Bott Periodicity(Matthew Pancia).pdf

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1、BOTTPERIODICITYMATTHEWPANCIA1INTRODUCTIONANDSOMEHISTORYConsiderthegroupsO(n),U(n).WehavefibrationsO(n)!O(n+1)!SnandU(n)!U(n+1)!S2n+1comingfromtheactionofO(n+1)andU(n+1)onRn+1andR2n+2respectivelywiththepointstabilizersO(n)andU(n).WecanalsothinkofthisasarisingfromtheinclusionsofO(n)andU(

2、n)inO(n+1)andU(n+1)astheuppercorner.Fromtheseinclusions,wecandefineO=colimO(n)U=colimU(n).Thelongexactsequenceofafibration(inhomotopy)impliesthatthehomotopygroupsofO(n)andU(n)stabilize,andsowecandefinethehomotopygroupspiOandpiU.Becausetherearesphereswrappedupinthesegroups,itwasexpectedth

3、atitwouldbedifficulttocalculatethesehomotopygroupsthestablehomotopygroupsofspheresarenotoriouslydifficulttocompute.However,thisturnsoutnottobethecase,andwehavethefollowingtheorem:Theorem1.1(Bott,1959).Thehomotopygroupsoftheclassicalgroupsareperiodic:piU=pi+2UpiO=pi+8O.Infact:Ω2U≃UΩ8O≃O.

4、Whyisthisimportant?Ayearlater,GrothendieckinventsalgebraicK-theory,whichinspiresAtiyahandHirzebruchtoinventtopologicalK-theory.BottPeriodicityisrequiredtoshowthattopologicalK-theorydefinesageneralizedcohomologytheory,whichisim-portant.ThereistheJhomomorphismthatisamapfromthehomotopyg

5、roupsofSOtothestablehomotopygroupsofspheres.TheanalysisoftheimageoftheJhomomor-phismismadeeasierbyBottPeriodicity,asitimpliesthatthereareonlyseveralcasestocheck.Itisimpressive,andthetechniquesareinteresting!Date:October19,2011.12OUTLINEOFPROOFTheprooftechniquethatBottusesisveryintere

6、sting,asitisdifferential-geometric.ItusesMorsetheory,whichhadbeendevelopedtwentyyearsearlierbyMarstonMorse.Thistheoryallowsonetostudythetopologicalpropertiesofmanifoldsbylookingatthepropertiesofnicereal-valuedfunctionsdefinedonthem.Oneofthethingsthatcanbedonewiththistheoryisbuildingupa

7、CWdecompositionofamanifoldbylookingatitscriticalpoints-attachingcellsthatcorrespondtotheindicesofthecriticalpointsofthefunction.Moregenerally,onecanconsidercriticalsubmanifolds,andnotjustpoints.Bott’sideawastousethisnotiontoanalyzethepathspaceofaRiemannianmanifold.KeyIdea:Wehavethatth

8、ereis