vector bundles and k-theory - a. hatcher

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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.ACellStructureonGrassmannManifolds

3、22.Appendix:Paracompactness24.Chapter2.ComplexK-Theory2.1.TheFunctorK(X).......................28RingStructure31.CohomologicalProperties32.2.2.BottPeriodicity........................38ClutchingFunctions38.LinearClutchingFunctions43.ConclusionoftheProof45.2.3.A

4、dams’HopfInvariantOneTheorem...........48AdamsOperations51.TheSplittingPrinciple55.2.4.FurtherCalculations.....................61TheThomIsomorphism61.Chapter3.CharacteristicClasses3.1.Stiefel-WhitneyandChernClasses............63AxiomsandConstruction64.Cohomolo

5、gyofGrassmannians69.Applicationsofw1andc172.3.2.TheChernCharacter.....................73TheJ–Homomorphism76.3.3.EulerandPontryaginClasses................83TheEulerClass87.PontryaginClasses90.1.BasicDefinitionsandConstructionsVectorbundlesarespecialsortsoffiberbu

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

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

8、evectorbundleE,lettingtherestofthedatabeimplicit.WecouldequallywelltakeCinplaceofRasthescalarfieldhere,obtainingthenotionofacomplexvectorbundle.Ifwemodifythedefinitionbydroppingallre