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1、LectureNotesinMathematicsAcollectionofinformalreportsandseminarsEditedbyA.Dold,HeidelbergandB.Eckmann,ZUrich3J.FrankAdamsDepartmentofMathematics,UniversityofManchesterStableHomotopyTheoryThirdEditionLecturesdeliveredattheUniversityofCaliforniaatBerkeley1961.NotesbyA.T.Vasque
2、z$Springer-VerlagBerlin.Heidelberg.NewYork1969NationalScienceFoundationGrant10700Allrightsreserved.NopartofthisbookmaybetranslatedorreproducedinanyformwithoutwrittenpermissionfromSpringerVerlag.©bySpringer-VerlagBerlin-Heidelberg1969LibraryofCongressCatalogCardNumber70-90867
3、PrintedinGermany.TitleNo.7323TABLEOFCONTENTSI.Introduction.................I2.Prlmaryoperatlons.(SteenrodSquares,Eilenberg-MacLanespaces,Milnor'sworkontheSteemrodalgebra.).......43.Stablehomotopytheory.(Constructionandpropertiesofacategoryofstableobjects.)..............224.A
4、pplicationsofhomologica!algebratostablehomgto~ytheory.(Spectralsequences,etc.).........385.Theoremsofperlodiclt~andapproximationinhomologicalalgebra....586.Commentsonprospectlveappllcationsof5,workinprosress,etc........69Bibliography..................7~Appeudix..............
5、.......75l)IntroductionBeforeIgetdowntothebusinessofexposition,I'dliketoofferalittlemotivation.Iwanttoshowthatthereareoneortwoplacesinhomotopytheorywherewestronglysuspectthatthereissomethingsystematic~oingon,butwherewearenotyetsurewhatthesystemis.Thefirstquestionconcernsthes
6、tableJ-homomorphlsm.Irecallthatthisisahomomorphismj:~r(S$)~~S(Sn)nlarge.r=~r+nItisofinteresttothedifferentialtopologists.SinceBott,weknowthat~r(SO)isperiodicwithperiod8:r=i23456789o,o~r(SO)=z20Z000ZZ2Z2...Ontheotherhand,wSrisnotknown,butwecanneverthelessaskaboutthebehaviorof
7、J.Thedifferentialtopologistsprove:2Theorem:Ifr=4k-i,sothat~r(S0)mZ,thenJ(~r(S0))ZmwheremisamultipleofthedenominatorofBk/~k(B~beinginthekthBernoullin~nher.)Conjecture:Theaboveresultisbestpossible,i.e.J(~r(S0))=Zmwheremisexactlythisdenominator.Statusofconjecture:Noproofinsight
8、.~pnJecture[Ifr=8kor8k+l,sothat~r(SO)=Z2,thenJ(~r(SO))=Z2.Statusofconjectur
