Transcription of paper.' - pnas.org
1 , "'Itisprovedbyconstructingan"abstract"ma ppingcylinderofXandtran-scribingintoalge braictermstheproofoftheanalogoustheoremo nCW-complexes.*Thisnotearosefromconsulta tionsduringthetenureofaJohnSimonGuggenhe imMemorialFellowshipbyMacLane.'Whitehead , ,"CombinatorialHomotopyIandII," ,55,214-245and453-496(1949).Werefertothe sepapersasCHIandCHII,respectively.'Byaco mplexweshallmeanaconnectedCWcomplex,asde finedin ,S.,"CohomologyTheoryinAbstractGroupsIII ," ,50,736-761(1949),referredtoasCTIII.'An( unpublished) ( )thestrongerhypothesisthatXBcontainsthec enterofA,butalltherelevantdevelopmentsth ereapplyundertheweakerassumption( ).
2 IEilenberg,S.,andMacLane,S.,"CohomologyT heoryinAbstractGroupsII," ,48,326-341(1947).8 Eilenberg,S.,andMacLane,S.," ,"thesePROCEEDINGS,32,277-280(1946).9 Blakers, ,"SomeRelationsBetweenHomologyandHomotop yGroups," ,49,428-461(1948), ,requiringthatv-(1)notbecyclic,canberead ilyrealizedbysuitablechoiceofthefreegrou pX,butthishypothesisisnotneededhere( ).11 Eilenberg,S.,andMacLane,S.,"HomologyofSp aceswithOperatorsII," ,65,49-99(1949); (k)hereistheC(K) (N) 'K'-K", , ,"SimpleHomotopyTypes."IfW=1,Theorem5fol lowsfrom(17:3) ,AlgebraicTopology,(NewYork,1942)andargu -mentsin ,"OnSimplyConnected4-DimensionalPolyhedr a"( ,22,48-92(1949)).
3 Howeverthisproofcannotbegeneralizedtothe caseW# ,JR.* ,November16,1949 Onemaydefineaconceptofann-persongameinwh icheachplayerhasafinitesetofpurestrategi esandinwhichadefinitesetofpaymentstothen playerscorrespondstoeachn-tupleofpurestr ategies, , ,thepay-offfunctionsaretheexpecta-tionso ftheplayers, ,oneforeachplayer, :-suchn-tuplecountersanotherifthestrateg yofeachplayerinthecounteringn-tupleyield sthehighestobtainableexpectationforitspl ayeragainst, :ifPi,P2,..andQi,Q2,..Qn,.. ,PnPandQ,,countersP,, ,weinferfromKakutani'stheorem'thatthemap pinghasafixedpoint( ,pointcontainedinitsimage).
4 "maintheorem"2andtheexistenceof, ,butthisneednotoccuringeneral.* ' 'Kakutani,S., ,8,457-459(1941).2 VonNeumann,J.,andMorgenstern,O.,TheTheor yofGamesandEconomicBe-haviour, ,PrincetonUniversityPress,Princeton, 'SNOTE"INEQUALITIESBETWEENTHETWOKINDSOFE IGENVALUESOFALINEARTRANSFORMATION"*ByGEO RGEPOLYADEPARTMENTOFMATHEMATICS, ,November25, (X)andconcerningtheeigenvaluesajofalinea rtransformationAandthose,Ki,ofA* , ,195049
