现代几何方法和应用（第3卷）

图书标准信息

• 作者 B.A.Dubrovin、A.T.Fomenko et al 编
• 出版社 世界图书出版公司
• 出版时间 1999-11
• 版次 1
• ISBN 9787506212649
• 定价 71.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 416页

【内容简介】

Inexpositionsoftheelementsoftopologyitiscustomaryforhomologytobegivenafundamentalrole.SincePoincare,wholaidthefoundationsoftopology,homologytheoryhasbeenregardedastheappropriateprimarybasisforanintroductiontothemethodsofalgebraictopology.Fromhomotopytheory,ontheotherhand,onlythefundamentalgroupandcovering-spacetheoryhavetraditionallybeenincludedamongthebasicinitialconcepts.Essentiallyallelementaryclassicaltextbooksoftopology（thebestofwhichis,intheopinionofthepresentauthors,SeifertandThrelfall'sATextbookofTopology）beginwiththehomologytheoryofoneoranotherclassofcomplexes.Onlyatalaterstage（andthenstillfromahomologicalpointofview）dofibre-spacetheoryandthegeneralproblemofclassifyinghomotopyclassesofmaps（homotopytheory）comeinforconsideration.However,methodsdevelopedininvestigatingthetopologyofdifferentiablemanifolds,andintensivelyelaboratedfromthe1930sonwards（byWhitneyandothers）,nowpermitawholesalereorganizationofthestandardexpositionOfthefundamentalsofmoderntopology.Inthisnewapproach,whichresemblesmorethatofclassicalanalysis,thesefundamentalsturnouttoconsistprimarilyoftheelementarytheoryofsmoothmanifolds,homotopytheorybasedonthese,andsmoothfibrespaces.Furthermore,overthedecadeofthe1970sitbecameclearthatexactlythiscomplexoftopologicalideasandmethodswereprovingtobefundamentallyapplicableinvariousareasofmodernphysics.

【作者简介】

     B. A. Dubrovin，俄罗斯著名数学家。

内容摘要

这套3卷集是以苏联莫斯科大学数力系的几何课讲义为基础形成的。它全面介绍现代几何学的基本概念和定理，并特别强调在数学其他分支以及理论物理中的应用。语言通俗易懂，尽量使物理工作者易于人门。 第3卷系统阐述了同调理论的基础知识。自从庞加莱奠定了拓扑学的基础之后，同调理论就被认为是学习代数拓扑学的基本入门知识，因此，本书对于广大研究生学好同调理论并进而研究拓扑学都是一本极好的教材。 本卷主要章节：同通和上同调计算的方法；光滑函数的临界点和同调理论；配边和光滑结构。

主编推荐

     B. A. Dubrovin，俄罗斯著名数学家。

【目录】

Contents
Preface
CHAPTER1HomologyandCohomology.ComputationalRecipes
1.CohomologygroupsasclassesofcloseddifferentialformsTheirhomotopyinvariance
2.Thehomologytheoryofalgebraiccomplexes
3.Simplicialcomplexes.TheirhomologyandcohomologygroupsTheclassificationofthetwo-dimensionalclosedsurfaces
4.Attachingcellstoatopologicalspace.Cellspaces.Theoremsonthereductionofcellspaces.Homologygroupsandthefundamentalgroupsofsurfacesandcertainothermanifolds
5.Thesingularhomologyandcohomologygroups.Theirhomotogyinvariance.Theexactsequenceofapair.Relativehomologygroups
6.Thesingularhomologyofcellcomplexes.Itsequivalencewithcellhomology.Poincaredualityinsimplicialhomology
7.Thehomologygroupsofaproductofspaces.Multiplicationincohomologyrings.ThecohomologytheoryofH-spacesandLiegroups.Thecohomologyoftheunitarygroups
8.Thehomologytheoryoffibrebundles（skewproducts）
9.Theextensionproblemformaps,homotopies,andcross-sectionsObstructioncohomologyclasses
9.1.Theextensionproblemformaps
9.2.Theextensionproblemforhomotopies
9.3.Theextensionproblemforcross-sections
10.Homologytheoryandmethodsforcomputinghomotopygroups.
TheCartan-Serretheorem.Cohomologyoperations.Vectorbundles
10.1.Theconceptofacohomologyopcration.Examples
10.2.CohomologyoperationsandEilenberg-MacLanecomplexes
10.3.Computationoftherationalhomotopygroups
10.4.Applicationtovectorbundles.Characteristicclasses
10.5.ClassificationoftheSteenrodoperationsinlowdimensions
10.6.Computationofthefirstfewnontrivialstablehomotopygroupsofpheres
10.7.Stablehomotopyclassesofmapsofcellcomplexes
11.Homologytheoryandthefundamentalgroup
12.ThecohomologygroupsofhyperellipticRiemannsurfaces.Jacobitori.eodesicsonmulti-axisellipsoids.Relationshiptofinite-gappotentials
13.ThesimplestpropertiesofKahlermanifoldsAbeliantori
14.Sheafcohomology

CHAPTER2CriticalPointsofSmoothFunctionsandHomologyTheory
15.Morsefunctionsandcellcomplexes
16.TheMorseinequalities
17.Morse-Smalefunctions.Handles.Surfaces
18.Poincareduality
19.CriticalpointsofsmoothfunctionsandtheLyusternik-Shnirelmancategoryofamanifold
20.CriticalmanifoldsandtheMorseinequalities.Functionswithsymmetry
21.Criticalpointsoffunctionalsandthetopologyofthepathspace（m）
22.Applicationsoftheindextheorem
23.Theperiodicproblemofthecalculusofvariations
24.Morsefunctionson3-dimensioalmanifoldsandHeegaardsplittings
25.UnitaryBottperiodicityandhigher-dimensionalvariationalproblems
25.1.Thetheoremonunitaryperiodicity
25.2.Unitaryperiodicityviathetwo-dimensionalcalculusofvariations
25.3.Onthogonalperiodicityviathehigher-dimensionalcalculusofvariations
26.Morsetheoryandcertainmotionsintheplanarn-bodyproblem

CHAPTER3CobordismsandSmoothStructures
27.Characteristicnumbers.Cobordisms.CyclesandsubmanifoldsThesignatureofamanifold
27.1.Statementoftheproblem.ThesimplestfactsaboutcobordismsThesignature
27.2.Thomcomplexes.Calculationofcobordisms（modulotorsion）Thesignatureformula.Realizationofcyclesassubmanifolds
27.3.Someapplicationsofthesignaturefonnula.Thesignatureandtheproblemoftheinvarianceofclasses
28.Smoothstructuresonthe7-dimensionalsphere.Theclassificationproblemforsmoothmanifolds（normalinvariants）.Reidemeistertorsionandthefundamentalhypothesis（Hauptvermutung）ofcombinatorialtopology
Bibliography
APPENDIX1（byS.P.Novikov）
AnAnalogueofMorseTheoryforMany-ValuedFunctionsCertainPropertiesofPoissonBrackets
APPENDIX2（byA.T.Fomenko）Plateau'sProblem.SpectralBordismsandGloballyMinimalSurfacesinRiemannianManifolds
Index
ErratatoParts1and11