现代几何方法和应用(第3卷)

现代几何方法和应用(第3卷)
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作者: ,
1999-11
版次: 1
ISBN: 9787506212649
定价: 71.00
装帧: 平装
开本: 24开
纸张: 胶版纸
页数: 416页
分类: 自然科学
22人买过
  • 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. 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
  • 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
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