拓扑学Ⅰ

图书标准信息

• 作者 诺维科夫 著
• 出版社 科学出版社
• 出版时间 2007-01
• 版次 1
• ISBN 9787030166739
• 定价 60.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 332页
• 字数 391千字

【内容简介】

　　本书作者是拓扑学领域最知名的专家之一，曾获菲尔兹奖和沃尔夫数学奖。本书对整个拓扑学领域作出最新综述。依照诺维科夫自己的观点，拓扑学在19世纪末被称为位置分析，随后分为组合拓扑、代数拓扑、微分拓扑、同伦拓扑、几何拓扑等不同领域。本书从基本原理开始，随之阐述当前的研究前沿，概述这些领域；第二章介绍纤维空间；第三章论述CW-复形、同调和同伦理论、配边理论、K-理论及亚当斯-诺维科夫谱序列；第四章全面讲座流形理论。附录阐述纽结和连接理论及低维拓扑中的最新进展。

【作者简介】

　　S.P.Novikov，InstituteforPhysicalScienceandTechnology，UniversityofMeryland，CollegePark，MD20742-2431，USA.e-mail：novikov@ipst.umd，edu.LandauInstituteforTheoreticalPhysicsoftheRussianAcademyofSciences，Kosyginstr2，117940Moscow，Russia.e-mail：novikov@landau.ac.ru.

【目录】

Introduction
IntroductiontotheEnglishTranslation
Chapter1.TheSimplestTopologicalProperties
Chapter2.topologicalSpaces.Fibrations.Homotopies
1.Observationsfromgeneraltopology.Terminology
2.Homotopies.Homotopytype
3.Coveringhomotopies.Fibrations
4.Homotopygroupsandfibrations.Exactsequences.Examples
Chapter3.SimplicialComplexesandCW-complexes.HomologyandCohomology.TheirRelationtoHomotopyTheory.Obstructions
1.Simplicialcomplexes
2.Thehomologyandcohomologygroups.Poincareduality
3.Relativehomology.Theexactsequenceofapair.Axiomsforhomologytheory.CW-complexes
4.Simplicialcomplexesandotherhomologytheories.Singularhomulogy.Coveringsandsheaves.Theexactsequenceofsheavesandcohomology
5.Homologytheoryofnon-simply-connectedspaces.Complexesofmodules.Reidemeistertorsion.Simpleshomotopytype
6.Simplicialandcellbundleswithastructuregroup.Obstructions.Universallbjects：universalfiberbundlesandtheuniversalpropertyofEilenberg-MacLanecomplexes.Cohomologyoperations.TheSteenrodalgebra.TheAdamsspectralsequence
7.Theclassicalapparatusofhomotopytheory.TheLarayspectralsequence.Thehomologytheoryoffiberbundles.TheCartan-Serremethod.ThePostnikovtower.TheAdamsspectralsequence
8.DefinitionandpropertiesofK-theory.TheAtiyah-Hirzebruchspectralsequence.Adamsoperations.AnaloguesoftheThomisomorphismandtheRiemann-Rochtheorem.EllipticoperatorsandK-theory.Transformationgroups.Four-dimensionalmanifolds
9.Bordismandcobordismtheoryasgeneralizedhomologyandcohomology.Cohomologyoperationsincobordism.TheAdams-Novikovspectralsequence.Formalgroups.Actionsofcyclicgroupsandthecircleonmanifolds
Chapter4.SmoothManifolds
1.Basicconcepts.Smoothfiberbundles.Connexions.Characteristicclasses
2.Thehomologytheoryofsmoothmanifolds.Complexmanifolds.Theclassicalglobalcalculusofvariations.H-spaces.Multi-valuedfunctionsandfunctionals
3.Smoothmanifoldsandhomotopytheory.Framedmanifolds.Bordisms.Thomspaces.TheHirzebruchformulae.Estimatesoftheordersofhomotopygroupsofspheres.Milnorsexample.Theintegralpropertiesofcobordisms
4.Classificationproblemsinthetheoryofsmoothmanifolds.Thetheoryofimmersions.Manifoldswiththehomotopytypeofasphere.RelationshipsbetweensmoothandPL-manifolds.IntegralPontryaginclasses
5.Theroleofthefundamentalgroupintopology.Manifoldsoflowdimension(n=2，3).Knots.Theboundaryofanopenmanifold.TheclassificationinvarianceoftherationalPontryaginclasses.Theclassificationtheoryofnon-simply-connectedmanifoldsofdimension>5.Highersignatures.HermitianK-theory.Geometrictopology：theconstructionofnon-smoothhomeomorphisms.Milnorsexample.Theannulusconjecture.TopologicalandPL-structures
ConcludingRemarks
Appendix.RecentDevelopmentsintheTopologyof3-manifoldsandKnots
1.Introduction：RecentdevelopmentsinTopology
2.Knots：theclassicalandmodernapproachestotheAlexanderpolynomial.Jones-typepolynomials
3.VassilievInvariants
4.Newtopologicalinvariantsfor3-manifolds.TopologicalQuantumFieldTheories
Bibliography
Index