vector bundles of k-theory

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1、VectorBundlesandK-TheoryVersion1.1,November2000AllenHatcherCopyrightc2000byAllenHatcherPaperorelectroniccopiesfornoncommercialusemaybemadefreelywithoutexplicitpermissionfromtheauthor.Allotherrightsreserved.TableofContentsChapter1.VectorBundles1.BasicDefinitionsandConst

2、ructions..............1Sections3.DirectSums4.PullbackBundles5.InnerProducts7.Subbundles8.TensorProducts9.AssociatedBundles11.2.ClassifyingVectorBundles..................12TheUniversalBundle12.VectorBundlesoverSpheres16.OrientableVectorBundles21.ACellStructureonGrassma

3、nnManifolds22.Appendix:Paracompactness24.Chapter2.ComplexK-Theory1.TheFunctorK(X)........................28RingStructure31.CohomologicalProperties32.2.BottPeriodicity.........................38ClutchingFunctions38.LinearClutchingFunctions43.ConclusionoftheProof45.3.Ad

4、ams’HopfInvariantOneTheorem............47AdamsOperations50.TheSplittingPrinciple54.Chapter3.CharacteristicClasses1.Stiefel-WhitneyandChernClasses.............59AxiomsandConstruction60.Examples64.Applicationsofw1andc167.Section1.1.BasicDefinitionsandConstructions1Chapte

5、r1.VectorBundles1.BasicDefinitionsandConstructionsVectorbundlesarespecialsortsoffiberbundleswithadditionalalgebraicstruc-ture.Hereisthebasicdefinition.Anndimensionalvectorbundleisamapp:E!B−1togetherwitharealvectorspacestructureonpbforeachb2B,suchthatthefollowinglocaltriv

6、ialityconditionissatisfied:ThereisacoverofBbyopensets−1nUforeachofwhichthereexistsahomeomorphismh:pU!URtaking−1npbtofbgRbyavectorspaceisomorphismforeachb2U.Suchanhiscalledalocaltrivializationofthevectorbundle.ThespaceBiscalledthebasespace,−1Eisthetotalspace,andthevec

7、torspacespbarethefibers.Oftenoneabbrevi-atesterminologybyjustcallingthevectorbundleE,lettingtherestofthedatabeimplicit.WecouldequallywelltakeCinplaceofRasthescalarfieldhere,obtainingthenotionofacomplexvectorbundle.Ifwemodifythedefinitionbydroppingallreferencestovectorspa

8、cesandreplacenRbyanarbitraryspaceF,thenwehavethedefinitionofafiberbundle:amapp:E!BsuchthatthereisacoverofBbyopensetsUforeachof