Introduction to Characteristic Classes

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1、UNIVERSITYOFCOPENHAGENFacultyofScienceDepartmentofMathematicalSciencesMauricioEstebanGomezLopezIntroductiontoCharacteristicClassesSupervisors:JesperMichaelMller,RyszardNest1AbstractThisprojectrepresentsanintroductiontothetheoryofcharacteristicclassesofrealandcomplexvectorbundles,thatis,cohomolo

2、gyclasseswhichareassignednaturallytothebasespacesofvectorbundlesandwhichencodeinformationofsuchbundles.Beginningintherstchapterwithasummaryofthebasicsofvectorbundles,theprojectthenturnsinchapter2totheclassicationofvectorbundlesonaparacompactspace,anecessaryresulttodenethenotionofacharacteristic

3、class.Oncewehaveproventhisresult,weinvestchapters3and4toasystematicstudyoftwoimportanttypesofcharacteristicclassesassociatedtorealvectorbundles,namely,theStiefel-WhitneyclassesandtheEulerclass.Thelastchaptertreatsmainlycharacteristicclassesofcomplexvectorbundles.Inparticular,inthischapterwedescrib

4、eawayofconstructingtheChernclasseswhichwillalsoyieldusanalternateconstructionoftheStiefel-Whitneyclasses.ThischapterandthewholeoftheprojectconcludeswiththedescriptionofthecohomologyringsoftheinnitecomplexandrealGrassmannianmanifoldsusingthecoecientsZandZ2respectively,andwithabriefintroductiontoP

5、ontrjaginclasses.ThroughoutthisprojectallspacesareassumedtobeHausdor.Contents1BasicsaboutVectorBundles31.1DenitionsandBasicExamples...................31.2ConstructionofVectorBundles...................51.3ContinuousFunctors.........................61.4HermitianandRiemannianMetrics................

6、.71.5SpecicsaboutComplexVectorBundles..............71.6BasicsaboutParacompactSpaces..................82ClassicationofVectorbundles92.1TheHomotopyTheorem.......................92.2GrassmannManifoldsandUniversalBundles...........112.3ClassicationofVectorBundles...................163Characteristic

7、ClassesofVectorBundles183.1DenitionofCharacteristicClasses.................183.2Stiefel-WhitneyClasses.......................193.3DenitionofTheStiefel-WhitneyClasses.............203.4VericationoftheAxioms.....