FINITE TYPE INVARIANTS OF THREE MANIFOLDS AND THE DIMENSION SUBGROUP PROBLEM

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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

Sujets

• Kyoto University
• Jennings
• Problem
• Central series
• Group ring

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

FINITE-TYPEINVARIANTSOFTHREE-MANIFOLDS ANDTHEDIMENSIONSUBGROUPPROBLEM GWE ´ NAE ¨ LMASSUYEAU Abstract.Foracertainclassofcompactoriented3-manifolds,M.GoussarovandK. Habirohaveconjecturedthattheinformationcarriedbyfinite-typeinvariantsshould becharacterizedintermsof“cut-and-paste”operationsdefinedbythelowercentral seriesoftheTorelligroupofasurface.Inthispaper,weobservethatthisisavariation ofaclassicalproblemingrouptheory,namelythe“dimensionsubgroupproblem.” Thisviewpointallowsustoprove,bypurelyalgebraicmethods,ananalogueofthe Goussarov–Habiroconjectureforfinite-typeinvariantswithvaluesinafixedfield.We deducethattheiroriginalconjectureistrueatleastinaweakerform. Contents 1.Introduction 2.Thedimensionsubgroupproblem 3.Homologycylindersandtheirfinite-typeinvariants 3.1.Finite-typeinvariantsof3-manifolds 3.2.Groupofhomologycylinders 3.3.TheGoussarov–Habiroconjecture 4.TheGoussarov–Habiroconjecturewithcoefficientsinafield 4.1.Takingthecoefficientsinafield 4.2.Increasingthedegree 5.Thealgebradualtofinite-typeinvariantsofhomologycylinders 5.1.Quillen’stheoremforanarbitraryN-series 5.2.Applicationtohomologycylinders 6.Appendix:Torelligroupsandclaspers References 1 4 6 6 8 01 11 11 31 41 41 71 71 02 1.Introduction TheGoussarov–Habirotheoryisaimedatunderstandinghow3-manifoldscanbe obtainedonefromtheotherbycut-and-pasteoperationsofacertainkind[3,6,4,1].In particular,itappliestothestudyoffinite-typeinvariantsintroducedbyOhtsuki[21]: Thelatterareinvariantsof3-manifoldswhich,inasense,behavepolynomiallywith respecttocertainsurgeryoperations. Wewillworkwithcompactoriented3-manifoldswhoseboundary,ifany,isidentified withafixedabstractsurface.Thosemanifoldsareconsidereduptohomeomorphisms thatpreservetheorientationandtheboundaryidentification.Thekindofsurgerymod- ificationsthatareusedintheGoussarov–Habirotheorycanbedescribedasfollows: Date :November13,2006. 2000 MathematicsSubjectClassification. 57M27,16S34. Keywordsandphrases. 3-manifold,finite-typeinvariant,groupring,N-series,dimensionsubgroup. 1

2 Givenacompactoriented3-manifoldM,ahandlebodyH⊂MandaTorelliauto- morphismhof∂H(thatis,h:∂H→∂Hisahomeomorphismwhichactstriviallyin homology),onecanformanewcompactoriented3-manifold: M h :=(M\intH)∪ h H. ThemoveM ❀ M h iscalledaTorellisurgery. Apossiblecharacterizationofafinite-typeinvariantisasfollows.Letfbeaninvariant ofcompactoriented3-manifoldswithvaluesinA,anAbeliangroup.Then,fisafinite- typeinvariantofdegreeatmostdif,foranymanifoldMandforanysetΓofd+1 pairwisedisjointhandlebodiesinM–eachcomingwithaTorelliautomorphismofits boundary–thefollowingidentityholds: X (1.1)(−1) |Γ′|
f(M Γ ′ )=0∈A. ′ Γ⊂Γ Here,M Γ ′ denotesthemanifoldobtainedfromMbythesimultaneousTorellisurgery alongthehandlebodiesbelongingtoΓ ′ . Thereareplentiesoffinite-typeinvariants[15]andthenaturalproblemisto“quan- tify”howfinetheyare,degreebydegree.Forthis,GoussarovandHabirohaveintro- duced,foreveryintegerk≥1,theY k -equivalence.Thisequivalencerelationamong manifoldscanbecharacterizedasfollows:Twocompactoriented3-manifoldsMand M ′ areY k -equivalentifM ′ canbeobtainedfromMbyaTorellisurgery M ❀ M h =M ′ suchthathbelongstothek-thtermofthelowercentralseriesoftheTorelligroupof∂H. Fact. IftwomanifoldsareY d +1 -equivalent,thentheyarenotdistinguishedbyfinite-type invariantsofdegreeatmostd. Theconversehasbeenprovedforintegralhomology3-spheresbyHabiroandGoussarov [6,4]whichprovides,inthisspecialcase,ageometriccharacterizationofthepowerof finite-typeinvariants.Unfortunately,theconversedoesnotholdingeneral. 1 Nevertheless,therearemanifoldsofaspecialtypewhichplayakeyroleinthe Goussarov–Habirotheory.LetΣbeaconnectedcompactorientedsurface.Recallfrom [6,3]thatahomologycylinderoverΣisacobordismMfrom−ΣtoΣ(withcorners, if∂Σ6= ∅ )whichcanbeobtainedfromthecylinderΣ×[−1,1]byaTorellisurgery. Homologycylinderscanbecomposed,andtheirmonoidisdenotedbyCyl(Σ). Conjecture (Goussarov–Habiro) . TwohomologycylindersoverΣareY d +1 -equivalent if,andonlyif,finite-typeinvariantsofdegreeatmostddonotseparatethem. Inthispaper,weidentifythealgebraunderlyingthe“Goussarov–Habiroconjecture” (orGHC,forshort)byobservingthatitisaspecialinstanceofageneralproblemin grouptheory,whichisknownasthe“dimensionsubgroupproblem”(orDSP,forshort). Classically,foragivengroupG,thequestionistodecidewhetherthei-thsubgroupof thelowercentralseriescoincideswiththei-thdimensionsubgroup,i.e.thissubgroup determinedbythei-thpoweroftheaugmentationidealof Z [G].Here,weconsiderthe DSPinthemoregeneralsettingwherethelowercentralseriesisreplacedbyanN-series inLazard’ssense[14].SeeSection2foraprecisestatementofthatproblem. 1 Wehaveinsertedacounter-examplebelow,inSection3.