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Topology:AVeryShortIntroduction
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VERYSHORTINTRODUCTIONSareforanyonewantingastimulatingandaccessiblewayintoanewsubject.Theyarewrittenbyexperts,andhavebeentranslatedintomorethan45differentlanguages. Theseriesbeganin1995,andnowcoversawidevarietyoftopicsineverydiscipline.TheVSIlibrarycurrentlycontainsover600volumes—aVeryShortIntroductiontoeverythingfromPsychologyandPhilosophyofSciencetoAmericanHistoryandRelativity—andcontinuestogrowineverysubjectarea.
VeryShortIntroductionsavailablenow:
ABOLITIONISM RichardS.NewmanTHEABRAHAMICRELIGIONS CharlesL.CohenACCOUNTING ChristopherNobesADAMSMITH ChristopherJ.BerryADOLESCENCE PeterK.SmithADVERTISING WinstonFletcherAESTHETICS BenceNanayAFRICANAMERICANRELIGION EddieS.GlaudeJrAFRICANHISTORY JohnParkerandRichardRathboneAFRICANPOLITICS IanTaylorAFRICANRELIGIONS JacobK.OluponaAGEING NancyA.PachanaAGNOSTICISM RobinLePoidevinAGRICULTURE PaulBrassleyandRichardSoffeALEXANDERTHEGREAT HughBowdenALGEBRA PeterM.HigginsAMERICANCULTURALHISTORY EricAvilaAMERICANFOREIGNRELATIONS AndrewPrestonAMERICANHISTORY PaulS.BoyerAMERICANIMMIGRATION DavidA.GerberAMERICANLEGALHISTORY G.EdwardWhiteAMERICANNAVALHISTORY CraigL.SymondsAMERICANPOLITICALHISTORY DonaldCritchlowAMERICANPOLITICALPARTIESANDELECTIONSL. SandyMaiselAMERICANPOLITICS RichardM.ValellyTHEAMERICANPRESIDENCY CharlesO.JonesTHEAMERICANREVOLUTION RobertJ.AllisonAMERICANSLAVERY HeatherAndreaWilliamsTHEAMERICANWEST StephenAronAMERICANWOMEN’SHISTORY SusanWareANAESTHESIA AidanO’DonnellANALYTICPHILOSOPHY MichaelBeaneyANARCHISM ColinWardANCIENTASSYRIA KarenRadnerANCIENTEGYPT IanShawANCIENTEGYPTIANARTANDARCHITECTURE ChristinaRiggsANCIENTGREECE PaulCartledgeTHEANCIENTNEAREAST AmandaH.PodanyANCIENTPHILOSOPHY JuliaAnnasANCIENTWARFARE HarrySidebottomANGELS DavidAlbertJonesANGLICANISM MarkChapmanTHEANGLO-SAXONAGE JohnBlairANIMALBEHAVIOUR TristramD.WyattTHEANIMALKINGDOM PeterHollandANIMALRIGHTS DavidDeGraziaTHEANTARCTIC KlausDoddsANTHROPOCENE ErleC.EllisANTISEMITISM StevenBellerANXIETY DanielFreemanandJasonFreemanTHEAPOCRYPHALGOSPELS PaulFosterAPPLIEDMATHEMATICS AlainGorielyARCHAEOLOGY PaulBahnARCHITECTURE AndrewBallantyneARISTOCRACY WilliamDoyleARISTOTLE JonathanBarnesARTHISTORY DanaArnoldARTTHEORY CynthiaFreeland
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ARTIFICIALINTELLIGENCE MargaretA.BodenASIANAMERICANHISTORY MadelineY.HsuASTROBIOLOGY DavidC.CatlingASTROPHYSICS JamesBinneyATHEISM JulianBagginiTHEATMOSPHERE PaulI.PalmerAUGUSTINE HenryChadwickAUSTRALIA KennethMorganAUTISM UtaFrithAUTOBIOGRAPHY LauraMarcusTHEAVANTGARDE DavidCottingtonTHEAZTECS DavídCarrascoBABYLONIA TrevorBryceBACTERIA SebastianG.B.AmyesBANKING JohnGoddardandJohnO.S.WilsonBARTHES JonathanCullerTHEBEATS DavidSterrittBEAUTY RogerScrutonBEHAVIOURALECONOMICS MichelleBaddeleyBESTSELLERS JohnSutherlandTHEBIBLE JohnRichesBIBLICALARCHAEOLOGY EricH.ClineBIGDATA DawnE.HolmesBIOGRAPHY HermioneLeeBIOMETRICS MichaelFairhurstBLACKHOLES KatherineBlundellBLOOD ChrisCooperTHEBLUES ElijahWaldTHEBODY ChrisShillingTHEBOOKOFCOMMONPRAYER BrianCummingsTHEBOOKOFMORMON TerrylGivensBORDERS AlexanderC.DienerandJoshuaHagenTHEBRAIN MichaelO’SheaBRANDING RobertJonesTHEBRICS AndrewF.CooperTHEBRITISHCONSTITUTION MartinLoughlinTHEBRITISHEMPIRE AshleyJacksonBRITISHPOLITICS AnthonyWrightBUDDHA MichaelCarrithersBUDDHISM DamienKeownBUDDHISTETHICS DamienKeownBYZANTIUM PeterSarrisC.S.LEWIS JamesComoCALVINISM JonBalserakCANCER NicholasJamesCAPITALISM JamesFulcherCATHOLICISM GeraldO’CollinsCAUSATION StephenMumfordandRaniLillAnjumTHECELL TerenceAllenandGrahamCowlingTHECELTS BarryCunliffeCHAOS LeonardSmithCHARLESDICKENS JennyHartleyCHEMISTRY PeterAtkinsCHILDPSYCHOLOGY UshaGoswamiCHILDREN’SLITERATURE KimberleyReynoldsCHINESELITERATURE SabinaKnightCHOICETHEORY MichaelAllinghamCHRISTIANART BethWilliamsonCHRISTIANETHICS D.StephenLongCHRISTIANITY LindaWoodheadCIRCADIANRHYTHMS RussellFosterandLeonKreitzmanCITIZENSHIP RichardBellamyCIVILENGINEERING DavidMuirWoodCLASSICALLITERATURE WilliamAllanCLASSICALMYTHOLOGY HelenMoralesCLASSICS MaryBeardandJohnHendersonCLAUSEWITZ MichaelHowardCLIMATE MarkMaslinCLIMATECHANGE MarkMaslinCLINICALPSYCHOLOGY SusanLlewelynandKatieAafjes-vanDoornCOGNITIVENEUROSCIENCE RichardPassingham
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THECOLDWAR RobertMcMahonCOLONIALAMERICA AlanTaylorCOLONIALLATINAMERICANLITERATURE RolenaAdornoCOMBINATORICS RobinWilsonCOMEDY MatthewBevisCOMMUNISM LeslieHolmesCOMPARATIVELITERATURE BenHutchinsonCOMPLEXITY JohnH.HollandTHECOMPUTER DarrelInceCOMPUTERSCIENCE SubrataDasguptaCONCENTRATIONCAMPS DanStoneCONFUCIANISM DanielK.GardnerTHECONQUISTADORS MatthewRestallandFelipeFernández-ArmestoCONSCIENCE PaulStrohmCONSCIOUSNESS SusanBlackmoreCONTEMPORARYART JulianStallabrassCONTEMPORARYFICTION RobertEaglestoneCONTINENTALPHILOSOPHY SimonCritchleyCOPERNICUS OwenGingerichCORALREEFS CharlesSheppardCORPORATESOCIALRESPONSIBILITY JeremyMoonCORRUPTION LeslieHolmesCOSMOLOGY PeterColesCOUNTRYMUSIC RichardCarlinCRIMEFICTION RichardBradfordCRIMINALJUSTICE JulianV.RobertsCRIMINOLOGY TimNewburnCRITICALTHEORY StephenEricBronnerTHECRUSADES ChristopherTyermanCRYPTOGRAPHY FredPiperandSeanMurphyCRYSTALLOGRAPHY A.M.GlazerTHECULTURALREVOLUTION RichardCurtKrausDADAANDSURREALISM DavidHopkinsDANTE PeterHainsworthandDavidRobeyDARWIN JonathanHowardTHEDEADSEASCROLLS TimothyH.LimDECADENCE DavidWeirDECOLONIZATION DaneKennedyDEMOCRACY BernardCrickDEMOGRAPHY SarahHarperDEPRESSION JanScottandMaryJaneTacchiDERRIDA SimonGlendinningDESCARTES TomSorellDESERTS NickMiddletonDESIGN JohnHeskettDEVELOPMENT IanGoldinDEVELOPMENTALBIOLOGY LewisWolpertTHEDEVIL DarrenOldridgeDIASPORA KevinKennyDICTIONARIES LyndaMugglestoneDINOSAURS DavidNormanDIPLOMACY JosephM.SiracusaDOCUMENTARYFILM PatriciaAufderheideDREAMING J.AllanHobsonDRUGS LesIversenDRUIDS BarryCunliffeDYNASTY JeroenDuindamDYSLEXIA MargaretJ.SnowlingEARLYMUSIC ThomasForrestKellyTHEEARTH MartinRedfernEARTHSYSTEMSCIENCE TimLentonECONOMICS ParthaDasguptaEDUCATION GaryThomasEGYPTIANMYTH GeraldinePinchEIGHTEENTH‑CENTURYBRITAIN PaulLangfordTHEELEMENTS PhilipBallEMOTION DylanEvansEMPIRE StephenHoweENERGYSYSTEMS NickJenkinsENGELS TerrellCarverENGINEERING DavidBlockley
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THEENGLISHLANGUAGE SimonHorobinENGLISHLITERATURE JonathanBateTHEENLIGHTENMENT JohnRobertsonENTREPRENEURSHIP PaulWestheadandMikeWrightENVIRONMENTALECONOMICS StephenSmithENVIRONMENTALETHICS RobinAttfieldENVIRONMENTALLAW ElizabethFisherENVIRONMENTALPOLITICS AndrewDobsonEPICUREANISM CatherineWilsonEPIDEMIOLOGY RodolfoSaracciETHICS SimonBlackburnETHNOMUSICOLOGY TimothyRiceTHEETRUSCANS ChristopherSmithEUGENICS PhilippaLevineTHEEUROPEANUNION SimonUsherwoodandJohnPinderEUROPEANUNIONLAW AnthonyArnullEVOLUTION BrianandDeborahCharlesworthEXISTENTIALISM ThomasFlynnEXPLORATION StewartA.WeaverEXTINCTION PaulB.WignallTHEEYE MichaelLandFAIRYTALE MarinaWarnerFAMILYLAW JonathanHerringFASCISM KevinPassmoreFASHION RebeccaArnoldFEDERALISM MarkJ.RozellandClydeWilcoxFEMINISM MargaretWaltersFILM MichaelWoodFILMMUSIC KathrynKalinakFILMNOIR JamesNaremoreTHEFIRSTWORLDWAR MichaelHowardFOLKMUSIC MarkSlobinFOOD JohnKrebsFORENSICPSYCHOLOGY DavidCanterFORENSICSCIENCE JimFraserFORESTS JabouryGhazoulFOSSILS KeithThomsonFOUCAULT GaryGuttingTHEFOUNDINGFATHERS R.B.BernsteinFRACTALS KennethFalconerFREESPEECH NigelWarburtonFREEWILL ThomasPinkFREEMASONRY AndreasÖnnerforsFRENCHLITERATURE JohnD.LyonsTHEFRENCHREVOLUTION WilliamDoyleFREUD AnthonyStorrFUNDAMENTALISM MaliseRuthvenFUNGI NicholasP.MoneyTHEFUTURE JenniferM.GidleyGALAXIES JohnGribbinGALILEO StillmanDrakeGAMETHEORY KenBinmoreGANDHI BhikhuParekhGARDENHISTORY GordonCampbellGENES JonathanSlackGENIUS AndrewRobinsonGENOMICS JohnArchibaldGEOFFREYCHAUCER DavidWallaceGEOGRAPHYJOHN MatthewsandDavidHerbertGEOLOGY JanZalasiewiczGEOPHYSICS WilliamLowrieGEOPOLITICS KlausDoddsGERMANLITERATURE NicholasBoyleGERMANPHILOSOPHY AndrewBowieGLACIATION DavidJ.A.EvansGLOBALCATASTROPHES BillMcGuireGLOBALECONOMICHISTORY RobertC.AllenGLOBALIZATION ManfredStegerGOD JohnBowkerGOETHE RitchieRobertsonTHEGOTHIC NickGroom
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GOVERNANCE MarkBevirGRAVITY TimothyCliftonTHEGREATDEPRESSIONANDTHENEWDEAL EricRauchwayHABERMAS JamesGordonFinlaysonTHEHABSBURGEMPIRE MartynRadyHAPPINESS DanielM.HaybronTHEHARLEMRENAISSANCE CherylA.WallTHEHEBREWBIBLEASLITERATURE TodLinafeltHEGEL PeterSingerHEIDEGGER MichaelInwoodTHEHELLENISTICAGE PeterThonemannHEREDITY JohnWallerHERMENEUTICS JensZimmermannHERODOTUS JenniferT.RobertsHIEROGLYPHS PenelopeWilsonHINDUISM KimKnottHISTORY JohnH.ArnoldTHEHISTORYOFASTRONOMY MichaelHoskinTHEHISTORYOFCHEMISTRY WilliamH.BrockTHEHISTORYOFCHILDHOOD JamesMartenTHEHISTORYOFCINEMA GeoffreyNowell-SmithTHEHISTORYOFLIFE MichaelBentonTHEHISTORYOFMATHEMATICS JacquelineStedallTHEHISTORYOFMEDICINE WilliamBynumTHEHISTORYOFPHYSICS J.L.HeilbronTHEHISTORYOFTIME LeofrancHolford‑StrevensHIVANDAIDS AlanWhitesideHOBBES RichardTuckHOLLYWOOD PeterDecherneyTHEHOLYROMANEMPIRE JoachimWhaleyHOME MichaelAllenFoxHOMER BarbaraGraziosiHORMONES MartinLuckHUMANANATOMY LeslieKlenermanHUMANEVOLUTION BernardWoodHUMANRIGHTS AndrewClaphamHUMANISM StephenLawHUME A.J.AyerHUMOUR NoëlCarrollTHEICEAGE JamieWoodwardIDENTITY FlorianCoulmasIDEOLOGY MichaelFreedenTHEIMMUNESYSTEM PaulKlenermanINDIANCINEMA AshishRajadhyakshaINDIANPHILOSOPHY SueHamiltonTHEINDUSTRIALREVOLUTION RobertC.AllenINFECTIOUSDISEASE MartaL.WayneandBenjaminM.BolkerINFINITY IanStewartINFORMATION LucianoFloridiINNOVATION MarkDodgsonandDavidGannINTELLECTUALPROPERTY SivaVaidhyanathanINTELLIGENCE IanJ.DearyINTERNATIONALLAW VaughanLoweINTERNATIONALMIGRATION KhalidKoserINTERNATIONALRELATIONS PaulWilkinsonINTERNATIONALSECURITY ChristopherS.BrowningIRAN AliM.AnsariISLAM MaliseRuthvenISLAMICHISTORY AdamSilversteinISOTOPES RobEllamITALIANLITERATURE PeterHainsworthandDavidRobeyJESUS RichardBauckhamJEWISHHISTORY DavidN.MyersJOURNALISM IanHargreavesJUDAISM NormanSolomonJUNG AnthonyStevensKABBALAH JosephDanKAFKA RitchieRobertsonKANT RogerScrutonKEYNES RobertSkidelskyKIERKEGAARD PatrickGardiner
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KNOWLEDGE JenniferNagelTHEKORAN MichaelCookLAKES WarwickF.VincentLANDSCAPEARCHITECTURE IanH.ThompsonLANDSCAPESANDGEOMORPHOLOGY AndrewGoudieandHeatherVilesLANGUAGES StephenR.AndersonLATEANTIQUITY GillianClarkLAW RaymondWacksTHELAWSOFTHERMODYNAMICS PeterAtkinsLEADERSHIP KeithGrintLEARNING MarkHaselgroveLEIBNIZ MariaRosaAntognazzaLEOTOLSTOY LizaKnappLIBERALISM MichaelFreedenLIGHT IanWalmsleyLINCOLN AllenC.GuelzoLINGUISTICS PeterMatthewsLITERARYTHEORY JonathanCullerLOCKE JohnDunnLOGIC GrahamPriestLOVE RonalddeSousaMACHIAVELLI QuentinSkinnerMADNESS AndrewScullMAGIC OwenDaviesMAGNACARTA NicholasVincentMAGNETISM StephenBlundellMALTHUS DonaldWinchMAMMALS T.S.KempMANAGEMENT JohnHendryMAO DeliaDavinMARINEBIOLOGY PhilipV.MladenovTHEMARQUISDESADE JohnPhillipsMARTINLUTHER ScottH.HendrixMARTYRDOM JolyonMitchellMARX PeterSingerMATERIALS ChristopherHallMATHEMATICALFINANCE MarkH.A.DavisMATHEMATICS TimothyGowersMATTER GeoffCottrellTHEMEANINGOFLIFE TerryEagletonMEASUREMENT DavidHandMEDICALETHICS MichaelDunnandTonyHopeMEDICALLAW CharlesFosterMEDIEVALBRITAIN JohnGillinghamandRalphA.GriffithsMEDIEVALLITERATURE ElaineTreharneMEDIEVALPHILOSOPHY JohnMarenbonMEMORY JonathanK.FosterMETAPHYSICS StephenMumfordMETHODISM WilliamJ.AbrahamTHEMEXICANREVOLUTION AlanKnightMICHAELFARADAY FrankA.J.L.JamesMICROBIOLOGY NicholasP.MoneyMICROECONOMICS AvinashDixitMICROSCOPY TerenceAllenTHEMIDDLEAGES MiriRubinMILITARYJUSTICE EugeneR.FidellMILITARYSTRATEGY AntulioJ.EchevarriaIIMINERALS DavidVaughanMIRACLES YujinNagasawaMODERNARCHITECTURE AdamSharrMODERNART DavidCottingtonMODERNCHINA RanaMitterMODERNDRAMA KirstenE.Shepherd-BarrMODERNFRANCE VanessaR.SchwartzMODERNINDIA CraigJeffreyMODERNIRELAND SeniaPašetaMODERNITALY AnnaCentoBullMODERNJAPAN ChristopherGoto-JonesMODERNLATINAMERICANLITERATURE RobertoGonzálezEchevarríaMODERNWAR RichardEnglishMODERNISM ChristopherButler
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MOLECULARBIOLOGY AyshaDivanandJaniceA.RoydsMOLECULES PhilipBallMONASTICISM StephenJ.DavisTHEMONGOLS MorrisRossabiMOONS DavidA.RotheryMORMONISM RichardLymanBushmanMOUNTAINS MartinF.PriceMUHAMMAD JonathanA.C.BrownMULTICULTURALISM AliRattansiMULTILINGUALISM JohnC.MaherMUSIC NicholasCookMYTH RobertA.SegalNAPOLEON DavidBellTHENAPOLEONICWARS MikeRapportNATIONALISM StevenGrosbyNATIVEAMERICANLITERATURE SeanTeutonNAVIGATION JimBennettNAZIGERMANY JaneCaplanNELSONMANDELA EllekeBoehmerNEOLIBERALISM ManfredStegerandRaviRoyNETWORKS GuidoCaldarelliandMicheleCatanzaroTHENEWTESTAMENT LukeTimothyJohnsonTHENEWTESTAMENTASLITERATURE KyleKeeferNEWTON RobertIliffeNIETZSCHE MichaelTannerNINETEENTH‑CENTURYBRITAIN ChristopherHarvieandH.C.G.MatthewTHENORMANCONQUEST GeorgeGarnettNORTHAMERICANINDIANS ThedaPerdueandMichaelD.GreenNORTHERNIRELAND MarcMulhollandNOTHING FrankCloseNUCLEARPHYSICS FrankCloseNUCLEARPOWER MaxwellIrvineNUCLEARWEAPONS JosephM.SiracusaNUMBERS PeterM.HigginsNUTRITION DavidA.BenderOBJECTIVITY StephenGaukrogerOCEANS DorrikStowTHEOLDTESTAMENT MichaelD.CooganTHEORCHESTRAD. KernHolomanORGANICCHEMISTRY GrahamPatrickORGANIZATIONS MaryJoHatchORGANIZEDCRIME GeorgiosA.AntonopoulosandGeorgiosPapanicolaouORTHODOXCHRISTIANITYA. EdwardSiecienskiPAGANISM OwenDaviesPAIN RobBoddiceTHEPALESTINIAN-ISRAELICONFLICT MartinBuntonPANDEMICS ChristianW.McMillenPARTICLEPHYSICS FrankClosePAUL E.P.SandersPEACE OliverP.RichmondPENTECOSTALISM WilliamK.KayPERCEPTION BrianRogersTHEPERIODICTABLE EricR.ScerriPHILOSOPHY EdwardCraigPHILOSOPHYINTHEISLAMICWORLD PeterAdamsonPHILOSOPHYOFBIOLOGY SamirOkashaPHILOSOPHYOFLAW RaymondWacksPHILOSOPHYOFSCIENCE SamirOkashaPHILOSOPHYOFRELIGION TimBaynePHOTOGRAPHY SteveEdwardsPHYSICALCHEMISTRY PeterAtkinsPHYSICS SidneyPerkowitzPILGRIMAGE IanReaderPLAGUE PaulSlackPLANETS DavidA.RotheryPLANTS TimothyWalkerPLATETECTONICS PeterMolnarPLATO JuliaAnnasPOETRY BernardO’DonoghuePOLITICALPHILOSOPHY DavidMillerPOLITICS KennethMinogue
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POPULISM CasMuddeandCristóbalRoviraKaltwasserPOSTCOLONIALISM RobertYoungPOSTMODERNISM ChristopherButlerPOSTSTRUCTURALISM CatherineBelseyPOVERTY PhilipN.JeffersonPREHISTORY ChrisGosdenPRESOCRATICPHILOSOPHY CatherineOsbornePRIVACY RaymondWacksPROBABILITY JohnHaighPROGRESSIVISM WalterNugentPROJECTS AndrewDaviesPROTESTANTISM MarkA.NollPSYCHIATRY TomBurnsPSYCHOANALYSIS DanielPickPSYCHOLOGY GillianButlerandFredaMcManusPSYCHOLOGYOFMUSIC ElizabethHellmuthMargulisPSYCHOPATHY EssiVidingPSYCHOTHERAPY TomBurnsandEvaBurns-LundgrenPUBLICADMINISTRATION StellaZ.TheodoulouandRaviK.RoyPUBLICHEALTH VirginiaBerridgePURITANISM FrancisJ.BremerTHEQUAKERS PinkDandelionQUANTUMTHEORY JohnPolkinghorneRACISM AliRattansiRADIOACTIVITY ClaudioTunizRASTAFARI EnnisB.EdmondsREADING BelindaJackTHEREAGANREVOLUTION GilTroyREALITY JanWesterhoffTHEREFORMATION PeterMarshallRELATIVITY RussellStannardRELIGIONINAMERICA TimothyBealTHERENAISSANCE JerryBrottonRENAISSANCEART GeraldineA.JohnsonREPTILES T.S.KempREVOLUTIONS JackA.GoldstoneRHETORIC RichardToyeRISK BaruchFischhoffandJohnKadvanyRITUAL BarryStephensonRIVERS NickMiddletonROBOTICS AlanWinfieldROCKS JanZalasiewiczROMANBRITAIN PeterSalwayTHEROMANEMPIRE ChristopherKellyTHEROMANREPUBLIC DavidM.GwynnROMANTICISM MichaelFerberROUSSEAU RobertWoklerRUSSELL A.C.GraylingRUSSIANHISTORY GeoffreyHoskingRUSSIANLITERATURE CatrionaKellyTHERUSSIANREVOLUTION S.A.SmithTHESAINTS SimonYarrowSAVANNAS PeterA.FurleySCEPTICISM DuncanPritchardSCHIZOPHRENIA ChrisFrithandEveJohnstoneSCHOPENHAUER ChristopherJanawaySCIENCEANDRELIGION ThomasDixonSCIENCEFICTION DavidSeedTHESCIENTIFICREVOLUTION LawrenceM.PrincipeSCOTLAND RabHoustonSECULARISM AndrewCopsonSEXUALSELECTION MarleneZukandLeighW.SimmonsSEXUALITY VéroniqueMottierSHAKESPEARE’SCOMEDIES BartvanEsSHAKESPEARE’SSONNETSANDPOEMS JonathanF.S.PostSHAKESPEARE’STRAGEDIES StanleyWellsSIKHISM EleanorNesbittTHESILKROAD JamesA.MillwardSLANG JonathonGreenSLEEP StevenW.LockleyandRussellG.FosterSOCIALANDCULTURALANTHROPOLOGY JohnMonaghanandPeterJust
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SOCIALPSYCHOLOGY RichardJ.CrispSOCIALWORK SallyHollandandJonathanScourfieldSOCIALISM MichaelNewmanSOCIOLINGUISTICS JohnEdwardsSOCIOLOGY SteveBruceSOCRATES C.C.W.TaylorSOUND MikeGoldsmithSOUTHEASTASIA JamesR.RushTHESOVIETUNION StephenLovellTHESPANISHCIVILWAR HelenGrahamSPANISHLITERATURE JoLabanyiSPINOZA RogerScrutonSPIRITUALITY PhilipSheldrakeSPORT MikeCroninSTARS AndrewKingStatisticsDavidJ. HandSTEMCELLS JonathanSlackSTOICISM BradInwoodSTRUCTURALENGINEERING DavidBlockleySTUARTBRITAIN JohnMorrillSUPERCONDUCTIVITY StephenBlundellSYMMETRY IanStewartSYNAESTHESIA JuliaSimnerSYNTHETICBIOLOGY JamieA.DaviesTAXATION StephenSmithTEETH PeterS.UngarTELESCOPES GeoffCottrellTERRORISM CharlesTownshendTHEATRE MarvinCarlsonTHEOLOGY DavidF.FordTHINKINGANDREASONING JonathanStB.T.EvansTHOMASAQUINAS FergusKerrTHOUGHT TimBayneTIBETANBUDDHISM MatthewT.KapsteinTIDES DavidGeorgeBowersandEmyrMartynRobertsTOCQUEVILLE HarveyC.MansfieldTOPOLOGY RichardEarlTRAGEDY AdrianPooleTRANSLATION MatthewReynoldsTHETREATYOFVERSAILLES MichaelS.NeibergTHETROJANWAR EricH.ClineTRUST KatherineHawleyTHETUDORS JohnGuyTWENTIETH‑CENTURYBRITAIN KennethO.MorganTYPOGRAPHY PaulLunaTHEUNITEDNATIONS JussiM.HanhimäkiUNIVERSITIESANDCOLLEGES DavidPalfreymanandPaulTempleTHEU.S.CONGRESS DonaldA.RitchieTHEU.S.CONSTITUTION DavidJ.BodenhamerTHEU.S.SUPREMECOURT LindaGreenhouseUTILITARIANISM KatarzynadeLazari-RadekandPeterSingerUTOPIANISM LymanTowerSargentVETERINARYSCIENCE JamesYeatesTHEVIKINGS JulianD.RichardsVIRUSES DorothyH.CrawfordVOLTAIRE NicholasCronkWARANDTECHNOLOGY AlexRolandWATER JohnFinneyWAVES MikeGoldsmithWEATHER StormDunlopTHEWELFARESTATE DavidGarlandWILLIAMSHAKESPEARE StanleyWellsWITCHCRAFT MalcolmGaskillWITTGENSTEINA.C. GraylingWORK StephenFinemanWORLDMUSIC PhilipBohlmanTHEWORLDTRADEORGANIZATION AmritaNarlikarWORLDWARII GerhardL.WeinbergWRITINGANDSCRIPT AndrewRobinsonZIONISM MichaelStanislawski
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Availablesoon:KOREA MichaelJ.SethTRIGONOMETRY GlenVanBrummelenTHESUN PhilipJudgeAERIALWARFARE FrankLedwidgeRENEWABLEENERGY NickJelley
Formoreinformationvisitourwebsitewww.oup.com/vsi/
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RichardEarl
TOPOLOGYAVeryShortIntroduction
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GreatClarendonStreet,Oxford,OX26DP,UnitedKingdomOxfordUniversityPressisadepartmentoftheUniversityofOxford.ItfurtherstheUniversity’sobjectiveofexcellencein
research,scholarship,andeducationbypublishingworldwide.OxfordisaregisteredtrademarkofOxfordUniversityPressintheUKandincertainothercountries
©RichardEarl2019Themoralrightsoftheauthorhavebeenasserted
Firsteditionpublishedin2019Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthepriorpermissioninwritingofOxfordUniversityPress,orasexpresslypermittedbylaw,by
licenceorundertermsagreedwiththeappropriatereprographicsrightsorganization.EnquiriesconcerningreproductionoutsidethescopeoftheaboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,attheaddressabove
YoumustnotcirculatethisworkinanyotherformandyoumustimposethissameconditiononanyacquirerPublishedintheUnitedStatesofAmericabyOxfordUniversityPress198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
BritishLibraryCataloguinginPublicationDataDataavailable
LibraryofCongressControlNumber:2019949429ISBN978–0–19–883268–3
ebookISBN978–0–19–256899–1PrintedinGreatBritainbyAshfordColourPressLtd,Gosport,Hampshire
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithandforinformationonly.Oxforddisclaimsanyresponsibilityforthematerialscontainedinanythirdpartywebsitereferencedinthiswork.
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InmemoryofDanLunn.
Friend,colleague,tutor.
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Contents
Acknowledgements
Listofillustrations
Whatistopology?
Makingsurfaces
Thinkingcontinuously
Theplaneandotherspaces
Flavoursoftopology
Unknotorknottobe?
Epilogue
Appendix
Historicaltimeline
Furtherreading
Index
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Acknowledgements
ThanksgotoMartinGalpin,AndyKrasun,NatalieLane,MarcLackenby,KevinMcGertyfortheircommentsonandhelpwithdraftchapters.ThanksespeciallytoMarcforhisencouragementtowritethebook.
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Listofillustrations
Londonundergroundmaps(a)ArchivePL/AlamyStockPhoto(b),©TfLfromtheLondonTransportMuseumcollection.
Examplesofpolyhedra
ManipulationsofacubetofinditsEulernumber
ThePlatonicsolids
MattParkerandhisfootballMattParker/YouTube.
Diagonalsinasquaremustintersect
Twonon-planargraphs
ComplicatedJordancurves(a)GetDrawings(b)DavidEppstein/WikimediaCommons/PublicDomain.
OriginalfigurefromFlatlandshowinghowASphereisperceivedbyASquareTheHistoryCollection/AlamyStockPhoto.
Theunknotandthetrefoil
Thetorus
Makingacylinder
Makingatorus
Gluinginstructionsforatriangle,pentagon,andsquare
Validandinvalidsubdivisionsofasphere
Connectedsumswithtori
TheMöbiusstrip(b)DottedYeti/Shutterstock.com.
MovinganorientedlooparoundaMöbiusstrip
TheKleinbottleandprojectiveplane
Thereallineandcomplexplane
VisualizingRiemannsurfaces
Distance,speed,andaccelerationonajourney
Examplesofgraphs
Continuousanddiscontinuousfunctions
Therigorousdefinitionofcontinuousanddiscontinuous
Theintermediatevaluetheorem
Theblancmangefunction
Visualizingdifferentmetrics
Graphsoffunctionsoftwovariables
Openballsintheplane
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Opensetsintheplane
Visualizingconvergence
Relatingtoadisconnectedsubspace
Relatingtopath-connectedness
Examplesofaminimum,maximum,andsaddlepoint
Criticalpointsoffunctionsonatorusandsphere
Vectorfieldsonthesphereandtorus
Examplescalculatingindicesofvectorfields
Loopsonatorus
Functionsonthedisc
Theunknotandtrefoils
ThethreeReidemeistermoves
Thegrannyandreefknots
Primeknotswithsevenorfewercrossings
Theknotgroupofatrefoil
Linksinvolvedintheskeinrelation
Simpleexamplesoflinks
CalculatingtheAlexanderpolynomialofatrefoil
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Chapter1Whatistopology?
Asyoureadthis,passengerstheworldoveraretravellingonmetro(orsubwayorunderground)trains.Therearearound60billionindividualjourneysmadeannuallyonsuchmetrosystems.ButwhetherthisbeinTokyo,London,SãoPaolo,NewYork,Shanghai,Paris,Cairo,Moscow,thosetravellersareperusingmapsfortheirjourneysthatarecruciallydifferentfrommapsinatlasesorseenongeographyclassroomwalls.Foremostinthemindsofthosepassengersaretheconnectionstheyneedtomake—gettingoutattherightstationandchangingtothecorrectnewline.Theyarenotinterestedinwhetherthemap’sleft–rightlinesdoindeedrunwest–east,orwhethertheyreallydidmakearightangleturnwhentheychangedlines,asdepictedonthemetromap.
TheoldestmetronetworkintheworldistheLondonUnderground.Whenfirstproduced,theundergroundmapssuperimposedthedifferenttrainlinesontoanactual(geographicallyaccurate)mapofLondon,asshowninFigure1(a).AfirstversionofthecurrentmapwasdesignedbyHarryBeckin1931asinFigure1(b).Beck’smap,andthecurrentundergroundmap,arenotwrong.Rathertheytransparentlyshowinformationimportanttotravellers—forexample,thevariousconnectionsbetweenlinesandthenumberofstopsbetweenstations.Itisanearlyexampleofatopologicalmapanddemonstratesthedifferentfocusoftopology—whichisallaboutshape,connection,relativeposition—comparedwiththatofgeometry(orgeography)whichisaboutmorerigidnotionssuchasdistance,angle,andarea.
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1.Londonundergroundmaps(a)Geographicallyaccurate1908map,(b)Beck’stopological1931map.
Topologyisnowamajorareaofmodernmathematics,soyoumaybesurprisedtolearnthatanappreciationoftopologycamelateinthehistoryofmathematics.Thewordtopology—meaning‘thestudyofplace’—wasn’tevencoineduntil1836.(‘Geometry’bycomparisonisanancientGreekwordand‘algebra’isanArabicword,withitsmathematicalmeaningdatingbacktothe9thcentury.)Justwhythiswasthecaseisnotasimplequestiontoaddress,thoughwewillseesomeaspectsoftopologydevelopedasmathematicianssoughttoputtheirsubjectonamorerigorousfooting.Topologyisahighlyvisualsubjectthatlendsitselftoaninformaltreatmentandthisbookwillgiveyouasenseoftopology’sideasanditstechnicalvocabulary.
Atopologist’salphabetAsafirstexample,toconveyhowdifferentlytopologistsandgeometersseeobjects,considerwhatcapitallettersatopologistwoulddeemtobethe‘same’.Usingthesansseriffont,thefourletters
EFTY
arealltopologicallythesame.Theyarenotcongruent,meaningthatnoneoftheletterscanbe
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pickedup,androtatedorreflected,andthenputdownasoneoftheotherletters.ButIhopeyoucanenvisage,ifallowedtobend,stretch,orshrinktheletters,howanyofthemmightbetransformedintooneoftheothers.
Toatopologistthesefourlettersarehomeomorphictooneanother.ThegeometerwouldnoticethattheanglemadebythearmsoftheYisdifferentfromanyanglefoundintheotherletters.Thetopologist,ontheotherhand,wouldbehappytoflattenthearmsoftheY,andstretchitsbodyalittle,togivetheTshape.LikewisetheEcouldhaveitsbottomrungbentaroundtothevertical,andthenshortenedsomewhat,tomaketheF.FinallydoingthesametothetopoftheFwouldmakeaTonitsside.Thesefourletterscanbecontinuouslydeformedintooneanotherandbackagain.Broadlyspeakingthisiswhatitistobehomeomorphic,tobetopologicallythesame.
Butwhatisitabouttheselettersthatmakesthemtopologicallydifferentfromotherletters?Anothercollectionoflettersthataretopologicallythesameasoneanotheris:
CGIJLMNSUVWZ.
Topologicallyalloftheseareequivalenttoalinesegmentandit’snothardtoimaginehoweachmightbeformedbybendingandstretchingasuitablymutableletterI.Sohopefullyyou’reconvincedthatthelettersinthesecondlistareallhomeomorphictooneanother,butwhatmakesthissecondcollectiontopologicallydifferentfromthefirstlist?
Note,foreachletterE,F,T,Y,thateverypointliesonadistortedbitoflinewithoneexception.IneachoftheselettersthereisasinglepointthatmightbedescribedasaT-junction.TheseT-junctionsarehighlightedbelow.
OnewayinwhichtheseT-junctionsarespecialisthat,ifremoved,theremainderoftheletterisdisconnectedintothreeparts;theremovalofanyotherpointwouldleavejusttwopartsremaining.InwhateverwayswemightbendanddeformanEthedeformedversionwouldstillincludeasingleT-junction.AsnoneofthesecondsetC…ZhassuchaT-junctionthennoneofthemcanbeadeformedversionofanE(oranF,T,orY ).
Thisgivesagenuinesenseofhowmathematiciansresolvethequestion:aretwoshapesthesametopologically?Thiseitheramountstofindingsomemeansofcontinuouslydeformingoneintotheother,orinvolvesfindingsometopologicalinvariantofonethatdoesnotapplytotheother.Thewordinvariantisusedindifferentcontextsinmathematics:forexample,ifyoushuffleapackofcards,therewillstillremainfifty-twocardsafterwardsandfoursuits,theseareinvariants;butthetopcardmayhavechangedandthejackofclubsmaynolongercomebeforetheeightofdiamonds,andsosuchfactsaren’tinvariantsofashuffle.Atopologicalinvariantissomethingimmutableaboutashape,nomatterhowwestretchanddeformit.IntheaboveexampleweusedthepresenceofaT-junctionasourtopologicalinvariant.YoumightnotethatanEincludesfourrightangleswhilstanFcontainsonlythree.ThepresenceoffourrightanglesisageometricinvariantandsoshowsthatEandFarenotcongruent(i.e.notgeometricallythesame),but—workingtopologically—wearepermittedtounbendtheserightanglesandsorightanglesarenotimportantfromatopologicalpointofview.Ratherthey’remutableaspectsofashapeandnottopologicalinvariants.
Theremainingtwenty-sixletters,groupedtopologically,breakdownas:
DO,KX,AR,B,PQ,H.
YoumightwanttotakeamomentthinkingaboutwhatmakesanAdifferentfromaPorOdifferentfromQ.Infact,theOintroducesanimportanttopologicalinvariantthatseparatesitfrombothIandE.TheshapeoftheOisdifferentasitmakesaloop.TechnicallyOisnotsimplyconnected,atopicwewilldiscussmoreinChapter5.
Euler’sformulaOneofthefirsttopologicalresultswasduetoLeonhardEuler(pronounced‘oil-er’),atitanof18th-centurymathematicsandoneofthemostprolificmathematiciansever;hisformuladatestoaround1750.Theresultrelates—atfirstglance—topolyhedra,three-dimensionalobjectssuchas
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cubesandpyramids(Figure2).Itisalsosofundamental—astraightforwardobservationatleast—thatitissurprisingancientGreekmathematiciansmissedit.
2.Examplesofpolyhedra(a)Acube,(b)ASquare-basedpyramid.
Lookingatthecube,wecanseethatitismadeupofvertices(thecornersofthecube—thesingularis‘vertex’),theseverticesbeingconnectedbyedgesandthattheseedgesthenbound(square)faces.ForthecubethenumberofverticesVequals8,thereareE=12edgesandF=6faces.Forthe(square-based)pyramidwehaveV=5,E=8,andF=5.Nopatternmaybeevidentimmediatelybutifweincludethefourotherso-calledPlatonicsolids—tetrahedron,octahedron,dodecahedron,icosahedron(Figure4)—andotherfamiliarpolyhedra,wecreateTable1.
Table1. Vertices,edges,facesforvariouspolyhedra
4.ThePlatonicsolids(a)Tetrahedron,(b)Cube,(c)Octahedron,(d)Dodecahedron,(e)Icosahedron.
(Weshallseesoonthatthetruncatedicosahedronisfamiliartous—justnotbythatname!)
ForallthegeometrythattheancientGreeksknew,itseemsstrikingthatthispatterneludedthem,butwewillprovenow—ormorehonestlysketchaproofof—Euler’sformulawhichstates,forapolyhedronwithVvertices,Eedges,andFfaces,that
Intheproofouraimwillbetobeginwithapolyhedronandmanipulateitincertainways—forexample,wemightremoveorsubdividefaces—butinallcaseswewillcarefullytracktheeffectourmanipulationhas(ifany)onthenumber .If,aftersuchmanipulations,wearrive
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••••
atasimplifiedsituationwhereweknowwhat equals,andweknowtheeffectsourmanipulationshadonthatnumber,thenwemaybeabletoworkbackwardstofindwhat
wasoriginally.
Webeginthenwithapolyhedron,andfirstremoveoneofthefaces.Thishastheeffectofreducingby1asFhasdecreasedby1.Nowthatthepolyhedronhasamissingface—
effectivelythepolyhedronhasbeenpunctured—itcanbeflattenedintotheplane,takingcarethatallthevertices,edges,facespresentonthepuncturedpolyhedronremainandareconnectedintheplaneinthesamemannertheywereonthepuncturedpolyhedron.Forexample,ifweremovedonefacefromacubeandflattenedtheremainingcubethenwewouldhavesomethinglikeFigure3(a).
3.ManipulationsofacubetofinditsEulernumber(a)Aflattened,puncturedcube,(b)Withflattenedfacestriangulated,(c)Removingatriangle,(d)Removingatriangle.
Thenextmanipulationistosubdivideeachoftheflattenedfacesintotriangles—ashasbeendonetotheflattenedcubeinFigure3(b).IntroducingasingletrianglehastheeffectofincreasingFby1—whatwasonefacebecomessplitintotwo—ofincreasingEby1—thenewedge,introducedtomakeatriangle—anddoesn’tchangeV.Sothereisnooveralleffectto aswekeepintroducingtriangles;theincreaseof1toF,atermthatisaddedintheformula,ispreciselybalancedbytheincreaseinEwhichisatermwesubtract.Whenthishasbeendoneforeachflattenedface(asinFigure3(b))then isstilljustonelessthanitwasoriginally.
Wenowremovethetrianglesoneatatime.Forexample,ifweremovethebottomtrianglefromFigure3(b)tomakeFigure3(c),thenweremoveoneedgeandonefaceand,bythesamereasoningasbefore,thishasnooveralleffecton .
Similarly,wemightthenremovetheright-mosttriangletocreateFigure3(d),themanipulationagainhavingnoeffecton .ButthebottomtriangleofFigure3(d)isconnecteddifferently.Ifweremovethattrianglethenweremove2edges,1face,and1vertex.Thealgebraisalittlemorecomplicatedthistime,butagainremoving1vertexand1facemeansgoesdown2butthisiscounteredbyremoving2edgesasEisatermwesubtract.Orifyouprefermoreformalalgebraicreasoning,wearejustsaying
Let’ssummarizewhat’shappenedsofar:
Weremovedafaceand decreasedby1.Weflattenedthepolyhedronintotheplane—allV,E,Fremain,sonochangeto .Wesubdividedtheflattenedfacesintotriangles—thishadnoeffecton .Wekeptremovingtrianglesfromtheedgeoftheflattenedpolygon—eachremovalhavingno
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effecton
Eventuallyonlyasingletrianglewillremain,havingremovedallothers.Atrianglehasasingleface,threevertices,andthreeedges,sothat equals .Thisisthevalueof thatwefinishwith.Theonlymanipulationthateverchanged wasthatveryfirstremovalofafacewhichreduceditbyone;initiallythenitwasthecasethat
This‘sketchproof’wasgivenbyAugustin-LouisCauchyin1811.It’sworthhighlightingthereareseveral‘i’sstilltobedottedtomakeaproofwithwhichaprofessionalmathematicianwouldbehappy,butalsonotinghowmuchoftheideaoftheproofisgenuinelyhere.Wedidn’ttakecaredescribinghowweremovedtrianglesfromtheboundaryoftheflattenedpolygon;ifwe’dbeencarelesswemighthaveremovedatrianglethatdisconnectedtheflattenedpolygonintotwoseparatepolygons,andweshouldhavetakentimetomakesuresuchanoccasioncanalwaysbeavoided.OtherissueswillbecomemoreapparentinChapter2butthese‘i’scanindeedbedotted.InProofsandRefutations,theHungarianphilosopherImreLakatosusedthespecificexampleofEuler’sformula,andhistoricaleffortstoproveit,tohighlighthowharditcansometimesbetogenerateawatertightproofandtoalsoraisethequestionofwhenatheoremproperlybecomespartofmathematicsorhasmathematicalcontent.
It’salsoworthmentioningthatRenéDescarteshad,overacenturybeforeEuler,demonstratedatheoremforpolyhedrathatisequivalenttoEuler’sformula;histheoremwasintermsof‘angulardefects’atvertices.Allofasuddenwearebackinthegeometricalworldandit’slessthanclearthatthereisagenuinelynewsubject,animportantlydifferentwayofmathematicalthinking,thatEuler’sfingertipswerebrushingagainst.Euler’sformulationencouragesappreciationoftheresultassomethingalittlenew—inEuler’sterminology,ratherthanDescartes’s,it’smuchclearerthattheconnectionofthevertices,edges,andfacesiswhatcounts,buthistoricallywearestillalongtimefromadeeperappreciationoftopologyasafundamentalmodeofmathematicalthinking.
TherearefivePlatonicsolidsPlatonicsolidsarepolyhedrawithregularfacesthatareallcongruent(geometricallythesame)andwhichmeetinthesamemannerateachvertex.Thereareinfinitelymanyregularpolygons—equilateraltriangles,squares,regularpentagons,etc.—butin3Ditturnsoutthattherearejustfiveregularsolidswhichhavebeenknownsinceantiquity.TheseareshowninFigure4andwithEuler’sformulawecanshowtherearejustthesefive.
ConsideraregularpolyhedronwithVvertices,Eedges,andFfaces.Asthesolidisregulartheneachfaceisboundedbythesamenumberofedges;let’scallthisnumbern.Likewisethereisacommonnumbermforhowmanyedgesmeetateachvertex.So,withthecube, (thefacesaresquares)and (threeedgesmeetateachvertex).Continuingwiththecubeasourexamplefornow,thinkabouthowwecanmakeacubebygluingtogethertheedgesofsixsquares.
Webeginwith6separatesquaressothat,beforeanygluinghappens,thereare6squares,24edges,and24vertices.Notethattomakethecubeittakestwo‘unglued’edgestomakeeachedgeofthecube(whichagreeswiththerebeing24/2=12edges)andittakesthree‘unglued’verticestomakeasinglevertexofthecube(againthereare24/3=8vertices).Moregenerally,whenwehaveFfaceseachwithnedges,wewouldhavenFedgesbeforeanygluing.Ittakestwooftheseungluededgestomakeasingleedgeofthepolyhedronwhichthenhasedges.Thereareasmanyungluedverticesasungluededges,namely ,andthesewillbegluedtogethertomake verticesonthepolyhedronasittakesmungluedverticestomakeonegluedvertexonthesolid.
PuttingtheseexpressionsforVandFintoEuler’sformulaweget
(Theequation hasbeenrearrangedtomakeFthesubjectoftheequationsothat.)Wecanthendividebothsidesoftheaboveequationby2Eandrearrangetofind
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As1/Eispositivethismeansthat
Somandncan’tbothbeverylargeasthen and wouldbeverysmallandtheirsumwouldnotexceed .Alsorecallmandnarepositivewholenumbers,sotherearen’tmanyoptionsandit’snothardtofindalltheirpossiblevalues.
It’simpossibleforbothmandntoexceed4asthen wouldbelessthan
.Soeither or or or (withperhapsmorethanoneof
thesebeingtrue).Forexample,if ,theonlynforwhichtheinequalityistrueare.If then equals orlessandtheinequalityis
nottrue.Forthethreecasesof and wehave
Asimilarcalculationfor leadsto , ,andwhen wefind ,.Inallthesefivecaseswecanusethepreviousformulastoworkoutthenumbersof
vertices andfaces .WecanputthefulldetailsintoTable2.
Table2. PossiblemandnvaluesforthePlatonicsolids
Ifweareseekingtoberigoroushere,weshouldreallypointoutthatthepreviouscalculationsshowthatthereareatmostfivepossiblepairsofvaluesthatm,ncantake.Thosecalculationslimitthepossibilities,butdonotnecessarilymeanthatthereisaPlatonicsolidforeachofthesecases,norprecludetherebeingmorethanonePlatonicsolidforpermittedmandn—itmightbethattherearetwodifferentPlatonicsolidswiththreepentagonsmeetingateachvertex.ListedinTable2arethefivePlatonicsolids,andsowecanseethatthereisatleastonesolidforpermittedm,n.Andit’snothardtoappreciatewhytherecanbeatmostone.Inthecasewhere ,
thenthreesquaresmeetateachvertex;seeminglythisonlytellsussomethingaboutpartsofthesolid,butifwefollowthisrecipeofattachingthreesquaresateachvertexthenthereisonlyonewaytoprogressbuildingupthesolid—it’snotclearthatthisrecipewillactuallyleadtoacompletesolid,butitdoesshowthattherecanbeatmostonePlatonicsolidforeachallowedm,n.
(Asanaside,youmayhavenoticedthat and , and , aresolutionsifwepermitEtobeinfinite.These‘solutions’correspondtotessellationsoftheplanewherefoursquaresmeetatavertex,wherethreeregularhexagonsmeetatavertex(aswith
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honeycombs),andwheresixequilateraltrianglesmeetatavertex.TherearealsosomepatternsapparentinTable2forthevaluesofV,E,Fforthecubeandoctahedron,andlikewisethedodecahedronandicosahedron.Thisisbecausethesesolidsaredualtooneanother—thismeansthatthemidpointsofthefacesofacubemakeanoctahedron,andviceversa;likewisethedodecahedronisdualtotheicosahedronandthetetrahedronisself-dualinthissense.)
FootballsIn2017themathematicspopularizerMattParkerbeganapetitionseekingtogetroadsignstofootballstadiacorrected.
Youmaynothavenoticedtheinaccuracyofsuchsignsinthepast,perhapsbeinghappyjusttoknowyou’retravellingtherightwayforthegame.Butit’sclear(Figure5)thatthesign’sfootballdoesnotresembletheactualfootballthatMattiscarrying.Afootball’ssurfaceismadefrompentagonsandhexagonsandtheeverydayfootballismoreformallyknownasatruncatedicosahedron.(Itcanbecreatedfromanicosahedronbyplaningdown,aroundeachvertex,thefiveedgesmeetingthere.Ifweplanedownone-thirdofeachofthosefiveedges,wecreateanewpentagonalfaceandcontinuedplaningeventuallyshrinksallthetriangularfacestohexagons.)IthinkthoughtheirksomeprincipleforMattwasnotthatthesign’sfootballwasbadlydrawn,itwasinfactimpossiblydrawn.
5.MattParkerandhisfootball.
Thereisnowaythataspherecanbemadebystitchingtogetherhexagonsasshownonthesign.Thatwouldbeanexamplewhere and ,usingthepreviousnotation,andwhilstwecancovertheplaneinthisway—whichmaybewhythesignlooksplausibleatfirstglance—makingafootballthiswayismathematicallyimpossible.
Infact,Euler’sformulashowsushowmanypentagonsandhexagonsthereareonafootball.Recallinghowtotruncateanicosahedronweseethereareasmanypentagonsasoriginalvertices(12)andhexagonsasoriginalfaces(20),butEuler’sformulacanshowthisistheonlywaytoconstructsuchafootball.SayafootballhasPpentagonalfacesandHhexagonalfaces.Then,beforegluingthesetogether,wehave5P+6Hungluededgesandthesamenumberofungluedvertices.LookingatMatt’sballwecanseethat(i)twoungluededgesareneededtomakeanedgeonthefootballand(ii)threeungluedverticesmakeavertexwith(iii)twohexagonsandonepentagonmeetingatavertex;from(iii)weseetherearetwiceasmany‘unglued’verticescollectivelyonthehexagonsasonthepentagons.So
IfweputthesevaluesintoEuler’sformula wefindthat
whichsimplifiesandrearrangestoP=12andtheequation10P=6HyieldsH=20.This,then,istheonlywaytomakeafootballifwefollowtherules(i),(ii),(iii).
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GraphtheoryLet’schangetackalittleandconsiderthefollowingproblem.ThesquarePQRSinFigure6hasdiagonallyoppositeverticesPandR,QandS.IfweweretodrawcurvesfromPtoR,andfromQtoS,curveswhichremainwithinthesquareasinFigure6,thensurelythosecurveswouldhavetocrossatleastonce.(InFigure6therearethreeintersections.)Thisseemsobvious—andistrue—buthowwouldyougoaboutprovingthis?
6.DiagonalsinasquaremustintersectCurvesPRandQSinthesquarePQRS.
Beforemoreissaid,itmightbeworthstressinghowcharacteristicofatopologicalquestionthisis.ThecurvesPRandQSneedtoconnecttheirendpoints.Thosecurvesdon’tneedtobepolygonal,orhavewell-definedgradients,orbedefinedbyspecificfunctions.Theyneedtoconnecttheendpointsinsomecontinuoussense—fullerdetailsinChapter3—buttheyareotherwisegeneralpathsfromPtoRandfromQtoSthatremaininthesquare.
Atfirstglance,thisproblemmightseemquiteremovedfromthepolyhedrawewerejustdiscussing.However,Figure6doesn’tlookthatdifferentfromFigures3(a)–(d).Wehavevertices(P,Q,R,SandanypointswherethecurvesPRandQSmeet),edgesrunningbetweenthesevertices(thoughadmittedlythey’renowcurved),andwehavefacesboundedbythoseedges.ItwascrucialtotheproofofEuler’sformulathat foreachofFigures3(a)–(d).Ifwealsoincludetheoutsideregionasaface—essentiallytheoneremovedsowecouldflattenthepolyhedron—thenwearrivebackatEuler’sformula .(Bythisreckoning ,
, inFigure6.)
Sosuppose,somehow,wecoulddrawcurvesPRandQSinthesquarePQRSwhichdon’tintersect.We’dthenfind
thefourcornersP,Q,R,S.thesquare’sfoursidesandthecurvesPR,QS.theoutsideofthesquare,abovePS,belowQR,rightofPQ,leftofSR.
Butthisleavesuswith andsosuchascenarioisimpossiblebyEuler’sformula.
Graphtheoryisanareaofmathematicsthatmodelsnetworksinawidesense:physical,biological,andsocialsystems,variouslyrepresentingtransportnetworks,computernetworks,websitestructure,evolutionofwordsacrosslanguagesandtimeinphilology,migrationsinbiology,etc.Agraphisacollectionofpointscalledvertices,withtheseverticesconnectedbyedges.Wewillalsoassumethatgraphsareconnected,meaningthatthereisawalkbetweenanytwoverticesalongtheedges.Thisdefinitionmaybeextendedtoincludeone-wayedges—directedgraphsordigraphs—andweightsmightbeintroducedtoedgesrepresentingthedifficulty—intermsoftime,distance,orcost—oftravellingalongaparticularedge.
Somegraphsareplanar,meaningthattheycanbedrawnintheplanewithouttheiredgescrossing(atpointsthataren’tvertices).ThetwographsK5andK3,3inFigure7areimportantlynotplanar.Thecompletegraphon5vertices,denotedK5,hasasingleedgebetweeneachpairofthe5verticesmaking10edges.YoumightthinkthatK5isplanarasit’sdrawninFigure7(a)—thepointisthat,sodrawn,manyoftheedges’crossingsdon’toccuratverticesandtodeemthese
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crossingsasverticeswouldmeanwewerenolongerconsideringK5whichhasonly5vertices.IfwecouldproperlydrawK5intheplanetherewouldbe10triangularfacesv1v2v3,v1v2v4,throughtov3v4v5.We’dthenhavethat equals andsoK5isnotplanar.
7.Twonon-planargraphs(a)ThecompletegraphK5,(b)ThecompletebipartitegraphK3,3.
K3,3isthecompletebipartitegraphbetweentwotriosofvertices.AsomewhatsubtlerargumentshowsK3,3isnotplanar.NotethatK3,3has verticesand edges.Ifdrawnintheplanethiswouldmean .ButafaceofK3,3wouldhaveatleastfouredgesasitsperimeternecessarilyrunsfromavtoawtoadifferentvandtoawandonlythenmayreturntotheoriginalv.So,countingtheedgesbygoingaroundallthefaces,wewouldgetatotalofatleast edges.However,asanedgecanboundatmosttwofacesthiswouldmeanwe’dhaveatleast edges,whichisourrequiredcontradiction.
ThePolishmathematicianKazimierzKuratowskiprovedin1930thatagraphisplanarpreciselywhenneitheracopyofK5norK3,3canbefoundwithinthegraph.WewillinduecourseseethatK3,3canbedrawnonothersurfacessuchasatorus(Figure13(c)).
NastysurprisesEulerarrivedathisformulaacenturybeforethewordtopologywascoined.Hisformulaischaracteristicofavisualsideoftopologynaturallyalignedwithgeometry.Buttopology,asasubject,woulddevelopalongvariousthemesandinparticularhadanimportantroleinthefoundationalworkmathematiciansweredoingaroundthestartofthe20thcentury.AsIhintedearlier,topology’srisemayhavebeenhamperedbyatraditionalmindsetthatsomeofitsquestionshadobviousanswers.ForexampleCamilleJordan,aslateasthe19thcentury,provedthefollowing:acurveintheplane,whichdoesnotcrossitselfandwhichfinishesbackwhereitbegan—acurvewhichwewouldnowcallaJordancurve—splitstheplaneintotworegions,thetechnicalphrasefortheseregionsbeingconnectedcomponents.Oneoftheseregionsisbounded,theinside,andtheotherisunbounded,theoutside,andthisistheJordancurvetheorem.Earliermathematicianswouldhavehappilythoughtthisobviousandthefirstrigorousproofdidn’tappearuntil1887.Youmayagreewiththoseearliermathematiciansthattheresultcanbesafelyassumed.MaybeeventhePollock-likeJordancurveinFigure8(a)doesnotswayyourviewoftheintuitivenessoftheresult.
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8.ComplicatedJordancurves(a)AmorecomplicatedJordancurve,(b)AJordancurvewithpositivearea.
InFigure8(b)istheKnopp–Osgoodcurvewhich,forallitsfractal-likeappearance,isaJordancurve.Astonishinglyithasapositivearea—thatisthecurveitselfhaspositivearea,we’renotreferringtosomeregionthatitbounds.Wouldyouhavesaidamomentagothatit’sobviousthatcurvescan’tthemselveshavearea?
Youshouldn’tworrytoomuchinthesensethatmostthingsthoseearlymathematiciansthoughttobetrueturnedouttobetrue,onceproperlyunderstoodandqualified,butmathematicianstowardstheendofthe19thcenturyweregettingnervousabouttherulesandassumptionsthatmathematicsreliedon.
Arelatedproblemwithintopologyatthattimewasrigorouslydefiningwhatdimensionmeans.Againthishadpreviouslybeentreatedasanintuitiveconcept,onlyformathematicianstobeginfindingspace-fillingcurvesthatpassthrougheverypointintheplaneorotherweird-and-wonderfulspacesthatcanreasonablybeassigneddimensionsthatarenotwholenumbers—spacesthatwouldnowbecalledfractals.
Anearlythemeoftopologywasthisgeneraltopologyorpoint-settopologyseekingtoaddresswhatitmeanstobeaset,tobeaspace,etc.Metricandtopologicalspaceswereintroduced—tobediscussedinChapter4—eachbeingattemptstodescribegeneralstructureswherecontinuitycouldbedefined.Settheorydealswithcollectionsthatareessentiallyjustthings-in-a-bag.Thisgeneraltopologysoughttodefinewaysinwhichobjectsmightbeconsidered‘close’toone
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another,withtheaimbeingtodefinecontinuityinabroadsetting.
AFlatlandmindsetThenovellaFlatland:ARomanceofManyDimensions,writtenin1884byEdwinAbbott,isasatireonVictorianmores.Thenarratoris‘ASquare’,aninhabitantofFlatland,aplanarworldhavingjusttwodimensions.ThecultureofFlatlandandthelogisticsoflivingintwodimensionsarefullydescribed,implicitlyhighlightingsomeofthenarrow-mindednessofVictorianculture—forexample,womenareone-ratherthantwo-dimensionalbeings.Thestorydoesn’texplicitlydiscusstopology,butinitsdescriptionofworldswithdifferentdimensionsandimplicationsfortheinhabitants,itprovidesausefulmetaphorforunderstandingcertainaspectsoftopology.
Forexample,ASquareisvisitedatonepointbyASphere.Beingathree-dimensionalobject,ASpherecanonlybeperceivedbyFlatlandersasacircularcross-section(Figure9).Bymovingupanddown—relativetoFlatland’splane—ASpherecangrow,shrink,andevendisappearentirely.Inasimilarmanner,totrulyunderstandthetopologyofaspace,wehavetobeginthinkinglikeinhabitantsofthatspace.
9.OriginalfigurefromFlatlandshowinghowASphereisperceivedbyASquare.
Topologyisoftencharacterizedasrubber-sheetgeometry.It’sasomewhatclichédmetaphor,butit’salsoslightlyinaccurate.Itgivesacorrectsenseoftopologybeingmoreaboutshapeandlessrigidthangeometryinitsfocus.Ontheotherhand,inChapter6wediscussknots,andasa(genuine)knot—likethetrefoil—andthe(unknotted)circle(Figure10)cannotbecontinuouslydeformedintooneanotherin3Dthenyoumightbetemptedtosaythecircleandtrefoilarenothomeomorphic,buttheyare.
10.Theunknotandthetrefoil(a)Theunknot,(b)Thetrefoilknot.
Theknottednessofthetrefoilsayssomethingaboutitspositionin3D.Infact,allknotsarehomeomorphictoacircle.Tobetterappreciatethis,youmightimaginelifeasanantlivingoneitherthecircleortrefoil.Astheantmovesaroundeithertheunknotortrefoilithasasenseofbeingonaloop,buttheanthasnonotionofwhetheritislivingonaknot.Itisonlybybeingabletoviewthingsfromoutsidethetwoloops,andlookingonfromapositionintheambientspace,thatweareabletorecognizeoneloopasknottedascomparedwiththeother.ThisFlatlandmindsetwillproveusefulagainlaterwhenwemeetsubspaces.
Topologywouldadvanceonvariousfrontsinthe19thand20thcenturies.Inparticular,BernhardRiemannwouldearlyonshowtheusefulnessofa‘topologicalmindset’,introducingRiemannsurfacesintothestudyofpolynomialequationsanddemonstratingsomedeepconnectionsbetweentopologyandmanyotherareasofmathematics.
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Chapter2Makingsurfaces
TheshapeofsurfacesRecallEuler’sformulastates forapolyhedron.Variousdetailsoftheproofwerebrushedunderthecarpet,themostsignificantofthesebeingtheclaimthat,onceafaceisremovedfromapolyhedron,theremainingpolyhedroncanbeflattenedintotheplane.Thiswastrueforthepolyhedrawewereconsidering,buttheclaimsayssomethingimportantabouttheshapeoftheremainingpolyhedronthatwasperhapsunintentional.Inanycase,thenextexamplewilleithermakeusquestionwhatwemeanbyapolyhedronorhaveuslookingtogeneralizeEuler’sformula.
ForFigure11(a)’s‘polyhedron’,acountofvertices,edges,andfacesshowsthat ,, ,giving whichseemstodisproveEuler’sformula.Weareleft
withafewalternatives:eithertheobjectinFigure11(a)shouldnotbeconsideredapolyhedron,orweneedtorestrictEuler’sformulatoacertaintypeofpolyhedron,orweneedtoadaptandgeneralizeEuler’sformulaintoaversionthatremainstrueforabroaderfamilyofpolyhedra.
11.Thetorus(a)Apolyhedronwithonehole,(b)Atorus.
Themostobviousissuewiththisnew‘polyhedron’istheholethroughitsmiddle.Thisisnotimmediatelyreasonenoughtoexcludeitasapolyhedron,butthisshape,onceafaceisremoved,doesnotleavearemainderthatcanbeflattenedintotheplane,makingourearlierproofinvalid.WeneedeithertorestrictEuler’sformulatopolyhedrawithoutholes,orweneedtoworkoutthecorrect valuesforpolyhedrawithholes.
Recallingtherubberynatureoftopology,wemightrecognizethatthepolyhedraofChapter1allhadthesameunderlyingsphericalshape.Ifallowedtosmoothoutthosepolyhedra—thepointyverticesandtheridgyedges—wecouldtransformeachofthosepolyhedratoasphere,coveredwithapatchworkofcurvedfaces,justlikeMattParker’sfootball(Figure5)wascoveredwithcurvedpentagonsandhexagons.Buthoweverwesmoothdownournewpolyhedronwecan’tmakeasphere,ratherwewouldmakeatorus,theshapeofadoughnutwithaholethroughit(Figure11(b)).
PerhapsthenalloftheexamplesofChapter1—includingtheproofofEuler’sformula—pointtotheEulernumberofthespherebeing2.AndFigure11(a)isafirstexamplesuggestingtheEulernumberofthetorusis0;thiswouldmeanthenumber equals0howeverwedivideupthetorus.Allthiscouldbecomequiteinvolvedunlesswehaveawayofefficientlydescribingsurfaces—includingmorecomplicatedonesthanthesphereortorus—andforsystematicallycalculatingtheirEulernumbers,thatisthe valuecommontoallsurfacesofacertainunderlyingshape.
Gluingsurfacestogether
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Ausefulwayofconstructingsurfacesistobeginwithapolygonandpairwisegluetogethertheedgesofthepolygon,thewayamodelkitmightdirectyouto‘gluetabAtotabB’.Howmightwemakeatorusinthismanner?Ifwebeginwitha(suitablyelastic)square(Figure12(a)),benditaround(Figure12(b)),andgluetheoppositeedgese1ande3sothattheverticesv1,v2getgluedrespectivelytov4,v3,andlikewiseforallotheroppositepointsofe1ande3—assignifiedbythetwoarrows—thenwewillmakethecylinderdrawninFigure12(c).Notethattheedgese2ande4ontheoriginalsquarehavebecomethetwocircularendsofthiscylinder.Wecanthengluetogetherthesecircularends;ifwedothissothattheoppositepointsoftheoriginale2ande4aregluedtogether,thenwemakeatorusasinFigure13(a).
12.Makingacylinder(a)Asquarewithidentifiededges,(b)Makingthecylinder,(c)Acylinder.
13.Makingatorus,(a)Toruswithvande1,e2drawn,(b)Squarewithgluinginstructions,(c)K3,3onthetorus.
Notethatthefourcornersoftheoriginalsquare—denotedv1,v2,v3,v4—haveallbeengluedtogethertomakeasinglepointvonthetorus.Similarly,theedgese1ande3havebeengluedtogethertoformacirclegoingaroundtheoutsideofthetorusande2ande4havebeengluedtogethertoformadifferentcirclegoingthroughtheholeofthetorus,thesetwocirclesmeetingatthepointv.
Importantly,thetorus’sshapeisfullydescribedbyasquarewithdirectionsforhowtheedgesaretobegluedtogether.Mathematicianswoulddrawthissquare-with-gluing-instructionsasshowninFigure13(b).Thesinglearrows—andimportantlythedirectionsinwhichthey’redrawn—showhowthosetwoedgesareglued,andthedoublearrows(againnotingdirections)tellushowtheotherpairofedgesisglued.
Asanaside,thegraphK3,3,whichwesawcan’tbedrawnintheplane(Figure7(b)),canbedrawnonthetorus(Figure13(c)).ThereasonK3,3canbedrawnonthetorusisbecauseatorushasalowerEulernumberof0and,arguingasbefore,itcanthenbeshownthatF=3.IfyoulookcarefullyatFigure13(c)youwillseethatthereareindeedthreefaces(quadrilateralsv1w2v2w1andv3w3v2w2andasinglehexagonalfacev1w2v3w1v2w3).
HowdoesallthishelpwithdeterminingEulernumbers?Withasinglesquareandgluingdirections,wehavebeenabletomakeanobjectwiththeshapeofatorusandthisiscertainlyaconciserdescriptionofatorusthanFigure11(a)forwhich , , .ButisthisgluedsquareenoughtoworkouttheEulernumberofatorus?Theanswerisyesifwe’recarefulwhenthinkingaboutjusthowtheoriginalverticesandedgesgluetogether.Ontheoriginalsquaretherewasjustoneface—thesquare’sinterior—fourungluededges(labellede1,e2,e3,e4),and
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fourungluedvertices(labelledv1,v2,v3,v4).However,oncewe’vefollowedthegluingdirections,thosefouredgeshavebecomethetwocircularedgesonthetorusandthefourverticeshavebecomeonepointv,asinFigure13(a).Andthereisstillone‘face’onthetorus—thesquare’sinteriorhasbeenstretchedtobecomeallofthetorusexceptthosecirclesandv.SowhenweusethisgluedsquaretocalculatetheEulernumberofthetoruswegettheanswer
whichagreeswithourcalculationfromFigure11(a).
IfyoupreferFigure13(a)showingthetoruswiththefourgluedverticesbecomingone,andthefouredgesbecometwoloopsonthetorus,thenFigure13(b)willonlyappearasanunfinishedDIYjob.Butthesinglesurfaceobtainedfromgluingthetriangle,pentagon,andsquareinFigure14,followingallthegluingdirectionsa, b,…faccordingtothearrows,mightreasonablystartstretchingyourvisualizationskills.ButwecanstillworkouttheEulernumberofthischimeraandseektounderstandjustwhatsurfacewearelookingat.Thistimetherearethreefaces(thetriangle,pentagon,andsquare)andthetwelveungluededgesofthepolygonsmakesixgluededgesa, b, c,…fonthesurface.Howmanyverticeswillweultimatelyhave?Thereweretwelveungluedverticesoriginallybutvariousofthesegetgluedtogetheraswemakethesurface.Forexample,v1andv5aregluedtogetherastheyarebothattherearendoftheedgemarkeda.Infact,wecanchasearoundthesegluingstoseejusthowmanyverticeswehave:
14.Gluinginstructionsforatriangle,pentagon,andsquare.
v1andv5areglued(rearendofa)v5andv9areglued(rearendoff )v9andv4areglued(rearendofd)v4andv12areglued(frontendoff )v12andv2areglued(rearendofc)v2andv7areglued(rearendofb)v7andv11areglued(frontendofe)v11andv1areglued(frontendofc)
Soeightdifferent(unglued)verticesv1,v2,v4,v5,v7,v9,v11,v12allgetgluedtogetherasasinglevertexonthesurface.Inasimilarfashionwecanseethattheremainingfourverticesv3,v6,v10,v8getgluedtogether(inthatorder).Sooncemade,thesurfacehas2vertices,6edges,and3facesgivinganEulernumber of .Justwhatsurfacehavewemade?
Gettingtherightanswer:subdivisionsWeneednowtomakeclearjustwhatsurfacesweareconsidering—closedsurfaces—andhowtheycanbedividedupintovertices,edges,andfaces.Aclosedsurfaceisonewithoutaboundary,suchasatorusorsphere,butnotthecylinderofFigure12(c).Ourprocessofmakingatorusbeginswithasquareandatthatpointoursurfacehasaboundaryconsistingofitsfouredges;whenwegluetwoedgestomakeacylinderthenthesurfacestillhasaboundary,namelyitstopandbottomcircles.Oncethetorushasbeenmade,noboundarypointsremainunglued.
Secondly,wecanonlycalculatethecorrectEulernumberofaclosedsurfaceifwearecarefuldividingitup.Cubes,footballs,dodecahedra,andpyramidsareallvalidwaysof‘subdividing’thesphereintovertices,edges,andfaces.IneachofthesecasesweobtainedanEulernumber
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of2.HoweverhereareotherwayswemightsubdividethespherethatseeminglyproducedifferingEulernumbers(Figure15).
15.Validandinvalidsubdivisionsofasphere(a)V=0,E=0,F=1,(b)V=0,E=1,F=2,(c)V=1,E=0,F=1,(d)V=1,E=1,F=2,(e)V=1,E=2,F=3,(f)V=2,E=0,F=1.
Inorderthevaluesof forthesixspheresinFigure15are1,1,2,2,2,3andweknowthecorrectEulernumberequals2.Soanyoldsubdivisionwillnotleadtoacorrectcalculationofthesphere’sEulernumber.Foracollectionofvertices,edges,andfacestomakeapermissiblesubdivisionthefollowingmustbetrue:
anedgemuststartandfinishinavertex;whentwoedgesmeet,theymustmeetinavertex;facesmustbe(distorted)polygons.
Lookingattheseso-calledsixsubdivisions,onlytwooftheseareinfactpermissible,15(c)and15(d).In15(a),15(e),15(f),thereisafacethatisnotadistortedpolygon;neitherthewholesphere(15(a))northepuncturedcummerbund(15(e))northetwice-puncturedsphere(15(f))aretopologicallythesameasapolygonandsonotpermissiblefaces.In15(b)and15(e),theedgesdonotbeginandendinavertex.15(e)isdeliberatelygiventoshowthatthecorrectEulernumbercanbeincorrectlycalculated.
LookingbackatthetorusinFigure13(a)andthesurfaceinFigure14,wecalculatedtheirEulernumbersusingsubdivisionsconsistentwiththeabovethreerules.Therefore,wecorrectlycalculatedtheirEulernumbersas0and–1respectively.
ConnectedsumsGiventwoclosedsurfacesS1andS2,thenwecancreatetheirconnectedsumS1#S2.Thisisawaytogluesurfacestogetherandausefulmeansofmakingnewsurfacesfromthefewwehavesofarmet.SayS1andS2eachhasasubdivisionthatincludesa‘triangular’faceboundedbythreeedges.(Theinvertedcommasherehintthat,thisbeingtopology,thefacesmaynotbethat
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recognizablytriangularintermsofhavingstraightedges.)TheconnectedsumS1#S2isthencreatedbyremovingthesetwotriangularfaces,somakingtwoholesinthesurfaces,andthengluingthetwosurfacestogetheralongtheboundariesoftheholes,pairingupthethreeverticesandthreeedgeswiththoseontheboundaryofthesecondremovedfaceas,forexample,inFigure16.
16.Connectedsumswithtori(a)Onetoruswitha“triangle”missing,(b)Atoruswithtwoholesasaconnectedsum.
HelpfullythereisaformulafortheEulernumberofS1#S2.Inmakingtheconnectedsum,weremovetwotriangularfaces,thesixdifferentverticesonthesetrianglesaregluedtomakethreeverticesontheconnectedsum,andlikewisesixedgesaregluedtomakethree.Sothetotalnumberoffaceshasgonedownby2andthetotalnumbersofedgesandverticeshaveeachgonedownby3.AsVandFareaddedintheformulafortheEulernumber,andEissubtracted,overallwehave
Or,ifyoupreferamorecarefulalgebraicproof,saytheoriginalsubdivisionofS1hasV1vertices,E1edges,andF1facesanddefineV2,E2,F2similarlyforS2.ThenumberofverticesV#,edgesE#,andfacesF#ontheconnectedsumisgivenby
Finally
ThinkingintermsofconnectedsumshelpsusworkouttheEulernumbersofsomemorecomplicatedsurfaces.Weknowthatatorus hasanEulernumberof0.Theconnectedsum
isatoruswithtwoholes(Figure16(b))andwesee
andsimilarlythetoruswiththreeholes, ,hasEulernumber
Infact,wecanseethateverytimewemakeaconnectedsumwith thesurfacegainsonemoreholeandtheEulernumberreducesby2.Sothetoruswithgholes—whichcanbeconsideredas
,theconnectedsumofgcopiesofthetorus —hasEulernumber
Thenumbergofholesinthesurface iscalledthegenusofthesurface.
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One-sidedsurfacesAtthispoint,westillcan’tidentifythepeculiarsurfacefromFigure14whichhasanEulernumberof–1.Sofarwe’veonlyconstructedsurfaceswithevenEulernumbersand–1isodd.Infact,withthetori ,we’veonlymethalfthestoryandhalfoftheclosedsurfaces.RecallhowinFigure12wemadeacylinderbygluingtwoedgesofasquare.Wecould,instead,havegluedthosetwosidesusingreversedarrows(Figure17(a)),introducingasingletwist.Sothepointsnearv2one1aregluedtothepointsnearv4one3andthosenearv1one1aregluedtothepointsnearv3one3.ThiswouldhavecreatedaMöbiusstrip,namedafterAugustMöbiuswhodiscovereditin1858.
17.TheMöbiusstrip(a)Asquarewithidentifiededges,(b)RunnersonaMöbiusstrip.
TheMöbiusstripisunusualinonlyhavingoneside—thisisapparentinFigure17(b)astherunnerscovertheentiretyofthestripratherthanjustonesideofitastheywouldifrunningaroundjusttheoutside(orinside)ofacylinder.Oryoucanimaginepaintingtheoutsideofacylinderblackandtheinsidewhite,butshouldyoubeginpaintingaMöbiusstriponecolouryouwouldfindyourselfcoveringtheentirestripinthatcolour.TheMöbiusstripisanexampleofanon-orientablesurface.Likethecylinderitisasurfacewithboundary,butnoteitsboundaryisasinglecircleratherthantwoseparateonesaswiththecylinder.
InFigures18(a)–(d)weseeanorientedloop—hereacircle—movingaroundaMöbiusstrip.ByanorientedloopImeanaloopwithagivensenseofdirection,hereinitially(18(a))appearingasclockwisetothereader.Butasthisloopmovesaroundthestrip(orequivalentlymovesleftinthesquare)weseethatwhenthecirclereturnstoitsoriginalposition(18(d))thatsensehasnowreversedandappearsanti-clockwise.Ifyouarehavingalittletroublevisualizingwhat’shappeningtotheloop,notein18(b)and18(c)howthepointslabelledParegluedtogetherandlikewisetheQs.In18(b)mostoftheloop(ontheleft)lookstobeclockwiserunningfromPtoQ,butastheloopappearsontherightandcontinuesfromQtoPthatsenseisbeginningtoappearasanti-clockwise.
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18.MovinganorientedlooparoundaMöbiusstrip.
Anysurfaceonwhichitispossibletoreversethesenseofanorientedloopiscallednon-orientable.Ifitisimpossibletoreversealoop’ssense,thenthesurfaceiscalledorientable.AnysurfacethatcontainsaMöbiusstripisnon-orientableaswecouldjustsendanorientedlooponcearoundthatstriptoreverseitssense.Asurfacewithaninsideandanoutsideisorientable.Toappreciatethis,imaginewalkingaroundtheoutsideofsuchasurface.Lookingdowntoyourfeetonthesurfaceyoucoulddrawacircleinaclockwisemanner.Asyouwanderaroundtheoutsideofthesurfaceyoucanconsistentlytakeyournotionofclockwiseacrossthewholesurface.Thismeans,inparticular,thatthetori ,whichwemetearlierandwhicheachhaveaninsideandoutside,areallexamplesoforientablesurfaces.
ReturningtoFigure17(a),apartlygluedsquaremakingaMöbiusstrip,thereremaintwoungluededgese2ande4.WecouldgluethesetogetherasinFigure19(a),butwhatsurfacewouldwemake?Certainlyanon-orientableoneasitcontainsaMöbiusstrip(theshadedregion).Ifinsteadwemakethissurfacebygluinge2ande4first,wefirstcreateacylinderwithe1ande3asitscircularends.Buttocompletethesurface,ratherthanbringingthosecircularendstogetheraswithatorus,onecircularendhastobegluedbackwardsontotheothercircularend—thisisbecauseofthereversearrowsone1ande3.Figure19(b)showshowwemighttrytodothis;wecouldtakeonecircularendbackintothecylinderandglueittotheotherendfrominside,andthiswaythereversearrowslineupproperly.ThesurfacemadeiscalledaKleinbottle,afterFelixKleinwhofirstdescribeditin1882.Beingnon-orientable,theKleinbottledoesnothaveaninsideandoutside.
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19.TheKleinbottleandprojectiveplane(a)AKleinbottle,(b)3DdepictionofaKleinBottle,(c)Aprojectiveplane.
ThereisasubtleproblemwiththeKleinbottleinFigure19(b).Whenwetakethecylinderbackintoitself,somesinglepointsinspaceactuallyrepresenttwodistinctpointsontheKleinbottle.SothisimageisnotaproperrepresentationorembeddingoftheKleinbottlein3D.Infact,itisimpossibletoconstructaKleinbottlein3Dwithoutsuchself-intersectionsasoccurwherethecylindercutsbackintoitself.TherelevantresultdemonstratingthisimpossibilitycanbeviewedasageneralizationoftheJordancurvetheorem.ThattheoremconcernedembeddingcirclesintheplanewithaJordancurvehavinganinsideandanoutside.Inalikemannerwhenaclosedsurfaceisembeddedin3D,thesurfaceagaindividestheremainingspaceintoaninsideandanoutsideandsotheclosedsurfacemustbeorientable.AstheKleinbottleisnon-orientable,itcannotbeembeddedin3D.
However,theKleinbottlecanbeembeddedin4Dandthisisn’ttoohardtoimagineifwetreatthefourthdimensionastime.TheKleinbottleistwo-dimensional(assurfacesare)andsofromthis4DviewpointitisimportanttoconsidertheKleinbottleasonlyexistingforaninstant,acertain‘now’;forittohaveapastorfuturewouldgiveitathirddimension.Sowhenfacedwithbringingthecylinderbackintoitself—whichwouldnormallycauseself-intersections—wecaninsteadmovethatbitofcylindergraduallyintothefuture(thefourthdimension),wheretheremainderoftheKleinbottledoesn’texistandthen,oncethecylinderhaspassedthroughthespaceitspresentselfoccupies,wecangraduallybringthatbitofthecylinderbackintothepresent.Theself-intersectionsnolongeroccur,asthedistinctpointsoftheKleinbottlethatbecamemergedinFigure19(b)insteadsitinthesamepointofspacebutcruciallyatdifferenttimes.
WecanalsodeterminetheEulernumberoftheKleinbottle,againbeingcarefultonotehowedgesandverticesaregluedtogether.Thesquareisouronlyface;e1ande3aregluedtogether,asaree2ande4,makingtworatherthanfouredges;finallyv1isgluedtov2whichisgluedtov4whichisgluedtov3andsowehavejustonevertex,giving ,theEulernumberoftheKleinbottle.Unfortunately,0isalsotheEulernumberofthetorus,soanyhopewemighthavehadthattheEulernumberaloneisinformationenoughtorecognizetheshapeofasurfacewassimplistic.ThetorusandKleinbottlearedifferentsurfaces—theformerisorientable(two-sided),thelatternot—andyettheybothhavethesameEulernumber.
Anotherimportantnon-orientablesurface,whichcanbeformedfromgluingasquare’sedgestogether,istheprojectiveplaneℙ.InFigure19(c)weassigne2ande4reversearrows(incontrastto19(a)).Thesurfaceformedisnon-orientable,asitagaincontainsaMöbiusstrip(theshadedregion),andwecancalculatetheEulernumberasbefore:again and butthistimev1andv3aregluedtogetherandseparatelyv2andv4areglued,sothat .HenceℙhasEulernumber .
TheclassificationtheoremClassificationisanimportantthemeinmathematics.Amathematicaltheoryoftenbeginswithdefinitionsandrulesaboutcertainmathematicalobjectsorstructures(sayfunctionsorcurves)andseekstoproveresultsaboutthemusingthoserules.It’snaturaltosearchforexamplessatisfyingthoserules,preferablyproducingacompletelistorclassificationofsuchobjects.
Wearenowclosetoclassifyingclosedsurfaces.Explicitly,weareseekingtogiveacompletelistofalltheclosedsurfaces,sothateveryclosedsurfaceishomeomorphicto(i.e.topologicallythesameas)oneofthesurfacesonthelist,andthelistcontainsnoduplicates—eachsurfaceonthe
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listcanbeshowntobetopologicallydifferentfromallothersonthelist.
ItturnsoutthattheEulernumbergoesalongwaytoseparatingoutthedifferentsurfaces,butwehaveseenthatthiscannotbethewholestoryasthetorusandKleinbottlehavethesameEulernumberwhilstbeingdifferentsurfaces—thefirstisorientable,thesecondnot.Theonlymissingingredientintheclassificationisthatnotionoforientability.
Sothefirsthalfoftheclassificationtheoremfortwo-sidedsurfacesstates:
Anorientableclosedsurfaceishomeomorphictopreciselyoneofthetori whereThesetoriarenottopologicallythesameasoneanotherastheyhavedifferent
Eulernumbers—theEulernumberof is2–2g.
Asimilarresultholdsforone-sidedclosedsurfaces.Justasthetorus isabuildingblockfortheorientablesurfaces,socantheprojectiveplaneℙbeusedtomakethenon-orientablesurfaces.RecallthattheprojectiveplaneℙhasEulernumber1.Sotheconnectedsumsℙ#ℙandℙ#ℙ#ℙhave
andmoregenerallykcopiesofℙinaconnectedsum,asurfacedenotedℙ#k,hasEulernumber2–k.
Andthesecondhalfoftheclassificationtheoremforone-sidedsurfacesstates:
Anon-orientableclosedsurfaceishomeomorphictopreciselyoneofℙ#kwhereThesesurfacesarenottopologicallythesameastheyhavedifferentEulernumbers—theEulernumberofℙ#kis2–k.
MakingaconnectedsumwithℙisequivalenttosewingaMöbiusstripintothesurface.ℙitselfcanbemadebyintroducingaMöbiusstripintoasphere;todothiswemightmakeatearinthesphereandthen,ratherthangluingthetearbacktogether,wecouldinsteadassignreversearrowstothetwosidesofthetear,thusintroducingaMöbiusstrip.Sothesurfaceℙ#kcanbethoughtofasaspherewithkMöbiusstripssewedin.
Overallthen,theclassificationtheoremsaysthatifweknowtheEulernumberofaclosedsurfaceandwhetheritisone-ortwo-sided,thenweknowitstopologicalshape.Ifyouwerewondering,wheretheKleinbottleisonthislist,weknowitsEulernumbertobe0andweknowittobeone-sided.Theonlysurfaceintheclassificationmatchingthesefactsisthe surfaceℙ#ℙandthisistopologicallythesameastheKleinbottle.Wemightcreateayetmorecomplicatedconnectedsumsuchas whichatfirstglanceisnotonourlist.Thissurfaceisone-sidedanditsEulernumberequals
sotopologicallyit’sthesamesurfaceasℙ#7.AndatlonglastweareabletoidentifythesurfaceweformedinFigure14.ThatsurfacehadEulernumber–1andsothesurfaceisℙ#ℙ#ℙ,thisbeingtheonlysurfaceonourlistwiththatEulernumber.
ComplexnumbersSurfacesareanaturaltwo-dimensionalextensionofone-dimensionalcurveswhichmathematicianshadlongbeeninterestedinbut,historically,surfacesandtheirtopologybecameofparticularimportancebecauseoftheworkof19th-centurymathematicians,mostnotablyBernhardRiemann.
TounderstandRiemann’smotivationforstudyingsurfaces,weneedtotakeabriefforayintothe
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worldofcomplexnumbers.Complexnumbershave,atfirstglance,nothingtodowithtopology,buttheneedtointroducethemhereisaconsequenceofthedeepinterconnectednessofmathematics.Inthemid-19thcenturymathematiciansfoundworthwhilereasonstothinkaboutoldermathematicsinnewtopologicalways.Itmightthenseemasthoughtopologywassomehowbornofpracticalnecessityforaddressingtheseolderproblems.HoweverI’dliketosuggestarosierpictureofhowmathematiciansthink:nothingwillputaglintintheeyesofagenerationofmathematicians,anitchtobethinkinghardabouttheessenceofmathematics,somuchasasenseoftherebeingsomethingprofoundjustaroundthecornerandadeeperunderstandingoftheirsubjecttantalizinglybeyondtheirfingertips.Andsoitwastoprove.
Theseso-called‘complex’numbersarose—somewhatuncertainly—fromtheworkofItalianmathematiciansduringtheRenaissance.Foralongtimemathematicianshadbeeninterestedinthesolutionsofpolynomialequations.Theseareequationsinvolvingpowersandmultiplesofanunknownquantity,sayx,suchas
Thisisadegree3equation,thatbeingthehighestpowerofx.Asolutionofanequationisavalueofxwhichmakesbothsidesequal.Wecanseethat solvesthisequationbecause
Youmightcheckthat isasolutionandsois .Andthat’sallofthem!Threesolutions.Otherpolynomials,though,seemtohavenosolutions.Forexample,thedegree2
equation
hasnorealnumbersassolutions.Ifyoutakeapositivenumberxthenitssquarex2isalsopositive(andsocannotequal–1);ifyoutakeanegativenumberthenitssquareisalsopositive;finally
.Sotherearenosolutions.Ifyoupreferamorepictorialapproachthenyoumightdrawthegraphsof and ,andthefactthatthesegraphsdon’tmeet(Figure20(a))isagainanotherwayofshowingthatnonumberxsolvestheequation .Basicallytheproblemisthatnegativenumbersdon’thaverealsquareroots.
20.Thereallineandcomplexplane(a)Graphsofy=x2andy=–1,(b)Therealline,(c)Thecomplexplane.
AndtherethestorymighthaveendedexceptthoseRenaissancemathematiciansfoundgoodreasonsto‘imagine’that doeshavesolutions,denotingasolutionasi.Thismayseemsomewhatludicrousatfirst,butaround1530amethodwasfoundforsolvingdegree3equations.Oneproblemwasthatthismethodnecessitatedcalculationswithsquarerootsofnegativenumbers,evenwhenalltheequation’ssolutionswererealnumbers.Theworthofthenumberibecametrulyapparentwiththeproofofthefundamentaltheoremofalgebrain1799byCarlGauss.Thistheoremshowsallthesolutionsofanypolynomialequationhavetheformwhereaandbarerealnumbers.Forexample,thenumber solvestheequation
asshownbythecalculation
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Numbersofthisform, whereaandbarerealnumbersand ,arecalledcomplexnumbersandthefundamentaltheoremofalgebrasaysthatapolynomialofdegreenhas(countingpossiblerepeats)nsolutionsamongstthecomplexnumbers.
Inthesamewaythatrealnumbersarecommonlyrepresentedontherealline(Figure20(b))thecomplexnumberscanberepresentedasaplane,thecomplexplane(Figure20(c)).Acomplexnumbersuchas canthennaturallybeidentifiedwiththepoint(1,2)asshown.Therealnumbersoccupythehorizontalaxis—denoted‘Re’—andtheverticalaxis‘Im’iscalledtheimaginaryaxis.
Complexnumbershavearichtheoryoftheirownwhich,formathematiciansatleast,isreasonenoughtowarranttheirstudy.Youmay,though,besurprisedtofindthatquantumtheory,thephysicaltheorythatsuccessfullymodelssubatomicphysics,isnaturallydescribedusingthelanguageofcomplexnumbersandsophysicists,chemists,andengineersallneedtobewellversedintheuseofcomplexnumbers.
RiemannsurfacesTheintroductionofcomplexnumbersledtoamuchrichertheoryconnectingalgebraandgeometry.InFigure20(a)weseethatthecurves and don’tmeet;iftheydidmeetatapoint(x,y)intherealxy-planethenwe’dhave andnosuchxexists.Butusingcomplexnumberstheydointersectattwopoints,at andat .Thefactthatadegree2curveandadegree1curvemeetin pointsinthiscaseisnotentirelycoincidental.Moregenerallyitisthecasethat,ifproperlycounted,adegreemcurveandadegreencurveintersectin points.Multiplecontactsneedtobecountedproperly—soalinetangentiallymeetingthecurve wouldcountasadoublecontact,sothattherearestill
intersections.Thefinalfinesse,whencountingintersections,istoincludepointsatinfinity.Forexample,twoparallellines—eachdegree1curves—arestilldeemedtomeetatapointatinfinitysothatthereis intersectionasexpected.
Usingrealnumbers,thegraphof isaone-dimensionalcurvelyinginthetwo-dimensional
xy-plane.Inthiscasexandyareeverydayrealnumbersandthecurveconsistsofallpoints(x,x2)wherexisarealnumber.Wemightinsteadconsiderthesameequationwherexandycannowbecomplexnumbers.Againallthepointssatisfying areoftheform(x,x2)butthistimexcanbeanycomplexnumber.Whenusingrealnumbers,theinputxrepresentssomepointofthex-axisandthecorrespondingoutputx2canbeplotteddistancex2abovethepoint(x,0)inthexy-plane(Figure20(a)).However,whenitcomestousingcomplexnumbers,theinput isitselftwo-dimensional.Thex-‘axis’isaversionofthecomplexplane,they-‘axis’asecondversion,andthecomplexxy-‘plane’isinfactfour-dimensional.‘Above’thepoint(x,0)isapoint(x,x2),andtogetherthepoints(x,x2)makeatwo-dimensionalsurfacesituatedinthefour-dimensionalcomplexxy-space.Allthepointssuchas(2,4)thatwereontheoriginalrealcurvearestillpresent,andmakeupacross-sectionofthecomplexsurface;presenttoonowarepointslike(i,–1)and(2+i,3+4i).Ifweseparateoutthesecomplexnumbersintotheirrealandimaginarydimensions,thenwemightinsteadrepresentthesepointsas
andtheirfour-dimensionalnatureisalittleclearer.
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Thecurve sitsintherealxy-planeasacurvedversionofthex-axis(Figure20(a));thecurveandaxisaretopologicallythesamewithahomeomorphismbetweenthetwojustpushingeachpoint(x,0)uptothepoint(x,x2).Ifweincludealsothecurve’spointatinfinitybringingtogetherthecurve’s‘ends’thenthecurvetopologicallybecomesacircle.Whenusingcomplexnumbers,thecurve sitsincomplexxy-spaceasacurvedversionofthex-axiswhich,remember,isitselfatwo-dimensionalcomplexplane.Whenweincludethepointatinfinitythisbringstogetherthiscurvedplaneasasphere.(ThisisthereverseprocessofpuncturingaspheretogettheplanethatwemetearlierintheproofofEuler’sformula.)
So,thecomplexversionof ,ifweincludeitspointatinfinity,istopologicallyasphere,asurface;thisiscalledtheRiemannsurfaceof .WemightsimilarlyconsidertheRiemannsurfacesofhigherdegreeequations.
InFigure21(a)wehavetherealcross-sectiondescribedbythedegree3equation.Ontheleftisaloop,andwhenweaddthepointatinfinitytothecurve
ontherightthentopologicallythisrealcross-sectionbecomestwoloops;soitmightnotbesurprisingthatthewholecomplexversion,theRiemannsurface,inthiscaseisatoruswithFigure21(a)justbeingacross-sectionofthattorus.InFigure21(b)wehavearealcross-sectionwithasingularpointwherethecurvecrossesitself.TopologicallythecomplexversionofthiscurveisapinchedtorusasinFigure21(c).
21.VisualizingRiemannsurfaces(a)Graphofy2=x(x–1)(x–2),(b)Graphofy2=x(x–1)2,(c)Apinchedtorus.
Providedtherearenosingularpoints,thenadegreedequationdefinesaRiemannsurfacewhichistopologicallyatoruswithgholes.Thereisaprofoundbuteasilydescribedconnectionbetweenthedegreeofacurve’sequationdandthegenusgofitsRiemannsurface.Thisisgivenbythedegree-genusformulawhichstatesthat
wheregisthegenusoftheRiemannsurfaceanddisthedegreeofthecurve’sequation.Rememberingtheexampleswehavemet,notethat for gives ,asphere,and
for gives ,atorus.Forcurveswithsingularpoints,theformulacanbegeneralizedincludingacorrectiontermforeachsingularity,asshownbyMaxNoetherin1884.
Thisisafirstglimpseatsomeofthedeepconnectionsbetweentopology,algebra,geometry,andcalculusthatwouldbeuncoveredinmathematics.QuotingtheFrenchmathematicianJeanDieudonné,‘Inthehistoryofmathematicsthetwentiethcenturywillremainasthecenturyoftopology.’Thecenturywouldseeafledglingsubject,intuitivelybutinformallyunderstood,goontobecomeoneofthecentralpillarsofmathematics.
Itisworthnotingthatthoseearlytopologists—Möbius,Klein,Riemann—didnotintheirtimehaveavailabletherigorousdefinitionsnecessarytoprovetheirresultstomodernstandards.In1861Möbiusgaveanearlysketchproofoftheclassificationtheoremfororientablesurfaces,andWaltherVonDyckgaveasketchproofforallclosedsurfacesin1888.Butwithouthavinganyformaldefinitionofwhatasurfaceis,theseproofscanatbestbeconsideredincomplete.Thisisnottorelegatesuchproofstothedustbin,nortoconsiderthemsimplywrong,assuchproofsoftencontainmostorallofthecrucialideasofaproof.Somewhatdifferentlyexpressedrigorous
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versionsoftheclassificationtheoremwouldbeprovedbyMaxDehnandPoulHeegaardin1907andbyRoyBrahanain1921.
Curvesandsurfacesareone-andtwo-dimensionalexamplesofmanifolds,spacesthatlook‘upclose’liketherealline,theplane,orsomehigherdimensionalequivalent.Itwasn’tuntil1936thatHasslerWhitneygavethemoderndefinitionofamanifold,andprovedanimportanttheoremshowingwhenmanifoldscanbeembeddinginspace(recall,forexample,howtheKleinbottlecannotbemadein3Dspacewithoutself-intersectionsbutcanbemadein4D).Animportantaspectofmoderngeometryconcernsthedifferenttypesofmathematicalstructure—continuous,smooth,complex,metric—thatcanbeputonthesemanifoldsandIwillsayalittlemoreonthisinChapter5whenwediscussdifferentialtopology.
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Chapter3Thinkingcontinuously
Givenjustonesentenceforthetask,manytopologistsmightchoosetodescribetheirsubjectasthestudyofcontinuity.Theword‘continuous’appearedafewtimesinChapter1butitwaslefttothereader’sintuitionastoquitewhatthewordentailed.Inmanywaysthisreflectshowmathematiciansusedtoregardcontinuity—historicallyitwasjustconsideredevidentwhatwasmeantby‘continuous’andasmany(butnotall)oftheresultsaboutcontinuousfunctionsthatare‘obvious’alsohappentobetrue,relativelylittleeffortwasspentprovidingfurtherclarity.Arigorousdefinitionofcontinuitydidnotappearuntilthe19thcentury.
Inyoureverydayroutinetherearecontinuousanddiscontinuousfunctionsaroundyou.Forexample,ifyoudrivetowork,thedistanceyouhavetravelledafteracertaintimewillbeacontinuousfunctionoftime—forthisnottobethecasewouldmeanthatatonemomentyourcarwasinacertainplaceonlyforittoimmediatelyafterwardsbeatanotherplacesomedistanceaway.Yourspeedonthejourneywillsimilarlybecontinuous.However,theaccelerationneednotbe;ifyouweresatatrest(sayattrafficlights)theaccelerationwouldbezerobutthenwouldjumptoacertainvalueonceyourfootwasontheaccelerator.ThegraphsinFigure22giveaplausible(ifsimplistic)modelforsomeone’sdrivetowork.
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22.Distance,speed,andaccelerationonajourney(a)Distance,(b)Speed,(c)Acceleration.
FromFigure22(b)wecanseethatthecarstopsatt3—wherethespeeds(t)becomeszero—andaftert4increasestothespeedlimit.Thedistancetravelledd(t)inFigure22(a)isacontinuousfunctionoftimet.Historicallythiswouldhavebeenunderstoodasmeaningitsgraphcouldbedrawnwithouttakingpenfrompaper,butwewillseektoprovideafullerunderstanding.Buttheaccelerationfunctiona(t)isnotcontinuousbecauseofthejumpsinthegraphinFigure22(c).Thetimest1,t2,…t6ofdiscontinuityintheaccelerationrelatetothedriver’sfootcomingofftheaccelerator,beingputonthebrake,comingoffthebrake,andthenthepatternrepeatsagain.
Myaiminthischapteristoprovideamorerigoroussenseofjustwhatcontinuityentailsforreal-valuedfunctionsofarealvariable.Thismeanswewillfocusonfunctionshavingasinglenumericalinputandasinglenumericaloutput.
Functions
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Theideaofafunctionisacentralonetomathematics,thoughthishasonlybeentruesincearoundthe17thcentury.OnceDescartesandFermatindependentlyintroducedtheideaofCartesiancoordinatesxandytodescribepositioninaplane,acurvecouldjustaseasilybedescribedbyanequationasbyitsgeometry.Forexample,thecurve isaparabola,acurvetheancientGreekswouldhaveinvestigatedsolelyusinggeometry.AsketchofthiscurveisgiveninFigure23(a)—thecurve’sequationgivesaruleforplotting,aboveeachpoint(x,0)ofthex-axis,apoint(x,x2).Notehowcertainalgebraicpropertiesofthefunctionarerepresentedintheshapeandpositionofthecurve—asx2⩾0forallx,thecurveliesentirelyonorabovethex-axis;as
,thecurveissymmetricaboutthey-axis.
23.Examplesofgraphs(a)Graphof ,(b)Graphofy=.
Forsomefunctions,wemightnaturallyhavetolimittheallowedinputs—orwemightchoosetodosoanyway.Forexample,if thenweatleastneedtoensurethatxisnon-zeroas
divisionbyzeroismeaningless;forthefunction ,thenwecannotpermitxtobenegative,asnorealnumberhasanegativesquare(Figure23(b)).
Moregenerally,afunctioncomeswithasetofinputs,knownasthedomain,andthereislikewisethecodomain,asetcontainingtheoutputs.Itisanimportant,ifsubtle,pointtoappreciatethatafunctionisthiswholepackage:thedomain,thecodomain,andtheruleassigningvalues.
SomefirstthoughtsaboutcontinuityLet’sfirsttrytounderstandwhatitmeansforafunction,withrealinputsandoutputs,tobecontinuous.Currentlywesortofintuitivelyknowcontinuitywhenweseeit.Certainly,lookingattwofunctionsinFigures24(a)and24(b),itseemsreasonabletosayf(x)iscontinuousandg(x)isnotcontinuous,andfurtherthatg(x)isdiscontinuousonlyat .(Thefulldisconthegraphshowswherethefunctiontakesitsvalue,sothat .)ButwhatdoesintuitionsayaboutFigure24(c)?Ish(x)continuousornot?Itseemsthat,ifh(x)isdiscontinuous,theonlypointofdiscontinuityis ,butthefunctionoscillatessowildlythere,wemaynowbethinkingthatourintuitiondidn’thavealltheanswers.
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24.Continuousanddiscontinuousfunctions(a)y=f (x)=sin x,(b) ,(c)
.
BacktoFigure24(b),whatisitaboutthefunction’sbehaviourat thatmakesusthinkg(x)isdiscontinuous?Forinputxalittlemorethan1,theng(x)hasmuchthesamevalueasg(1);howeverforinputxalittlelessthan1theng(x)isnoticeablydifferentfromg(1).Itisthisjumpinoutputat1thatiscrucialtog(x)beingdiscontinuousat .
Atfirst,wemightbetemptedtothinkthisisbecauseg(1)isdifferentfromthevalueofg(x)achievedimmediatelybeforewegettoxequalling1.Butthereareallsortsofproblemswiththisthinking.First,thereisnorealnumberxthatis‘immediatelybefore’1.Givenanumberlike0.999,closeto1,thenwecanalwaysimproveonthatandsee0.9999isalittlecloser.Orwemightsuggestusing0.999…(wheretheellipsismeansthatthereareinfinitelymanyrecurring9s)butthisisjustanotherdecimalexpansionfor1.Morerigorously,foranyinputx<1then(1+x)/2islessthan1butcloserto1thanxis.Insteadwemightbetemptedtotalkaboutaninputthatisinfinitesimallycloseto1butthen—whateverwemeanbythis—wearenolongertalkingabouttherealnumbersandhavejustreplacedresolvingonedefinitionwithresolvingadifferentone.
Weneedanotherapproachthatcanbecomfortablyexpressedentirelyintermsofrealnumbers.
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Thisproblemwasindependentlyresolvedinthe19thcenturybyBernardBolzanoandKarlWeierstrass.Wefeelthatg(x)isdiscontinuousat becauseg(x)isnoticeablydifferentfromg(1)forsomeinputsxnearbyto1.
Thereisstillquiteabitofsubtletyneededtofullycapturewhatthismeans.Inourexample,g(x)hasajumpof1fromoutputvaluesnear2(justbefore )tooutputvaluesnear1(justafter
).Thesizeofthatjumpwasunimportant,thepresenceofanyjumpatallwassufficient.Andthenotionof‘nearbyinputs’shouldnotbeinterpretedasseveralinputsthatareinsomesensecloseto1;ratherwemeanthereareinputsxarbitrarilycloseto1suchthatg(x)isnoticeablydifferentfromg(1).Necessarilythismeansthatwearetalkingaboutinfinitelymanysuchinputsx,notjustseveralx.Bywayofexample,itisenoughtonotethat:
Thisrigorouslyshowsthatg(x)isdiscontinuousat .Thesequenceofinputs0.9,0.99,0.999,0.9999,…getsarbitrarilycloseto1.Whatthismeansis:howeverdemanding‘nearbyto1’isrequiredtobe,thereareinputsfromthissequencethatareatleastthatclose.
Whilstwestillhaven’tquitedefinedjustwhatwemeanbydiscontinuous,wehavemadesomeprogresswithregardtothefunctionh(x)(Figure24(c)).Thisfunctiondoesnotappeartohaveanynoticeable‘jump’inoutputs,butitdoesseemtomeetthedefinition
h(x)isnoticeablydifferentfromh(0)forsomeinputsxarbitrarilynearto0.
Near thefunctionh(x)isvaryingcrazily.Fromthegraphwecanseethatthereareinputsx,arbitrarilycloseto0,where whilstwehave .Itnowseemsclearbyouremergingsenseofcontinuitythath(x)isdiscontinuousat .
AnexampleindetailWestillneedtobecarefulturningthesenascentthoughtsintoarigorousdefinition.We’llconsiderindetailthefunction whichiscontinuousforallinputsx.Iff(x)iscontinuousataninput then—basedonourpreviousthoughts—weneedthat
f(x)isnotnoticeablydifferentfromf(a)forallinputsxsuitablyneartoa.
Takeamomenttoappreciatewhyweneedallinputsxsuitablyneartoatoproducenotnoticeablydifferentoutputsf(x)tof(a).Ifsome—butonlyone—nearbyinputxtoaresultedinnoticeablydifferentoutputsf(x)andf(a),thenwecouldjusttightenournotionof‘suitablynear’toexcludetheprobleminputx.Infact,ifwecanneverget‘suitablynear’withourinputs,thenthismeansthattherewerearbitrarilycloseproblematicinputsxtoawheref(x)wasnoticeablydifferentfromf(a)—so,f(x)wouldbediscontinuousat .
Tobegin,whatdoesitmeanfor tobecontinuousat ?Isittruethat
x2isnotnoticeablydifferentfrom forallinputsxsuitablynearto0?
WetryoutsomevaluesinTable3.
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Table3. Sampleinputandoutputvaluesfor
Itseems—admittedlyonlyonthebasisoffivechoicesofx—thatx2iscloserto0thanxisto0,andsomequickalgebrachecksthatsmallnumbersgenerallysquaretosmallernumbers(inmagnitude).Wecannotfindinputsxcloseto0wheretheoutputsx2and0arenoticeablydifferent.
Nowwecanhangsomerigorousmathematicsontheseinitialthoughts:whateverpotential‘noticeabledifference’intheoutputsx2and weconsider,representedbyapositivenumbere,thenthereneedtobe‘suitablyclose’inputsxto0,representedbyapositivenumberd,suchthat
ifinputsxand0differbylessthandthenoutputsx2and0differbylessthane.
Astheoutputshereareclosertooneanotherthantheinputsare—thatis,asx2iscloserto0thanxisto0—thenwecanjustchoosedtoequale.Soifinputsdifferbyeorlesssodotheoutputs.Wehavethenshownthat iscontinuousat .
Whataboutcontinuityatadifferentinput,say ?WecancreateasimilartabletoTable3(seeTable4).
Table4. Moresampleinputandoutputvaluesfor
Thefunction isgrowingmuchmorerapidlyat thanitisat .Achangeofaround0.1intheinputsleadstoachangeintheoutputsofaround200;achangeof0.01intheinputsstillleadstoadifferenceofaround20intheoutputs.Thismayleadyoutothinkthattheoutputsare‘noticeablydifferent’here,butamorecarefulcheckofotherinputswouldshowthattheselargedifferenceshavebeenincrementallyachieved.Allthisisaconsequenceofthefunctionchangingmorerapidlynear ,andwhatneedstighteningisournotionof‘suitablynear’.Asthefunctionisgrowingmorerapidly,smallchangesintheinputwillleadtorelativelylargechanges,butstillinacontinuousfashion.Ifweconsidertheinput,alittlelargerthantheinput1000,thenthedifferenceintheoutputsequals
asd2<dwhend<1.Soashiftininputsbydresultsinashiftofoutputsroughly2000timeslarger.(NotesimilarbehaviourinTable4.)Thisis,initself,notaproblembutitdoesmeanthatifwewanttheoutputstodifferbynomorethanethenweshouldonlyallowtheinputstodifferbynomorethate/2001.Thisstillshowsthecontinuityof at ,wejustneeded
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atightersenseof‘suitablynear’withtheinputsasthefunctionwasgrowingsofast.Forcontinuityatyetlargerinputsthatnotionwouldhavetobecomeyetmorestringent,butwewouldalwaysbeabletofindsomesmallwiggleroomaboutaninputforwhichtheoutputsdon’tdifferbeyondthedesiredamounte.
ArigorousdefinitionPuttingallthisthinkingtogethergivesusarigorousdefinitionofcontinuity.I’dsuggestreadingthedefinitionandseekingtounderstandhowthismeansthatthefunctioninFigure25(a)iscontinuousandtheoneinFigure25(b)isn’t,butifyoufindthegeneralityofthedefinitionandthetechnicallevelofthelanguagedifficultthenmoveontothenextsectiononthepropertiesofcontinuousfunctions.Andbereassured,asittookgenerationsofmathematicianstofinallygetthisdefinitionright,andcurrentandpastgenerationsofmathematicsundergraduatesstillwrestlewithproofsinvolvingthisdefinitionintheiranalysiscourses.
25.Therigorousdefinitionofcontinuousanddiscontinuous(a)Acontinuousfunction,(b)Adiscontinuousfunction.
Formally,then,afunctionwithrealinputsxandrealoutputsf(x)iscontinuousataninputif:
foranypositiveethereissomepositived
suchthatthedifferencebetweentheoutputsf(x)andf(a)islessthane
whenthedifferencebetweentheinputsxandaislessthand.
InFigure25(a),wearefocusingondemonstratingthecontinuityoff(x)atinput .Aparticularchoiceofe>0hasbeenmadeandourtasknowistomakesurethattheoutputsdon’tdifferfromf(a)bymorethanthise.Sotheoutputshavetoremainbelowf(a)+eandabovef(a)–e(asshownonthey-axis).Andthishastohappenforinputsxinsomerangea–d<x<a+d.WecanseefromFigure25(a)thatsomesuchintervalhasbeenfound,asshownonthex-axis—therangeofoutputsonthisintervalareboundedbythedashedlinesandthesefallwithinthepermittedrangefortheoutputs.Toshowcontinuityoff(x)ataninput we’dhavetoshowthatthiscanbedoneforalle>0,howeversmall;toshowcontinuityofthefunctionf(x)we’dhavetodothisforallinputsx.
Thereareseveralimportantpointstonotehere:
werequirethattheoutputscanbeconstrainedinacertainwayiftheinputsareappropriatelyconstrained;weneedtobeabletodothisforallconstraintseintheoutputs;foreachchoiceofewewillneedachoiceofdthatmeetstherequirement;forasmallerchoiceofethendwillusuallyneedtobesmalleraswell;givenapositivee,thenanypositivedthatmeetstherequirementisfine—we’renotlookingforalargestsuchd,say;thefasterthefunctionf(x)ischangingata,thesmallerdwillneedtoberelativetoe.
Afunctionisthensaidtobecontinu
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