introduction to algebraic topology and algebraic geometry - u. bruzzo

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1、INTERNATIONALSCHOOLFORADVANCEDSTUDIESTriesteU.BruzzoINTRODUCTIONTOALGEBRAICTOPOLOGYANDALGEBRAICGEOMETRYNotesofacoursedeliveredduringtheacademicyear2002/2003Lafilosofia`escrittainquestograndissimolibrochecontinuamentecistaapertoinnanziagliocchi(iodicol’universo),

2、manonsipu`ointendereseprimanonsiimparaaintenderlalingua,econoscericaratteri,ne’quali`escritto.Egli`escrittoinlinguamatematica,eicaratterisontriangoli,cerchi,edaltrefiguregeometriche,senzaiqualimezi`eimpossibileaintenderneumanamenteparola;senzaquesti`eunaggirars

3、ivanamenteperunoscurolaberinto.GalileoGalilei(from“IlSaggiatore”)iPrefaceThesenotesassemblethecontentsoftheintroductorycoursesIhavebeengivingatSISSAsince1995/96.Originallythecoursewasintendedasintroductionto(complex)algebraicgeometryforstudentswithaneducationi

4、ntheoreticalphysics,tohelpthemtomasterthebasicalgebraicgeometrictoolsnecessaryfordoingresearchinalgebraicallyintegrablesystemsandinthegeometryofquantumfieldtheoryandstringtheory.Thismotivationstilltranspiresfromthechaptersinthesecondpartofthesenotes.Thefirstpart

5、onthecontraryisabriefbutrathersystematicintroductiontotwotopics,singularhomology(Chapter2)andsheaftheory,includingtheircohomology(Chapter3).Chapter1assemblessomebasicsfactinhomologicalalgebraanddevelopsthefirstrudimentsofdeRhamcohomology,withtheaimofprovidingan

6、exampletothevariousabstractconstructions.Chapter4isanintroductiontospectralsequences,aratherintricatebutverypower-fulcomputationtool.Theexamplesprovidedherearefromsheaftheorybutthiscom-putationaltechniquesisalsoveryusefulinalgebraictopology.Ithankallmycolleagu

7、esandstudents,inTriesteandGenovaandotherlocations,whohavehelpedmetoclarifysomeissuesrelatedtothesenotes,orhavepointedoutmistakes.InthisconnectionspecialthanksareduetoFabioPioli.MostofChapter3isanadaptationofmaterialtakenfrom[2].IthankmyfriendsandcollaboratorsC

8、laudioBartocciandDanielHern´andezRuip´erezforgrantingpermissiontousethatmaterial.IthankLotharG¨ottscheforusefulsuggestionsandforpointingoutanerrorandthestudentsofthe2002/2003course