Introduction to Lawson homology

Chris Peters

Description

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

Sujets

• Peters
• Lawson
• Algebraic variety
• Équivalent

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

Extrait

IntroductiontoLawsonhomology ChrisPETERS,email:[e-mail] SiegmundKOSAREWemail:[e-mail] DepartmentofMathematics,UniversityofGrenobleI UMR5582CNRS-UJF,38402-Saint-Martind’He`res France Abstract Lawsonhomologyhasquiterecentlybeenproposedasaninvariant foralgebraicvarieties.Variousequivalentdefinitionshavebeensug- gested,eachwithitsownmerit.Herewediscusstheseforprojective varietiesandwealsoderivesomebasicpropertiesforLawsonhomol- ogy.ForthegeneralcasewerefertoPauloLima-Filho’slecturesin thisvolume. Keywords:Lawsonhomology,cyclespaces MSC2000classification:14C25,19E15,55Qxx Introduction ThispaperismeanttoserveasaconciseintroductiontoLawsonhomology ofprojectivevarieties.Foranotherintroductionthereadershouldconsult .]41[ Itisorganizedasfollows.Inthefirstsectionwerecallsomebasictopolog- icaltoolsneededforafirstdefinitionofLawsonhomology.Thensomebasic examplesarediscussed.Inthesecondsectionwediscussthetopologyofthe so-called“cyclespaces”inmoredetailinordertounderstandfunctoriality ofLawsonhomology.Inthethirdandfinalsectionwerelatevariousequiv- alentdefinitions.Herethelanguageofsimplicialspacesisneededandwe onlysummarizesomecrucialresultsfromthevastliteratureonthishighly technicalsubject. Finallywewanttothanktherefereeforhissuggestionstoimprovethe exposition. 1

1BasicNotions 1.1Homotopygroups Westartbyrecallingthedefinitionandthebasicpropertiesofthehomotopy groups.Foranytwopairsoftopologicalspaces(X,A)and(Y,B)weuse thenotation[(X,A),(Y,B)]forthesetofhomotopyclassesofmapsX → Y sendingAtoB(anyhomotopyissupposedtosendAtoBaswell).Then, fixingapointsonthek-sphereS k ,wehave π k (X,x)=[(I k ,∂I k ),(X,x)]=[(S k ,s),(X,x)]. Thereisanaturalproductstructureonthesesets(divideI k intwoanduse thefirstmapononehalfandthesecondmapontheotherhalf).Thismakes π k (X,x)intoagroup,whichturnsouttobeabelianfork ≥ 2. HomotopyandhomologyarerelatedthroughtheHurewiczhomomor- msihp h k :π k (Y,y) → H k (Y), definedbyassociatingtotheclassofamapf:S k → Ytheimageunderf ∗ ofageneratorofH k (S k ).Thefollowingimportantresulttellsuswhenthe Hurewiczhomomorphismactuallyisanisomorphism: Theorem1(Hurewicz’Theorem)Supposethat(X,x)is(k − 1)-connected, i.e.π s (X,x)=1,s=0,...,k − 1.Thenh k isanisomorphism. Onecanshowthathomotopicmapsinducethesamemapinhomology. Hurewicz’theoremtellsusthatanymapinducingisomorphismsontheho- motopygroupswillalsoinduceisomorphismsonthehomologygroups.This motivatesthefollowingdefinitions. Definition2 (1)Acontinuousmapf:X → Yisahomotopyequivalenceifthereisa continuousmapg:Y → Xsuchthatf ◦ gandg ◦ farehomotopictothe identity. (2)Acontinuousmapf:X → Yisaweakhomotopyequivalenceifthe inducedmapsonthehomotopygroupsareallisomorphisms. (3)Twotopologicalspacesare(weakly)homotopicallyequivalentifthereexist a(weak)homotopyequivalencebetweenthem. Example3AspaceXissaidtobeanEilenberg-MaclaneK(π,k)-spaceif itsonlynon-trivialhomotopygroupisπ k (X)=π.Henceanyspacehomo- topyequivalenttoaK(π,k)-spaceisagainaK(π,k)-space.ForinstanceS 1 isaK( Z ,1),andtheinductiveunionofprojectivespaces, P ∞ isaK( Z ,2). 2