| In expositions of the elements of topology it is customary for homology to be given a fundamental role. Since Poincare, who laid the foundations of topology, homology theory has been regarded as the appropriate primary basis for an introduction to the methods of algebraic topology. From homotopy theory, on the other hand, only the fundamental group and covering-space theory have traditionally been included among the basic initial concepts. Essentially all elementary classical textbooks of topology (the best of which is, in the opinion of the present authors, Seifert and Threlfall's A Textbook of Topology) begin with the homology theory of one or another classof complexes. Only at a later stage (and then still from a homological point of view) do fibre-space theory and the general problem of classifying homotopy classes of maps (homotopy theory) come in for consideration. However, methods developed in investigating the topology of differentiable manifolds, and intensively elaborated from the 1930s onwards (by Whitney and others), now permit a wholesale reorganization of the standard exposition Of the fundamentals of modern topology. In this new approach, which resembles more that of classical analysis, these fundamentals turn out to consist primarily of the elementary theory of smooth manifolds, homotopy theory based on these, and smooth fibre spaces. Furthermore, over the decade of the 1970s it became clear that exactly this complex of topological ideas and methods were proving to be fundamentally applicable in various areas of modern physics. 本书为英文版。 |
| Contents Preface CHAPTER1HomologyandCohomology.ComputationalRecipes 1.Cohomologygroupsasclassesofcloseddifferentialforms Theirhomotopyinvariance 2.Thehomologytheoryofalgebraiccomplexes 3.Simplicialcomplexes.Theirhomologyandcohomologygroups Theclassificationofthetwo-dimensionalclosedsurfaces 4.Attachingcellstoatopologicalspace.Cellspaces.Theoremson thereductionofcellspaces.Homologygroupsandthefundamental groupsofsurfacesandcertainothermanifolds 5.Thesingularhomologyandcohomologygroups.Theirhomotogy invariance.Theexactsequenceofapair.Relativehomologygroups 6.Thesingularhomologyofcellcomplexes.Itsequivalencewithcell homology.Poincaredualityinsimplicialhomology 7.Thehomologygroupsofaproductofspaces.Multiplicationin cohomologyrings.ThecohomologytheoryofH-spacesandLie groups.Thecohomologyoftheunitarygroups 8.Thehomologytheoryoffibrebundles(skewproducts) 9.Theextensionproblemformaps,homotopies,andcross-sections Obstructioncohomologyclasses 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.Computationofthefirstfewnontrivialstablehomotopygroupsofspheres 10.7.Stablehomotopyclassesofmapsofcellcomplexes 11.Homologytheoryandthefundamentalgroup 12.ThecohomologygroupsofhyperellipticRiemannsurfaces.Jacobi tori.Geodesicsonmulti-axisellipsoids.Relationshiptofinite-gap potentials 13.ThesimplestpropertiesofKahlermanifoldsAbeliantori 14.Sheafcohomology CHAPTER2CriticalPointsofSmoothFunctionsandHomologyTheory 15.Morsefunctionsandcellcomplexes 16.TheMorseinequalities 17.Morse-Smalefunctions.Handles.Surfaces 18.Poincareduality 19.CriticalpointsofsmoothfunctionsandtheLyusternik-Shnirelmancategoryofamanifold 20.CriticalmanifoldsandtheMorseinequalities.Functions withsymmetry 21.Criticalpointsoffunctionalsandthetopologyofthe pathspace(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.Cyclesandsubmanifolds Thesignatureofamanifold 27.1.Statementoftheproblem.Thesimplestfactsaboutcobordisms Thesignature 27.2.Thomcomplexes.Calculationofcobordisms(modulotorsion) Thesignatureformula.Realizationofcyclesassubmanifolds 27.3.Someapplicationsofthesignaturefonnula.Thesignatureand theproblemoftheinvarianceofclasses 28.Smoothstructuresonthe7-dimensionalsphere.Theclassification problemforsmoothmanifolds(normalinvariants).Reidemeistertorsion andthefundamentalhypothesis(Hauptvermutung)ofcombinatorial topology Bibliography APPENDIX1(byS.P.Novikov) AnAnalogueofMorseTheoryforMany-ValuedFunctions CertainPropertiesofPoissonBrackets APPENDIX2(byA.T.Fomenko) Plateau'sProblem.SpectralBordismsandGloballyMinimal SurfacesinRiemannianManifolds Index ErratatoParts1and11 |
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