c.WedgesandtheIntersectionPairing
18d.DeRhamTheoryonSurfaces
PARTX
RIEMANNSURFACES
CHAPTER19
RiemannSurfaces
19a.RiemannSurfacesandAnalyticMappings
19b.BranchedCoverings
19c.TheRiemann-HurwitzFormula
CHAPTER20
RiemannSurfacesandAlgebraicCurves
20a.TheRiemannSurfaceofanAlgebraicCurve
20b.MeromorphicFunctionsonaRiemannSurface
20c.HolomorphicandMeromorphic1-Forms
20d.Riemann'sBilinearRelationsandtheJacobian
20e.EllipticandHyperellipticCurves
CHAPTER21
TheRiemann-RochTheorem
21a.SpacesofFunctionsand1-Forms
21b.Adeles
21c.Riemann-Roch
21d.TheAbel-JacobiTheorem
PARTXI
HIGHERDIMENSIONS
CHAPTER22
TowardHigherDimensions
22a.HolesandFormsin3-Space
22b.Knots
22c.HigherHomotopyGroups
22d.HigherDeRhamCohomology
22e.CohomologywithCompactSupports
CHAPTER23
HigherHomology
23a.HomologyGroups
23b.Mayer-VietorisforHomology
23c.SpheresandDegree
23d.GeneralizedJordanCurveTheorem
CHAPTER24
Duality
24a.TwoLemmasfromHomologicalAlgebra
24b.HomologyandDeRhamCohomology
24c.CohomologyandCohomologywithCompactSupports
24d.SimplicialComplexes
APPENDICES
APPENDIXA
PointSetTopology
A1.SomeBasicNotionsinTopology
A2.ConnectedComponents
A3.Patching
A4.LebesgueLemma
APPENDIXB
Analysis
B1.ResultsfromPlaneCalculus
B2.PartitionofUnity
APPENDIXC
Algebra
C1.LinearAlgebra
C2.Groups;FreeAbelianGroups
C3.Polynomials;Gauss'sLemma
APPENDIXD
OnSurfaces
D1.VectorFieldsonPlaneDomains
D2.ChartsandVectorFields
D3.DifferentialFormsonaSurface
APPENDIXE
ProofofBorsuk'sTheorem
HintsandAnswers
References
IndexofSymbols
Index