The embedding of two-spheres in the four-sphere Gluck

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1、THEEMBEDDINGOFTWO-SPHERESINTHEFOUR-SPHEREBYHERMANGLUCK1CommunicatedbyA.W.Tucker,June23,19611.Introduction.Anembeddingh,ofS2inS4willbecalledsmoothifhcanbeextendedtoanembeddingofS2XE2inS4,whereE2istheopenunitdiscintheplaneandS2isidentifiedwith52Xorigin.Theembeddinghwillbecalledsemi-lin

2、earifhisasimplicialhomeomor-phismofsomerectilinearsubdivisionoftheboundaryofa3-simplexwithasubcomplexofsomerectilinearsubdivisionoftheboundaryofa5-simplex.IfS2issemi-linearlyembeddedinS4andvisavertexofS2ythenthelinkofvonS2isasimpleclosedcurve,whilethelinkofvinS4isathree-sphere.Ifthis

3、simpleclosedcurveisunknottedinthisthree-sphere,then52issaidtobelocallyflatatv.IfS2(ZS*islocallyflatatallofitsvertices,thenitissaidtobelocallyflat.Noguchihasshownin[l,Theorem3]thateverylocallyflatsemi-linearS2in54issmoothlyembedded,andthatitsregularneighbor•hoodsarehomeomorphictoS2XE2

4、.LetS2haveaneighborhoodS2XE2inS4.LetE2'betheopendiscofradius1/2.Then^4=54-(S2XE2')isanorientablefour-manifoldwhosesingleboundarycomponentishomeomorphictoS2XS1.A4iscalledanexteriorofS2inS4.IftheboundarycomponentofA4isremoved,theremainingopenmanifoldisclearlyhomeomorphicto54—S2.Idonotk

5、nowwhethertwodifferentexteriorsofasmooth52in54arenecessarilyhomeomorphic.Inthelocallyflatsemi-linearcase,however,wecandefinetheexteriortobethecomplementinS4ofanyopenregularneighborhoodofS2.Thentheexterioriswell-defined,foronecanalwaysfindasemi-linearhomeomorphismofS*ontoitselftakingo

6、neregularneighborhoodofS2ontoanyother(see[2]).Twopairs(S4,S2)and(S4',52/)aresaidtobeequivalent(orhomeo•morphic)ifthereisahomeomorphismhof54ontoS4'whichcarriesS2ontoS2'.Wecantheninanycaseaskhowstronganinvariantofthepair(54,S2)wegetbyconsideringthetopologicaltypeofanex•teriorA*ofS2inS4

7、.Inotherwords,howmanynonequivalentsmoothembeddingsofS2in54canhaveanexteriorhomeomorphicto^4?1TheauthorisaNationalScienceFoundationPredoctoralFellowandwishestothankProfessorRalphFoxforhishelpinthepreparationofthispaper.Detailedproofsoftheresultsstatedherearecontainedintheauthor'sthesi

8、s.586THEEMBE