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FINITE TYPE INVARIANTS OF THREE MANIFOLDS AND THE DIMENSION SUBGROUP PROBLEM

20 pages
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


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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.Homologycylindersandtheirnite-typeinvariants3.1.Finite-typeinvariantsof3-manifolds3.2.Groupofhomologycylinders3.3.TheGoussarovHabiroconjecture4.TheGoussarovHabiroconjecturewithcoecientsinaeld4.1.Takingthecoecientsinaeld4.2.Increasingthedegree5.Thealgebradualtonite-typeinvariantsofhomologycylinders5.1.QuillenstheoremforanarbitraryN-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,ahandlebodyHMandaTorelliauto-morphismhof∂H(thatis,h:∂H∂Hisahomeomorphismwhichactstriviallyinhomology),onecanformanewcompactoriented3-manifold:Mh:=(M\intH)hH.ThemoveMMhiscalledaTorellisurgery.Apossiblecharacterizationofafinite-typeinvariantisasfollows.Letfbeaninvariantofcompactoriented3-manifoldswithvaluesinA,anAbeliangroup.Then,fisafinite-typeinvariantofdegreeatmostdif,foranymanifoldMandforanysetΓofd+1pairwisedisjointhandlebodiesinM–eachcomingwithaTorelliautomorphismofitsboundary–thefollowingidentityholds:X(1.1)(1)|Γ|f(MΓ)=0A.ΓΓHere,MΓdenotesthemanifoldobtainedfromMbythesimultaneousTorellisurgeryalongthehandlebodiesbelongingtoΓ.Thereareplentiesoffinite-typeinvariants[15]andthenaturalproblemisto“quan-tify”howfinetheyare,degreebydegree.Forthis,GoussarovandHabirohaveintro-duced,foreveryintegerk1,theYk-equivalence.Thisequivalencerelationamongmanifoldscanbecharacterizedasfollows:Twocompactoriented3-manifoldsMandMareYk-equivalentifMcanbeobtainedfromMbyaTorellisurgeryMMh=Msuchthathbelongstothek-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]thatahomologycylinderoverΣisacobordismMfromΣtoΣ(withcorners,ifΣ6=)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.