FINITE TYPE INVARIANTS OF THREE MANIFOLDS AND THE DIMENSION SUBGROUP PROBLEM

pages

English

Documents

Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

pages

English

Ebook

Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Publié par

pefav

Nombre de lectures

Langue

English

ar X iv :m at h/ 06 05 49 7v 3 [m ath .G T] 1 3 D ec 20 06 FINITE-TYPE INVARIANTS OF THREE-MANIFOLDS AND THE DIMENSION SUBGROUP PROBLEM GWENAEL MASSUYEAU Abstract. For a certain class of compact oriented 3-manifolds, M. Goussarov and K. Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of “cut-and-paste” operations defined by the lower central series of the Torelli group of a surface. In this paper, we observe that this is a variation of a classical problem in group theory, namely the “dimension subgroup problem.” This viewpoint allows us to prove, by purely algebraic methods, an analogue of the Goussarov–Habiro conjecture for finite-type invariants with values in a fixed field. We deduce that their original conjecture is true at least in a weaker form. Contents 1. Introduction 1 2. The dimension subgroup problem 4 3. Homology cylinders and their finite-type invariants 6 3.1. Finite-type invariants of 3-manifolds 6 3.2. Group of homology cylinders 8 3.3. The Goussarov–Habiro conjecture 10 4. The Goussarov–Habiro conjecture with coefficients in a field 11 4.1. Taking the coefficients in a field 11 4.2. Increasing the degree 13 5.

torelli surgery

lower central

∆3 ≥

∂h ? ∂h

∂h

problem

central series

group ring

has been proved

Voir

Publié par

pefav

Nombre de lectures

Langue

English

FINITE-TYPEINVARIANTSOFTHREE-MANIFOLDSANDTHEDIMENSIONSUBGROUPPROBLEMGWE´NAE¨LMASSUYEAUAbstract.Foracertainclassofcompactoriented3-manifolds,M.GoussarovandK.Habirohaveconjecturedthattheinformationcarriedbyﬁnite-typeinvariantsshouldbecharacterizedintermsof“cut-and-paste”operationsdeﬁnedbythelowercentralseriesoftheTorelligroupofasurface.Inthispaper,weobservethatthisisavariationofaclassicalproblemingrouptheory,namelythe“dimensionsubgroupproblem.”Thisviewpointallowsustoprove,bypurelyalgebraicmethods,ananalogueoftheGoussarov–Habiroconjectureforﬁnite-typeinvariantswithvaluesinaﬁxedﬁeld.Wededucethattheiroriginalconjectureistrueatleastinaweakerform.Contents1.Introduction2.Thedimensionsubgroupproblem3.Homologycylindersandtheirﬁnite-typeinvariants3.1.Finite-typeinvariantsof3-manifolds3.2.Groupofhomologycylinders3.3.TheGoussarov–Habiroconjecture4.TheGoussarov–Habiroconjecturewithcoeﬃcientsinaﬁeld4.1.Takingthecoeﬃcientsinaﬁeld4.2.Increasingthedegree5.Thealgebradualtoﬁnite-typeinvariantsofhomologycylinders5.1.Quillen’stheoremforanarbitraryN-series5.2.Applicationtohomologycylinders6.Appendix:TorelligroupsandclaspersReferences146680111113141417171021.IntroductionTheGoussarov–Habirotheoryisaimedatunderstandinghow3-manifoldscanbeobtainedonefromtheotherbycut-and-pasteoperationsofacertainkind[3,6,4,1].Inparticular,itappliestothestudyofﬁnite-typeinvariantsintroducedbyOhtsuki[21]:Thelatterareinvariantsof3-manifoldswhich,inasense,behavepolynomiallywithrespecttocertainsurgeryoperations.Wewillworkwithcompactoriented3-manifoldswhoseboundary,ifany,isidentiﬁedwithaﬁxedabstractsurface.Thosemanifoldsareconsidereduptohomeomorphismsthatpreservetheorientationandtheboundaryidentiﬁcation.Thekindofsurgerymod-iﬁcationsthatareusedintheGoussarov–Habirotheorycanbedescribedasfollows:Date:November13,2006.2000MathematicsSubjectClassiﬁcation.57M27,16S34.Keywordsandphrases.3-manifold,ﬁnite-typeinvariant,groupring,N-series,dimensionsubgroup.1

2Givenacompactoriented3-manifoldM,ahandlebodyH⊂MandaTorelliauto-morphismhof∂H(thatis,h:∂H→∂Hisahomeomorphismwhichactstriviallyinhomology),onecanformanewcompactoriented3-manifold:Mh:=(M\intH)∪hH.ThemoveM❀MhiscalledaTorellisurgery.Apossiblecharacterizationofaﬁnite-typeinvariantisasfollows.Letfbeaninvariantofcompactoriented3-manifoldswithvaluesinA,anAbeliangroup.Then,fisaﬁnite-typeinvariantofdegreeatmostdif,foranymanifoldMandforanysetΓofd+1pairwisedisjointhandlebodiesinM–eachcomingwithaTorelliautomorphismofitsboundary–thefollowingidentityholds:X(1.1)(−1)|Γ′|f(MΓ′)=0∈A.′Γ⊂ΓHere,MΓ′denotesthemanifoldobtainedfromMbythesimultaneousTorellisurgeryalongthehandlebodiesbelongingtoΓ′.Thereareplentiesofﬁnite-typeinvariants[15]andthenaturalproblemisto“quan-tify”howﬁnetheyare,degreebydegree.Forthis,GoussarovandHabirohaveintro-duced,foreveryintegerk≥1,theYk-equivalence.Thisequivalencerelationamongmanifoldscanbecharacterizedasfollows:Twocompactoriented3-manifoldsMandM′areYk-equivalentifM′canbeobtainedfromMbyaTorellisurgeryM❀Mh=M′suchthathbelongstothek-thtermofthelowercentralseriesoftheTorelligroupof∂H.Fact.IftwomanifoldsareYd+1-equivalent,thentheyarenotdistinguishedbyﬁnite-typeinvariantsofdegreeatmostd.Theconversehasbeenprovedforintegralhomology3-spheresbyHabiroandGoussarov[6,4]whichprovides,inthisspecialcase,ageometriccharacterizationofthepowerofﬁnite-typeinvariants.Unfortunately,theconversedoesnotholdingeneral.1Nevertheless,therearemanifoldsofaspecialtypewhichplayakeyroleintheGoussarov–Habirotheory.LetΣbeaconnectedcompactorientedsurface.Recallfrom[6,3]thatahomologycylinderoverΣisacobordismMfrom−ΣtoΣ(withcorners,if∂Σ6=∅)whichcanbeobtainedfromthecylinderΣ×[−1,1]byaTorellisurgery.Homologycylinderscanbecomposed,andtheirmonoidisdenotedbyCyl(Σ).Conjecture(Goussarov–Habiro).TwohomologycylindersoverΣareYd+1-equivalentif,andonlyif,ﬁnite-typeinvariantsofdegreeatmostddonotseparatethem.Inthispaper,weidentifythealgebraunderlyingthe“Goussarov–Habiroconjecture”(orGHC,forshort)byobservingthatitisaspecialinstanceofageneralproblemingrouptheory,whichisknownasthe“dimensionsubgroupproblem”(orDSP,forshort).Classically,foragivengroupG,thequestionistodecidewhetherthei-thsubgroupofthelowercentralseriescoincideswiththei-thdimensionsubgroup,i.e.thissubgroupdeterminedbythei-thpoweroftheaugmentationidealofZ[G].Here,weconsidertheDSPinthemoregeneralsettingwherethelowercentralseriesisreplacedbyanN-seriesinLazard’ssense[14].SeeSection2foraprecisestatementofthatproblem.1Wehaveinsertedacounter-examplebelow,inSection3.

Voir

Univers
Ebooks
Livres audio
Presse
Podcasts
BD
Documents

FINITE TYPE INVARIANTS OF THREE MANIFOLDS AND THE DIMENSION SUBGROUP PROBLEM

YouScribe

Le catalogue

Le service

Les conditions