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1、DIFFERENTIALTOPOLOGYbyC.T.C.WallIntroductionThesenotesarebasedonaseminarheldinCambridge1960-61.Inwritingup,ithasseemeddesirabletoelaboratetheroundationsconsiderablybeyondthepointrromwhichthelecturesstarted,andthenoteshaveexpandedaccordingly;thisisonlytherirstset.Itisdividedintothreepar
2、ts:0,onanalyticalroundations,I,ongeometricalroundations,and11,ontheoremsortransversalityandgeneralposition.(Noindexisincludedsincenumerationandpagina•tionarebychapters).Wehopetohavegivenath0roughtreatmentorthebasictheoremsoruseininvestigatingsmoothmanirolds;theonlyotherstomyknowledgear
3、eapaperbyJ.Cerr(Bull.Soc.Math.France89,pp.,227-380(1961))andarorthcomingbookbyS.Lang(onunboundedmaniroldsonly).Itisintendedthatsubsequentpartsorthesenotesshallbeasrollows:imbeddingsandimmersions,cobordismtheory,theh-cobordismtheorem,andsurgery;however,thisissomewhatoptimistic.Itisperha
4、psappropriatetocommenthereonarewpointswhichwereonlynoticedwhenthenotesweretypedout.Part0,4.2(theImplicitFunctionTheorem)isnotneeded;aproorcanbegivenasin1.2.2.Proorsor0,4.1and0,4.4canberoundinanygoodbookonanalysis.TheproororI,4.5iscooked:IshouldhaveextendedthemethodorproorofI,4.2.Thepro
5、ofsinI,5ofuniquenessoftubularnbdscanbeusedtogivealocalpiecing'together,andhonceproveexistencealso:thisavoidsthedifficultiesinI,6.2,andisthemethodadoptedbyCerrandLang.Ihaveusedamoredirectgeometricalconstructionbyprererence;theothermethodis,however,stronger,andremovestherestrictiontocomp
6、actsubmanifolds,thusanswering,rorexample,theproblemofI,7.1.Byanoversight,theexistencepartoftheprooforI,6.hwasomitted-itisverySimple,thereaderwilleasilysupplyitforhimself.-2-IamindebtedtoalltheCambridgetopologyresearchstudentsoflastyearforparticipatingintheseminar;inparticulartoP.Baxand
7、allfortakingnotesonthefirst6seminars9andtoSteveGerstenfordoingtherest,andforconsiderableassistanceinwritingup.NOTATIONS.etc.Weassumeknownacertainamountofanalysis,andafewtermsandresultsfromanalyticaltopology-forexample9Ifnbd"meansneighbourhood,If
