Geometric Topology- Localization, Periodicity and Galois Symmetry

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1、GeometricTopologyLocalization,Periodicity,andGaloisSymmetry(The1970MITnotes)byDennisSullivanEditedbyAndrewRanickiiiDennisSullivanDepartmentofMathematicsTheGraduateCenterCityUniversityofNewYork3655thAveNewYork,NY10016-4309USAemail:dsullivan@gc.cuny.eduMathematicsDepartmentStonyBrookUniversity

2、StonyBrook,NY11794-3651USAemail:dennis@math.sunysb.eduAndrewRanickiSchoolofMathematicsUniversityofEdinburghKing'sBuildingsMay¯eldRoadEdinburghEH93JZScotland,UKemail:a.ranicki@ed.ac.ukContentsEDITOR'SPREFACEviiPREFACEix1.ALGEBRAICCONSTRUCTIONS12.HOMOTOPYTHEORETICALLOCALIZATION313.COMPLETIONSI

3、NHOMOTOPYTHEORY514.SPHERICALFIBRATIONS895.ALGEBRAICGEOMETRY1136.THEGALOISGROUPINGEOMETRICTOPOLOGY187REFERENCES241GALOISSYMMETRYINMANIFOLDTHEORYATTHEPRIMESReprintfromProc.1970NiceICM251POSTSCRIPT(2004)261vEditor'sPrefaceTheseminal`MITnotes'ofDennisSullivanwereissuedinJune1970andwerewidelycirc

4、ulatedatthetime.Thenoteshadama-jorin°uenceonthedevelopmentofbothalgebraicandgeometrictopology,pioneeringthelocalizationandcompletionofspacesinhomotopytheory,includingp-local,pro¯niteandrationalhomotopytheory,lead-ingtothesolutionoftheAdamsconjectureontherelationshipbetweenvectorbundlesandsph

5、erical¯brations,theformulationofthe`Sullivanconjecture'onthecontractibilityofthespaceofmapsfromtheclassifyingspaceofa¯nitegrouptoa¯nitedimensionalCWcomplex,theactionoftheGaloisgroupoverQofthealgebraicclosureQeofQonsmoothmanifoldstructuresinpro¯nitehomotopytheory,theK-theoryorientationofPLman

6、ifoldsandbundles.SomeofthismaterialhasbeenalreadypublishedbySullivanhim-self:inanarticle1intheProceedingsofthe1970NiceICM,andinthe1974AnnalsofMathematicspapersGeneticsofhomotopytheoryandtheAdamsconjectureandThetransversalitycharacter-isticclassandlinkingcyclesinsurgerytheory2.Manyoftheideaso

7、riginatinginthenoteshavebeenthestartingpointofsubsequent1reprintedattheendofthisvolume2jointwithJohnMorganviiviiidevelopments3.However,thetextitselfretainsaunique°avourofitstime,andoftherangeofSullivan'sideas.AsWallwroteinsec-tion17FSullivan'sresul