vector bundles of k-theory

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1、Version1.3,July2001AllenHatcherCopyrightc2001byAllenHatcherPaperorelectroniccopiesfornoncommercialusemaybemadefreelywithoutexplicitpermissionfromtheauthor.Allotherrightsreserved.TableofContentsChapter1.VectorBundles1.1.BasicDefinitionsandConstructions.......

2、.....1Sections3.DirectSums5.PullbackBundles5.InnerProducts7.Subbundles8.TensorProducts9.AssociatedBundles11.1.2.ClassifyingVectorBundles.................12TheUniversalBundle12.VectorBundlesoverSpheres16.OrientableVectorBundles21.ACellStructureonGrassmannMan

3、ifolds22.Appendix:Paracompactness24.Chapter2.ComplexK-Theory2.1.TheFunctorK(X).......................28RingStructure31.CohomologicalProperties32.2.2.BottPeriodicity........................39ClutchingFunctions38.LinearClutchingFunctions43.ConclusionoftheProo

4、f45.2.3.Adams’HopfInvariantOneTheorem...........48AdamsOperations51.TheSplittingPrinciple55.2.4.FurtherCalculations.....................61TheThomIsomorphism61.Chapter3.CharacteristicClasses3.1.Stiefel-WhitneyandChernClasses............64AxiomsandConstructio

5、n65.CohomologyofGrassmannians70.Applicationsofw1andc173.3.2.TheChernCharacter.....................74TheJ–Homomorphism77.3.3.EulerandPontryaginClasses................84TheEulerClass88.PontryaginClasses91.1.BasicDefinitionsandConstructionsVectorbundlesarespeci

6、alsortsoffiberbundleswithadditionalalgebraicstruc-ture.Hereisthebasicdefinition.Anndimensionalvectorbundleisamapp:E!B−1togetherwitharealvectorspacestructureonpbforeachb2B,suchthatthefollowinglocaltrivialityconditionissatisfied:ThereisacoverofBbyopensets−1nUfor

7、eachofwhichthereexistsahomeomorphismh:pU!URtaking−1npbtofbgRbyavectorspaceisomorphismforeachb2U.Suchanhiscalledalocaltrivializationofthevectorbundle.ThespaceBiscalledthebasespace,−1Eisthetotalspace,andthevectorspacespbarethefibers.Oftenoneabbrevi-atestermi

8、nologybyjustcallingthevectorbundleE,lettingtherestofthedatabeimplicit.WecouldequallywelltakeCinplaceofRasthescalarfieldhere,obtainingthenotionofacomplexvectorbundle.Ifwemodifythedefinitionbydroppingallre