vector bundles and k-theory

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1、Version2.0,January2003AllenHatcherCopyrightc2003byAllenHatcherPaperorelectroniccopiesfornoncommercialusemaybemadefreelywithoutexplicitpermissionfromtheauthor.Allotherrightsreserved.TableofContentsIntroduction..............................1Chapter1.Ve

2、ctorBundles....................41.1.BasicDefinitionsandConstructions............6Sections8.DirectSums10.InnerProducts12.TensorProducts14.AssociatedFiberBundles16.1.2.ClassifyingVectorBundles.................18PullbackBundles18.ClutchingFunctions22.The

3、UniversalBundle27.CellStructuresonGrassmannians31.Appendix:Paracompactness35.Chapter2.K–Theory.......................382.1.TheFunctorK(X).......................39RingStructure40.TheFundamentalProductTheorem41.2.2.BottPeriodicity......................

4、..51ExactSequences51.DeducingPeriodicityfromtheProductTheorem53.ExtendingtoaCohomologyTheory54.ElementaryApplications58.2.3.DivisionAlgebrasandParallelizableSpheres......59H–Spaces59.AdamsOperations62.TheSplittingPrinciple65.2.4.BottPeriodicityintheR

5、ealCase[notyetwritten]2.5.VectorFieldsonSpheres[notyetwritten]Chapter3.CharacteristicClasses..............733.1.Stiefel-WhitneyandChernClasses............74AxiomsandConstructions74.CohomologyofGrassmannians80.3.2.EulerandPontryaginClasses............

6、....84TheEulerClass87.PontryaginClasses90.Chapter4.TheJ–Homomorphism..............944.1.LowerBoundsonImJ....................95TheChernCharacter96.TheeInvariant98.ThomSpaces99.BernoulliDenominators102.4.2.UpperBoundsonImJ[notyetwritten]Bibliography...

7、..........................106PrefaceTopologicalK–theory,thefirstgeneralizedcohomologytheorytobestudiedthor-oughly,wasintroducedaround1960byAtiyahandHirzebruch,basedonthePeriodic-ityTheoremofBottprovedjustafewyearsearlier.InsomerespectsK–theoryismoreel

8、ementarythanclassicalhomologyandcohomology,anditisalsomorepowerfulforcertainpurposes.Someofthebest-knownapplicationsofalgebraictopologyinthetwentiethcentury,suchasthetheoremofBottandMilnorthattherearenodivisionalgebrasaftertheCayleyoctonions,orAdams’