combinatorial homotopy 1(Whitehead).pdf

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1、COMBINATORIALHOMOTOPY.IJ.H.C.WHITEHEAD1.Introduction.Thisisthefirstofaseriesofpapers,whoseaimistoclarifythetheoryof"nuclei"and"w-groups"anditsrelationtoReidemeister's1Überlagerungen.Herewegiveanewdefinitionof"^-groups,"orn-typesaswenowproposetocallthem.

2、Thisisstatedintermsof(»—l)-homotopytypes,whichwereintroducedbyR.H.Fox.2Inalaterpaperweshallshowthatthisisequivalenttothedefinitionintermsofelementarytransformations,whichwasgivenin[l].Theseriesofw-types(w=l,2,•••)isahierarchyofhomot-opy,andafortioriofto

3、pologicalinvariants.Thatistosay,iftwocomplexes,3K,L,areofthesamew-type,thentheyareofthesamera-typeforanym

4、peif,andonlyif,theirfundamentalgroupsareisomorphic.Moreoverany(discrete)groupisisomorphictothefundamentalgroupofasuitablyconstructedcomplex.Thereforetheclassificationofcomplexesaccordingtotheir2-typesisequivalenttotheclassifica•tionofgroupsbytherelation

5、ofisomorphism.Thusthew-type(n>2)isanaturalgeneralizationofageometricalequivalentofanabstractgroup.4Followingupthisideawelookforapurelyalgebraicequivalentofanw-typewhenn>2.Animportantrequirementforsuchanalgebraicsystemis"realizability,"intwosenses.Inthef

6、irstinstancethismeansthatthereisacomplexwhichisintheappropriaterelationtoagivenoneofthesealgebraicsystems,justasthereisacomplexwhosefunda•mentalgroupisisomorphictoagivengroup.Thesecondkind,whoseimportanceisunderlinedbytheoremsin[5;6]andinthispaper,isthe

7、"realizability"ofhomomorphisms,chainmappings,etc.,bymapsofthecorrespondingcomplexes.Thusrealizabilitymeansthatthealgebraicrepresentationisnotsubjecttoconditionswhichcanonlybeexpressedgeometrically.AnaddressdeliveredbeforethePrincetonMeetingoftheSocietyo

8、nNovember2,1946,byinvitationoftheCommitteetoSelectHourSpeakersforEasternSec•tionalMeetings;receivedbytheeditorsJuly19,1948.1See[l],[3]and[8,p.177],Numbersinbracketsrefertothereferencescitedattheendofthepaper.2See[9,p.343]and[10,p