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BourbakiandAlgebrai cTop ology

byJohnMcClear y Thepr incipalaimoftheBourbakigr oup(L'AssociationdesCollaborat eurs deNicolas Bourbaki)istoprovide asolidfoundationfor thewhol ebodyofmo dernmathematics.The methodofexposit ionis axiomaticandabstract,logicallyco heren tandrigorous,proc eeding normallyfromthegeneral totheparticular,astylefound tobe notaltogetherc ongenial toman yreaders.Theongoing seriesofbooksbeganwith

El´ementsdeMath´ematiqu esin

1939,andoth erbo oksonalgebra, settheory,topology, ando thertopicshavefollowed.

Manybo oksintheserieshavebecome standard referenc es,though som emathematicians arecriticaloftheir austerelyabstractp ointofview . fromhttp://www.encyclop edia.com/html/B/BourbakiN1.asp,Dec.3,2004 Itisn owm orethan70yearsago thatthefoundersofLeComit ´eder´edactiondu trait´ed'analysemetinParis atthe Caf´eA.Capoulade,63b oulevardS aint-Michel,to discussthedraftingofa textbook onanalysis.Thismeetingin cluded (recent centenarian) HenriCart an(1904-), ClaudeChevalley(1909-1984),JeanDelsarte(1903-

1968),JeanDieudonn´e(1906-1992),Ren´edePo ssel(1905-1974),and Andr´eWeil

(1906-1998).Thef ateofth isproject isthe storyofthe Bourbaki,orshou ldIsa y,the storyofth ech aracterNicolasBou rbaki,author of

El´ementsdemath´emati que,as eries

ofinflu entialexpositionsofthebasic notionsofmodernmathematics. Thistal kisbasedona wild goosechas eafte rado cument.Theproj ectwassupported bytheGabriel SnyderBeckF undatVassar Collegethatfun dsresearchonanythingF rench. Inearly2000 Ilearne data meetingin Oberwohlfachthatan archiv eofp apersandint ernal documentsoftheBourbakiwassoontob eope nedtosc holarsinParis.TheBec kfund providedmethemeanstovisitthear chiv e.Themanagersof thisarchive,Lili aneBeau lieu andChristian Houzel,sho wedmegreathospitalit yduringmyvisittoParisinJuly2003, andm adeitpossiblefor meto rummagethroughtheBourbakipapers. Historicalresearchposes questions,towhichvariousmeth odsmaybe applied. My interestsincludethehistoryofalgebraictopol ogy,asub jec twhosedevelopmentduring thetwenti ethcenturyinfluencedagreatdealofthatce ntury'smathematics.They ears followingtheSecondW orldW arrepresentahighpoi ntinthisstory ,andseveral important membersofBourbakicontribu tedto thisdevelopment.How ever,algebraictopologydoes notappear amongthetopicstr eatedin El´ements - admittedlymanyotherimportanttop ics werealsoomitted.Theinvolv ementofso manypi oneering topologistsmakesthisomission standout. Whileagraduatestud ent, Icollected arumorthattherewasamanuscri pt,200p ages long,prepared for ElementsbyCartan,Kosz ul,Eilenberg,andChev alley,treatingalgebraic topology.Further more,thisdocumentwasbasedontheuseofdi ff erentialforms,that is,algebraic topologychez ElieCar tan(1869-1951)(leper ed'Henri).Accord ingto therumor,the manuscriptwas abandonedwh enthedoctoralthesesofJean-Pierre Serre(1926-)an dArmandBorel (1923-2003)we republished.Serre'sandBorel's subsequentpapersd idchangethefocu sofrese archintopology,a wayfromd i ff erential geometricmethodstomore algebraicmethods,principallythes pectr als equencean dthe Steenrodalgebra,makingthemanuscri ptobsolete.Sowhatwas inthismanus cript? Could 1 Igeta lookat it?Thehi storiansaliv atesat thec hanceto lookatthestateofaffairsbefore andaftera ke yeve nt. Well,themanuscriptwas n'tthere ,if,infact,itexistsatall.Th earchivalworkIwas abletodo,ho weve r, o ff eredman yinsightsintothew orkingsandspirit ofBourbakiandI willrel atesomefindingsinthisr eport. Asmystoryunfu rls,Iw anttocons idertheallure oftheaxiomatic metho db eforeandafterBourbaki,oneofth efeaturesofthei rexposition thathasreceiv edcriticism.

WhoisBourbaki?

Hisname isGreek,his nationalit yisFrenchandh ishis toryiscurious.Heis oneofthe mostinfluential mathematicians ofthe20thcentury.Thelegendsabouthimar eman y,and theyar egrowing everyday....Thestra ngestfactabouthim,however,is thathe doesn't exist.

PaulHalmos,1957

Andr´eWeilwasonthe facultyattheUniver sityofStr asbour gin 1934,togetherwith HenriCar tan.Theywereresp onsi bleforthecourseonthedi ff erentialandintegralcalculus, oneofthr ees tandardcoursesrequ iredforthelicensedemath´ematiques,along withgeneral physicsandrationalmec hanics.Thestandard textwasCoursd'Analy semath´ematiqueby EduoardGoursa t(1858-1936),writte nbeforetheFirstW orldWar.Cartanfoundit wanting,incompletewheregeneralizationsw ere known,andsimplynottheb estwayto presentthesetopics.An explicitexample,onewithastory ofitsown,istheform ulation ofStok es'sTheorem.Itmayb ewritten X X d whereωisad ifferentialform,dωitsex teriorderivative,Xthedomainofintegrati onand Xtheboundaryof X.Whenev erythi nginsightissmooth,theproofisclear, butthe importanceofthisformulai nthe caseofmoregenerald omainsofintegration istheconten t ofthe cele bratedtheoremofGeorgesDeRham(1903-1990),pr ovedin1931toanswer aque stionofElieCartanrelatingin varian tintegrals onLiegroupsto thetopologyofsuc h manifolds. PersistentbadgeringbyCartanle dWeiltosugge stthattheywrite atex tbookthat theycouldbes atisfiedwith.Weil writesthathetold Cartan,"Whydon'twegettogether andsettle suchmatter sonceandfor all,andyouwon'tplaguemewithyour questionsany more?" Thefirstm eetingon10 December1934i nPar isto planthebookoccurredaftera meetingofleSeminaire Julia,a notherofWeil'sandCartan'seffortstofill the gaple ftin FrenchmathematicsafterWorld WarI,whichWeilcalled"hectatombof1914-1918which hadslaught eredvirtuallyanentiregeneration"ofFren chmathematician s. Theseminar, organizedbytheseyou ngturcsinimitationof theseminarsin Germany,neededasp onsor inord ertogetaroomat theS orbonne .GastonJulia(1893-1978)hadb een theyoungest oftheirt eachers atthe EcoleNormaleSup ´erieureandhesteppe duptospons orthem.The seminartreatedatopicayear, begin ningin 1933-34withgroup sand algebras,goingonto 2 Hilbertspaces,the ntopology.Thes eminarcontinueduntil 1939whenitwassupe rseded bytheSeminar Bourbaki. Thecomm ittee'sfirstplanswerefora textinanalysis ,thatwould, accordingto Weil,"fixthecurr iculumfor 25yearsfordifferentialandintegra lc alculus."This text shouldbeaussimoderne quepossible ,untra it´eutile`atous,andfi nally, aussirobustes etaussi universelsquepossible .Weil alreadyknewapotential publis her inhi sfriend EnriquesFreymann,aMexican diplomatwhomarriedthedaught erof thef ounderof MaissonHermann,asc ien tificpublisher.Freymannbecamethechiefed itorandmanager ofthe publishing house. Amongth einnovations ofthistextwasthesuggestion,insisted onby Delsarte ,that itbe writtencollectivel ywithoutexpertleadership.The initial expectationwasthatth e textwou ldrunto1000-1200pagesandb edoneinab outs ixmon ths.The initial group ofsix wasexpanded toninemem bersinJanuary1935,w ithPaulDubreil(1904-1994), JeanLeray(1906-1998)andSzolemMandelbr ojt(1899-1983)added. Dubreiland Leraywerere placedbyJeanCoulombandCharlesEhresmann(1905-1979)be fore thefirst summerworkshopinJuly, 1935. Thefirs tBourbakicongress washeldinBesse-en-Chandesseint heVosgesmountains. Atthisworkshop ,theproposal wasmadetoexpandth eproject toaddapaquetabstrait, treatingabstract(n ewandmodern)notions thatwouldsup portanalysis. Theseincluded abstractsettheory,algebra,es pecial lydi ff erentialforms,andtopology,withparti cular emphasisonexistencetheore ms(Leray). ThepaqueteventuallybecametheFasciculedeR´esultats ,as ummar yofusefulresults presentedinsuchawaythat acomp etentmathematiciancoulds eewhe readesiredresult mightbefound ,andpro videtheresultthems elvesifthey neededit.Infact,thelast publication,FasciculeXXXVI,part twoofVari´et´esdiff´erentiellesetanalytiques,issucha summary.Bytheway, itisinFasciculeXXXVIthatthest atementofStok es'sTheorem founditsplace. Duringthefirst conference,wi thagroupofyoung,eage r,andablemathematicians inonep lace,a newresultonme asures onatopologicalsp acewasproved.Anotewas writtenuptosubmitt oComptes-Rendues.Thename ofBourbakiforthegrou pwas basedon astoryout ofscho ol:In1923, Delsarte, Cartan,an dWeilweremembe rsof thenewlymatricu latedcl assat EcoleNormaleSup erieure,whe ntheyreceiv edalecture noticebyaprofessor withavaguel ySc andinavianname,f orwhich attendan cewasstrongly recommended.Thespeakerwasaprankster, RaoulHusson,wearin gafalsebeard and speakingwithanundefi nableac cent.T akingo ff fromclassical functiontheory,thetalk haditsclimax inBourbaki'sTheoremleavingtheaudience"sp eechle sswithamazemen t." (ThisBourbakiwas thegeneralwho traveledwithNap oleon.)W eilrecalledthisst ory andthe familynamewas adopted.But whyNicolas?For thesubmission ofthepaper, theauthorneed edaprenom.Itw asWeil' swifeEveline whochristenedthen ewBourbaki Nicolas.Thenot ewas handledattheAcad´emiedesSciencesby

ElieCartanw hostood

upforth eunfor tunatePoldevianmath ematician.Thenotewasaccepted andpublished.

Toproduce theconstituentpartsofles

El´ements,the methodofediti ngadoptedby

theBourbakiem phasizedcommunalin volvement.Atextwas broughtbeforeameeti ng andpresen ted,pagebypage,linebyline,toth egr oupwhothen expres sedanyandall 3 criticism.Arevisi onwas handedovertoanothermemberof thegroupandthe process repeatedwhenanewdraftw asav ailable .Afte renoughi terationstoobtain unanimous approval - eitherforthestrengthofthetext orthefati gueofthegroup withthetopic - the textwou ldbefinalized(usuallyb yDieudonn´e) andsentto thepublisher.

Digression:TheAxiomaticMetho d

Inspiteof thehighp edagogic valueo ftheg eneticmethod,theaxiomaticmethodhasthe advantageofprovidinga concl usiveexpositionandfull lo gicalconfidencetothecontents ofour knowledge.

DavidHilbert,1900

Duringhis'apprenticeshi p'(documen tedin[Weil]),Weiltraveledextensively ,spending timeinGerman ywhile theriseofNati onalSocialismtopo wertookp lace.Ashewas interestedinnumbertheory, headmiredth emathematicsoftheGermanschools,especially theaxiomatic approachledbythew orkofDavidHIlbert(1862-1943)and theG¨ottingen school.French mathematicsthroughthenineteenthcentury andintothetwentiethw as dominatedbyanalys is.Evenr esultsofanumber-theoreticnaturewer eprovedth rough analyticmeans. ThesuccessofHilbert'sideas inmanyfieldsattractedmath ematicians everywhereandso,whenlooki ngfor amodel toshapetheirproject, themembersof

Bourbakiturned totheaxiomaticmethod.

Thisphenome nonwasnotwithoutpreceden t.WhenE.H.Moore (1862-1932)came tole adtheUniversityof Chicagomathematicsd epartmentaround1900,heconsciously adoptedthestyleof Hilber t'sGrundlagenderGeometrieasmod ern,precise,andamodel tobe imitated. HisearlieststudentsatChicagoincludedOswaldVeblen (1880-1960),quotesdbs_dbs2.pdfusesText_2

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