馆藏图书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.

【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