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Introduction to Lawson homology

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Niveau: Supérieur, Doctorat, Bac+8
Introduction to Lawson homology Chris PETERS, email: Siegmund KOSAREW email: Department of Mathematics, University of Grenoble I UMR 5582 CNRS-UJF, 38402-Saint-Martin d'Heres France Abstract Lawson homology has quite recently been proposed as an invariant for algebraic varieties. Various equivalent definitions have been sug- gested, each with its own merit. Here we discuss these for projective varieties and we also derive some basic properties for Lawson homol- ogy. For the general case we refer to Paulo Lima-Filho's lectures in this volume. Keywords: Lawson homology, cycle spaces MSC2000 classification: 14C25, 19E15, 55Qxx Introduction This paper is meant to serve as a concise introduction to Lawson homology of projective varieties. For another introduction the reader should consult [14]. It is organized as follows. In the first section we recall some basic topolog- ical tools needed for a first definition of Lawson homology. Then some basic examples are discussed. In the second section we discuss the topology of the so-called “cycle spaces” in more detail in order to understand functoriality of Lawson homology. In the third and final section we relate various equiv- alent definitions. Here the language of simplicial spaces is needed and we only summarize some crucial results from the vast literature on this highly technical subject.

  • group

  • basic properties

  • projective variety

  • equivalent

  • hurewicz map

  • space homo- topy equivalent

  • federer topology


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IntroductiontoLawsonhomologyChrisPETERS,email:chris.peters@ujf-grenoble.frSiegmundKOSAREWemail:siegmund.kosarew@ujf-grenoble.frDepartmentofMathematics,UniversityofGrenobleIUMR5582CNRS-UJF,38402-Saint-Martind’He`resFranceAbstractLawsonhomologyhasquiterecentlybeenproposedasaninvariantforalgebraicvarieties.Variousequivalentdefinitionshavebeensug-gested,eachwithitsownmerit.HerewediscusstheseforprojectivevarietiesandwealsoderivesomebasicpropertiesforLawsonhomol-ogy.ForthegeneralcasewerefertoPauloLima-Filho’slecturesinthisvolume.Keywords:Lawsonhomology,cyclespacesMSC2000classification:14C25,19E15,55QxxIntroductionThispaperismeanttoserveasaconciseintroductiontoLawsonhomologyofprojectivevarieties.Foranotherintroductionthereadershouldconsult.]41[Itisorganizedasfollows.Inthefirstsectionwerecallsomebasictopolog-icaltoolsneededforafirstdefinitionofLawsonhomology.Thensomebasicexamplesarediscussed.Inthesecondsectionwediscussthetopologyoftheso-called“cyclespaces”inmoredetailinordertounderstandfunctorialityofLawsonhomology.Inthethirdandfinalsectionwerelatevariousequiv-alentdefinitions.Herethelanguageofsimplicialspacesisneededandweonlysummarizesomecrucialresultsfromthevastliteratureonthishighlytechnicalsubject.Finallywewanttothanktherefereeforhissuggestionstoimprovetheexposition.1
1BasicNotions1.1HomotopygroupsWestartbyrecallingthedefinitionandthebasicpropertiesofthehomotopygroups.Foranytwopairsoftopologicalspaces(X,A)and(Y,B)weusethenotation[(X,A),(Y,B)]forthesetofhomotopyclassesofmapsXYsendingAtoB(anyhomotopyissupposedtosendAtoBaswell).Then,fixingapointsonthek-sphereSk,wehaveπk(X,x)=[(Ik,∂Ik),(X,x)]=[(Sk,s),(X,x)].Thereisanaturalproductstructureonthesesets(divideIkintwoandusethefirstmapononehalfandthesecondmapontheotherhalf).Thismakesπk(X,x)intoagroup,whichturnsouttobeabelianfork2.HomotopyandhomologyarerelatedthroughtheHurewiczhomomor-msihphk:πk(Y,y)Hk(Y),definedbyassociatingtotheclassofamapf:SkYtheimageunderfofageneratorofHk(Sk).ThefollowingimportantresulttellsuswhentheHurewiczhomomorphismactuallyisanisomorphism:Theorem1(Hurewicz’Theorem)Supposethat(X,x)is(k1)-connected,i.e.πs(X,x)=1,s=0,...,k1.Thenhkisanisomorphism.Onecanshowthathomotopicmapsinducethesamemapinhomology.Hurewicz’theoremtellsusthatanymapinducingisomorphismsontheho-motopygroupswillalsoinduceisomorphismsonthehomologygroups.Thismotivatesthefollowingdefinitions.Definition2(1)Acontinuousmapf:XYisahomotopyequivalenceifthereisacontinuousmapg:YXsuchthatfgandgfarehomotopictotheidentity.(2)Acontinuousmapf:XYisaweakhomotopyequivalenceiftheinducedmapsonthehomotopygroupsareallisomorphisms.(3)Twotopologicalspacesare(weakly)homotopicallyequivalentifthereexista(weak)homotopyequivalencebetweenthem.Example3AspaceXissaidtobeanEilenberg-MaclaneK(π,k)-spaceifitsonlynon-trivialhomotopygroupisπk(X)=π.Henceanyspacehomo-topyequivalenttoaK(π,k)-spaceisagainaK(π,k)-space.ForinstanceS1isaK(Z,1),andtheinductiveunionofprojectivespaces,PisaK(Z,2).2