CUP PRODUCTS, LOWER CENTRAL SERIES, AND HOLONOMY LIE ALGEBRAS 7 ALEXANDERI.SUCIU1ANDHEWANG 1 0 Abstract. WegeneralizebasicresultsrelatingtheassociatedgradedLiealgebraandtheholonomy 2 Liealgebrafromfinitelypresented,commutator-relatorsgroupstoarbitraryfinitelypresentedgroups. n In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2- a complex,andanalgorithmforcomputingthecorrespondingholonomyLiealgebra,usingaMagnus J expansionmethod.Weillustrateourapproachwithexamplesdrawnfromavarietyofgroup-theoretic 6 andtopologicalcontexts,suchaslinkgroups,one-relatorgroups,andfundamentalgroupsofSeifert 2 fiberedmanifolds. ] T G . Contents h t a 1. Introduction 1 m 2. Expansions forfinitelypresented groups 4 [ 3. Echelonpresentations andcellularchaincomplexes 7 1 4. Grouppresentations and(co)homology 9 v 5. Apresentation fortheholonomyLiealgebra 12 8 6. Lowercentral seriesandtheholonomy Liealgebra 15 6 7. Mildnessandgraded-formality 18 7 7 8. One-relatorgroups 20 0 9. Seifertfiberedspaces 23 . 1 References 25 0 7 1 : v 1. Introduction i X Throughout, we will let G be a finitely generated group. Our main focus will be on the cup- r product in the rational cohomology of the 2-complex associated to a presentation of G, and on a severalrationalLiealgebras attachedtosuchagroup. 1.1. Magnusexpansionsandcupproducts. Thenotionofexpansionofagroup,whichgoesback toW.Magnus (cf.[17]),hasbeengeneralized andusedinmanyways. Forinstance, apresentation fortheMalcevLiealgebraofafinitelypresentedgroupwasgivenbyS.Papadima[24]andG.Mas- suyeau [19], while X.-S.Lin [16]studied expansions of fundamental groups of smooth manifolds. 2010MathematicsSubjectClassification. Primary20F40,57M05. Secondary17B70,20F14,20J05. Keywordsandphrases. Lowercentralseries,derivedseries,holonomyLiealgebra,gradedformality,Magnusexpan- sion,cohomologyring,ChenLiealgebra,linkgroup,one-relatorgroup,Seifertmanifold. 1SupportedinpartbytheNationalSecurityAgency(grantH98230-13-1-0225)andtheSimonsFoundation(collabo- rationgrantformathematicians354156). 1 2 ALEXANDERI.SUCIUANDHEWANG Recently, D. Bar-Natan [2] has generalized the notion of expansion and has introduced the Taylor expansionofanarbitraryring. Inturn,weexploredin[33]variousrelationshipsbetweenexpansions andformalityproperties ofgroups. We go back here to the classical Magnus expansion, and adapt it for our purposes. Let G be a group withafinitepresentation G = F/R = hx ,...,x | r ,...,r i. Thereexiststhena2-complex 1 n 1 m K = K associated tosuchapresentation. Inthecase whenG isacommutator-relators group, i.e., G whenall relators r belong to thecommutator subgroup [F,F], R.Fennand D.Sjervecomputed in i [8]thecup-product map (1) µ : H1(K;Z)∧H1(K;Z) // H2(K;Z) , u∧v 7→ u∪v, K usingtheMagnusexpansion M: ZF → ZhhxiifromthegroupringofthefreegroupF = hx ,...,x i 1 n to the power series ring in n non-commuting variables, which is the ring morphism defined by M(x)= 1+ x. i i Ourfirstobjective inthisworkistogeneralize thisresult ofFennandSjerve,fromcommutator- relators groups to arbitrary finitely presented groups. We will avoid possible torsion in the first homology ofG by working overthe fieldof rationals. Tothat end, westart by defining aMagnus- likeexpansion κ = κ relativetosuchagroupGasthecomposition G (2) QF M // T(H (F;Q)) T(ϕ∗)// T(H (G;Q)), 1 1 b where ϕ: F ։ G is the canonical probjection. We then sbhow in Proposition 3.3 that there exists a group G admitting a ‘row-echelon’ presentation, G = hx ,...,x | w ,...,w i, and a map e e 1 n 1 m f: K → K inducing anisomorphism incohomology. Ge G Using the κ-expansion ofG , we determine in Theorem 4.3 the cup-product map for K , from e Ge which we obtain in Theorem 4.4 a formula for computing the cup-product map µ , with Q-coef- K ficients. Let b = b (G) be the first Betti number of G, and let {u ,...,u } and {β ,...,β } be 1 1 b n−b+1 m basesforH1(K;Q)andH2(K;Q),transferredfromsuitablebasesintherationalcohomologyofK Ge viatheisomorphism f∗: H∗(K ;Q) → H∗(K ;Q). Ourresultthenreadsasfollows. Ge G Theorem 1.1. Let K be a presentation 2-complex for a finitely presented group G. In the bases described above, thecup-product mapµ : H1(K;Q)∧H1(K;Q) → H2(K;Q)isgivenby K m u ∪u = κ(w ) β , for1 ≤ i, j≤ b. i j k i,j k k=Xn−b+1 Let us note that the map µ depends on the chosen presentation forG, and may differ from the K cup-product map µ in a classifying space for G. However, the two maps share the same kernel, G which makes the algorithm for finding µ useful in other contexts, for instance, in computing the K first resonance variety of G (see, e.g. [21, 37]), or finding a presentation for the holonomy Lie algebrah(G),aprocedure thatwediscuss next. 1.2. HolonomyLiealgebras. Theholonomy Liealgebra ofafinitelygenerated groupG,denoted byh(G),isthequotientofthefreeLiealgebraon H (G;Q)bytheLieidealgenerated bytheimage 1 of the dual to the cup-product map µ . It is easy to see that h(G) is a graded Lie algebra over Q G which admits a quadratic presentation depending only on kerµ . Moreover, this construction is G functorial. The holonomy Lie algebra was introduced by T. Kohno in [10], building on work of Chen[6],andhasbeenfurtherstudied inanumberofpapers, including [18,25,33]. CUPPRODUCTS,LOWERCENTRALSERIES,ANDHOLONOMYLIEALGEBRAS 3 Our next objective is to find a presentation for the holonomy Lie algebra h(G). We start by showing in Proposition 5.4 that there is homomorphism from a finitely presented group G to G f inducing an isomorphism on holonomy Lie algebras. Hence, without loss of generality, we may assumethatG admitsafinitepresentation. LetG = hx ,...,x | w ,...,w ibeagroupwithrow-echelon presentation, andletρ: G →G e 1 n 1 m e bethehomomorphism induced onfundamental groups bytheaforementioned map, f: K → K . Ge G It follows from Corollary 5.3 that the induced map, h(ρ): h(G ) → h(G), is an isomorphism of e graded Lie algebras. Using the computation of the cup-product map µ from Theorem 1.1, we KG describeinTheorem5.5analgorithm forfindingapresentation fortheholonomyLiealgebrah(G). Furthermore,weobtaininTheorem5.11apresentationforthederivedquotientsofthisLiealgebra, h(G)/h(G)(i). Wesummarizeourresults, asfollows. Theorem 1.2. LetG be a finitely presented group. The holonomy Liealgebra h(G) is the quotient of the free Q-Lie algebra with generators y = {y ,...,y } in degree 1 by the ideal I generated by 1 b κ (w ),...,κ (w ),whereκ isthedegree2partoftheMagnusexpansion ofG . Furthermore, 2 n−b+1 2 m 2 e foreachi≥ 2,thesolvable quotient h(G)/h(G)(i) isisomorphic tolie(y)/(I +lie(i)(y)). In the special case when G admits a presentation with only commutator relators, presentations for these Lie algebras were given by Papadima and Suciu in [25]. For arbitrary finitely generated groups G, the metabelian quotient h(G)/h(G)′′, also known as the holonomy Chen Lie algebra of G, is closely related to the first resonance variety ofG, a geometric object which has been studied intensely frommanypointsofview,seeforinstance [21,28,29,35,36,37]andreferences therein. 1.3. Lower central series, graded formality, and mildness. In the 1930s, P. Hall, W. Magnus, andE.Witt(cf.[17])introduced theassociated gradedLiealgebra, gr(G),asthefinitelygenerated graded Lie algebra whose graded pieces are the successive quotients of the lower central series of G, tensored with Q, and whose Lie bracket is induced from the group commutator. The quintes- sential exampleistheassociated graded Liealgebra ofthefreegrouponngenerators, F ,whichis n isomorphic tothefreeLiealgebralie(Qn). As we recall in §6.2, there is a natural epimorphism Φ : h(G) ։ gr(G). Thus, h(G) may be G viewed as a quadratic approximation to the associated graded Lie algebra of G. We say that G is graded-formal if the mapΦ isan isomorphism of graded Liealgebras. Amuch stronger require- G mentisthatGbe1-formal,acondition werecallin§2.1. Formuchmoreonthesenotions,werefer to[25,26,33]. InPropositions 6.2and 6.3, wecompare the holonomy Liealgebra ofG withthe holonomy Lie algebras of the nilpotent quotients G/ΓG and the derived quotients G/G(i). In Corollary 6.6, we i use Theorem 1.2 and a result from [33] to give explicit presentations for the graded Lie algebras gr G/G(i) inthecasewhenG isafinitelypresented, 1-formalgroup. SomeofthemotivationforourstudycomesfromtheworkofJ.Labute[11,12]andD.Anick[1], (cid:0) (cid:1) who gave presentations for the associated graded Liealgebra gr(G) in the case whenG is ‘mildly’ presented. We revisit this topic in §7, where we relate the notion of mild presentation to that of graded formality, and derive some consequences, especially, we study these notions in the context oflinkgroups. 1.4. Further applications. We illustrate our approach with several classes of finitely presented groups, including 1-relator groups in §8 and fundamental groups of orientable Seifert fibered 3- manifolds in §9. We give here presentations for the holonomy Lie algebras h(G) and the Chen 4 ALEXANDERI.SUCIUANDHEWANG Lie algebras gr(G/G′′) of such groups G. We also compute the Hilbert series of these graded Lie algebras, anddiscuss theformalityproperties ofthesegroups. This work was motivated by a desire to generalize some of the results of [8] and [25], from commutator-relators groups toarbitrary finitelygenerated groups. In[33],westudied theformality propertiesoffinitelygeneratedgroups,focusingonthefiltered-formalityand1-formalityproperties. Inrelated work, weapply thetechniques developed inthispaperandin[33]tothestudy ofseveral families of “pure-braid like” groups. For instance, we investigate in [35] the pure virtual braid groups,andweinvestigatein[36]theMcCoolgroups,alsoknownasthepureweldedbraidgroups. Asummaryoftheseresults, aswellasfurther motivationandbackground canbefoundin[34]. 2. Expansionsforfinitelypresentedgroups In this section, we introduce and study a Magnus-type expansion relative to a finitely presented group. Westartbyreviewingsomebasicnotions. 2.1. Completedgroup algebras andexpansions. LetG beafinitely generated group. Asshown by Quillen in [31, Appendix A], the group-algebra QG has a natural Hopf algebra structure, with comultiplication∆: QG → QG⊗ QGgivenby∆(g) = g⊗gforg∈G,andcounittheaugmentation Q map ε: QG → Q given by ε(g) = 1. The powers of the augmentation ideal I = kerε form a descending, multiplicative filtrationofQGbyHopfideals. The associated graded algebra, gr(QG) = Ik/Ik+1, comes endowed with the degree filtra- k≥0 tion, F = Ij/Ij+1. The correspondinLg completion, gr(QG), is again an algebra, endowed k j≥k withtheinveLrselimitfiltration. TheI-adiccompletionofthegroup-algebra, QG = lim QG/Ik,also b ←−−k comesequipped withadescending filtration. Applying the I-adic completion functor tothemap∆ d yieldsacomultiplication map∆,whichmakesQGintoacompleteHopfalgebra. An element x in a Hopf algebra is called primitive if ∆x = x⊗1 + 1⊗x. The set m(G) of all b d primitive elements in QG, with bracket [x,y] = xy−yx, is a complete, filtered Lie algebra, called theMalcevLiealgebraofG. Thesetofallprimitiveelementsingr(QG)formsagradedLiealgebra, d whichisisomorphic totheassociated gradedLiealgebra (3) gr(G):= (Γ G/Γ G)⊗Q, k k+1 Mk≥1 where {Γ G} is the lower central series of G, defined inductively by Γ G = G and Γ G = k k≥1 1 k+1 [Γ G,G]for k ≥ 1. As shown by Quillen in [30], there is an isomorphism of graded Lie algebras, k gr(m(G)) (cid:27) gr(G). ThegroupGissaidtobefiltered-formal ifitsMalcevLiealgebraisisomorphic(asafilteredLie algebra) to the degree completion of its associated graded Lie algebra. The group G is said to be 1-formal if its Malcev Lie algebra admits a quadratic presentation (see [26, 33] for more details). Forinstance, allfinitelygenerated freegroupsandfreeabelian groupsare1-formal. Anexpansion for agroupG isa filtration-preserving algebra morphism E: QG → gr(QG)with the property that gr(E) = id (see [16, 2, 33]). As shown in [33], a finitely generated group G is filtered-formalifandonlyifithasanexpansionE whichinducesanisomorphismofcombpleteHopf algebras, E: QG → gr(QG). b d b CUPPRODUCTS,LOWERCENTRALSERIES,ANDHOLONOMYLIEALGEBRAS 5 2.2. The Magnus expansion for a free group. Let F be a finitely generated free group, with generating set x = {x ,...,x }, and let ZF be its group-ring. Then the degree completion of the 1 n associated graded ring, gr(ZF), can be identified with the completed tensor ring T(F ) = Zhhxii, ab thepowerseriesringoverZinnnon-commutingvariables,bysending[x −1]to x. Thereisawell i i knownexpansion M: ZFb→ Zhhxii,calledtheMagnusexpansion, givenby b (4) M(x) = 1+ x and M(x−1) = 1− x + x2− x3+··· . i i i i i i TheFoxderivativesaretheringmorphisms∂ : ZF → ZF definedbytherules∂(1) = 0,∂(x )= i i i j δ , and∂(uv) = ∂(u)ε(v)+u∂(v)foru,v ∈ ZF,whereε: ZF → Zisthe augmentation map. The ij i i i higher Fox derivatives ∂ are then defined inductively. We refer to [17, 8, 21] for more details i1,...,ik andreferences onthesenotions. The Magnus expansion can be computed in terms of the Fox derivatives, as follows. Given an elementy ∈ F,ifwewrite M(y)= 1+ a x ,thena = ε (y),whereI = (i ,...,i ),andε = ε◦∂ I I I I 1 s I I is the composition of the augmentatioPn map with the iterated Fox derivative ∂I: ZF → ZF. For eachk ≥ 1,let M bethecomposite k Mk )) (5) ZF M // T(F ) grk // gr (T(F )) . ab k ab ForeachyinF,wehavethat M1(y) = bni=1εi(y)xi,whileforbeachyinthecommutatorsubgroup [F,F],wehave P (6) M (y) = ε (y)(x x − x x). 2 i,j i j j i Xi<j ThetensoralgebraT(F )ontheQ-vectorspaceF = F ⊗QhasanaturalgradedHopfalgebra Q Q ab structure, with comultiplication ∆ and counit ε given by ∆(a) = a ⊗ 1 + 1 ⊗ a and ε(a) = 0 for a ∈ F . The set of primitive elements in T(F ) is the free Lie algebra lie(F ) = {v ∈ T(F ) | Q Q Q Q ∆(v) = v ⊗ 1 + 1 ⊗ v}, with Lie bracket [v,w] = v ⊗ w − w ⊗ v. Notice that, if y ∈ [F,F], then M (y) is a primitive element in the degree 2 piece of the Hopf algebra T(F ) = gr(T(F )), which 2 Q Q corresponds tothedegree2element ε (y)[x,x ]inthefreeLiealgebralie(F ). i<j i,j i j Q b P 2.3. The Magnus expansion relative to a finitely generated group. Given a finitely generated group G, there exists an epimorphism ϕ: F ։ G from a free group F of finite rank to G. Let ϕ : F ։G betheinducedepimorphism betweentherespective abelianizations. ab ab ab Definition 2.1. The Magnus κ-expansion for F relative toG, denoted by κ (or κ for short), is the G composite κ (7) (( ZF M // T(F ) T(ϕab) // T(G ) , ab ab b whereMistheclassicalMagnusexpansionfobrthefreegroupFb,andthemorphismT(ϕ ): T(F ) ։ ab ab T(G )isinduced bytheabelianization mapϕ . ab ab b b b In particular, if the group G is a commutator-relators group, i.e., if all the relators of G lie in the commutator subgroup [F,F], then the projection ϕ identifies F with G , and the Magnus ab ab ab expansion κcoincides withtheclassicalMagnusexpansion M. 6 ALEXANDERI.SUCIUANDHEWANG Moregenerally,letGbeagroupgeneratedbyx= {x ,...,x },andletF bethefreegroupgener- 1 n atedbythesameset. Therational Magnusκ-expansion, stilldenoted byκ orκ,isthecomposition G (8) QF M // T(F ) T(π) // T(G ) , Q Q b where π = ϕab ⊗Q = H1(ϕ,Q) is the inducebd epimorphism ibn homology from FQ := H1(F;Q) to G := H (G;Q). Pickabasis y = {y ,...,y }forG , andidentify T(G )with Qhhyii. Letκ(r) be Q 1 1 b Q Q I thecoefficient ofy := y ···y inκ(r),for I = (i ,...,i ). Thenwecanwrite I i1 is 1 s b (9) κ(r) = 1+ κ(r) ·y . I I XI Lemma2.2. Ifr ∈ Γ F,thenκ(r) = 0,for|I|< k. Furthermore, ifr ∈ Γ F,thenκ(r) = −κ(r) . k I 2 i,j j,i Proof. Since M(r) = ε (r) = 0for|I| < k(seeforinstance [21]),wehavethatκ(r) = 0for|I| < k. I I I To prove the second assertion, identify the completed symmetric algebras Sym(F ) and Sym(G ) Q Q with the power series rings Q[[x]] and Q[[y]], respectively, in the following commuting diagram of d d Q-linearmaps. (10) QF M // T(F ) α1 // Sym(F ) Q Q ❉ ❉ ❉ ❉❉❉κ❉b T(π) d Sym(π) ❉"" (cid:15)(cid:15) (cid:15)(cid:15) T(Gb) α2 // Sym(Gd). Q Q When r ∈ [F,F], we have that α ◦ κb(r) = Sym(π)d◦ α ◦ M(r) = 1. Thus, κ(r) = 0 and 2 1 i κ(r) +κ(r) = 0. (cid:3) i,j j,i d Lemma2.3. Ifu,v ∈ F satisfyκ(u) = κ(v) = 0forall|J|< s,forsome s≥ 2,then J J κ(uv) = κ(u) +κ(v) , for|I|= s. I I I Moreover, theaboveformulaisalwaystruefor s= 1. Proof. We have that κ(uv) = κ(u)κ(v) for u,v ∈ F. If κ(u) = κ(v) = 0 for all |J| < s, then J J κ(u) = 1+ κ(u) y uptohigher-order terms,andsimilarlyforκ(v). Then |I|=s I I P (11) κ(uv) = κ(u)κ(v) = 1+ (κ(u) +κ(u) )y +higher-order terms. I I I |XI|=s Therefore, κ(uv) = κ(u) +κ(v),andsoκ(uv) = κ(u) +κ(v) . (cid:3) i i i I I I 2.4. Truncating the Magnus expansions. Recall that we defined in (5) truncations M of the k Magnus expansion M ofafree group F. Inasimilar manner, wecan also definethe truncations of theMagnusexpansion κforanyfinitelygenerated groupG. Lemma2.4. Foreachk ≥ 1,thefollowingdiagram commutes: QF M // T(F ) grk // gr (T(F )) kQn (12) ❋❋❋❋❋❋κ❋❋❋""b (cid:15)(cid:15) TQ(π) k b(cid:15)(cid:15) grkQ(T(π)) N (cid:15)(cid:15) ⊗kπ T(Gb) grk // gr (T(G b)) kQb. Q k Q N b b CUPPRODUCTS,LOWERCENTRALSERIES,ANDHOLONOMYLIEALGEBRAS 7 Proof. The triangle on the left of diagram (12) commutes, since it consists of ring morphisms, by the definition of the Magnus expansion for a group. The morphisms in the two squares are homomorphisms betweenQ-vectorspaces. Thesquares commute,sinceπisalinearmap. (cid:3) Indiagram (12),letusdenotethecomposition ofκandgr byκ . Wethenobtainthediagram k k κk (13) )) QF κ // T(G ) grk // gr (T(G )) . Q k Q Inparticular, κ (r) = b κ(r)y forr ∈ F.bByLemma2.2,ifrb∈ [F,F],then 1 i=1 i i P (14) κ (r) = κ(r) (yy −y y). 2 i,j i j j i 1≤Xi<j≤b Noticethatκ (r)isaprimitiveelementintheHopfalgebraT(G ),whichcorrespondstotheelement 2 Q κ (r)[y,y ]inthefreeLiealgebra lie(G ). i<j i,j i j Q Thenext lemmaprovides a close connection between the Magnus expansion κ and the classical P Magnusexpansion M. Lemma2.5. Let(a )betheb×nmatrixassociated tothelinearmapπ: F →G ,andletr ∈ F i,s Q Q beanarbitrary element. Then,foreach1≤ i, j ≤ b,wehavethat n n κ(r) = a ε (r) and κ(r) = a a ε (r). i i,s s i,j i,s j,t s,t Xs=1 sX,t=1 Proof. Byassumption, π(x ) = b a y. ByLemma2.4(fork = 1),wehave s i=1 i,s i P n n b κ (r) = π◦M (r) = π ε (r)x = a ε (r)y, 1 1  s s i,s s i whichgivestheclaimedformulaforκ(r). ByXs=L1emma2.4(fXso=r1kXi==12),wehave i n n b κ (r) = π⊗π◦ M (r) = π⊗π ε (r)x ⊗ x = ε (r)a a y ⊗y , 2 2  s,t s t s,t i,s j,t i j whichgivestheclaimedformulaforκ(r)sX,t.=1  sX,t=1iX,j=1 (cid:3) i,j 3. Echelonpresentationsandcellularchaincomplexes Inthissection weassociate toeveryfinitelypresented groupG an“echelon approximation”, G , e suchthattheyhaveisomorphiccohomology ontheirrespective2-complexes. 3.1. Presentation 2-complex. Westartwithabriefreviewofthecellular chaincomplexes associ- ated toapresentation 2-complex ofagroup, following theexposition from [4,8,21,27]. LetG be a group with a finite presentation P = hx | ri, where x = {x ,...,x } and r = {r ,...,r }. Then 1 n 1 m G = F/R,where F isthefreegroup ongenerating setxandRisthe(free) subgroup of F normally generated bythesetr ⊂ F. Let K bethe2-complex associated tothispresentation ofG,consisting ofa0-celle0,one-cells P {e1,··· ,e1}correspondingtothegenerators,andtwo-cells{e2,...,e2}correspondingtotherelators. 1 n 1 m 8 ALEXANDERI.SUCIUANDHEWANG The 2-complex K depends on the presentation P for the group G. However, if the presentation is P understood, wemayalsodenotethis2-complexby K . G The (integral) cellular chain complex C = C (K ;Z) is of the form C −−d→2 C −−d→1 C , where ∗ ∗ P 2 1 0 C are the free abelian groups on the specified bases. Furthermore, d = 0, whilethe matrix of the j 1 boundary mapd : C (K ;Z) →C (K ;Z)isthem×nJacobianmatrix J = (ε(r )). 2 2 P 1 P P i k Next, let p: K → K be the universal cover of the presentation 2-complex, and fix a lift e˜0 of P P the basepoint e0. The cells ei of K lift to cells e˜i at the basepoint e˜0. LetC = C (K ;Z) be the f j P j ∗ ∗ P (equivariant) cellular chain complex of the universal cover. This is a chain complex of free ZG- e f modulesoftheformC −−d→˜2 C −−d→˜1 C ,withC = ZG,C = (ZG)n generatedbytheset{e˜1,...,e˜1}, 2 1 0 0 1 1 n and C = (ZG)m generated by the set {e˜2,...,e˜2}. The differentials in this chain complex are the 2 e e e 1 e m e ZG-linearmapsgivenby e m (15) d˜ (e˜1)= x −1, d˜ (e˜2) = ϕ(∂ r )e˜1, 1 i i 2 j k j k Xk=1 whereϕ: F ։Gisthepresenting homomorphism forourgroup. 3.2. Echelon presentations. We now introduce a special type of group presentations which will playanimportantroleinthesequel. Definition 3.1. Let G be a group with a finite presentation P = hx | wi, where x = {x ,...,x } 1 n and w = {w ,...,w }. We say P is an echelon presentation if the augmented FoxJacobian matrix 1 m (ε(w ))isinrow-echelon form. i k LetK bethe2-complexassociated totheabovepresentation forG. Supposethepivotelements G of the m× n matrix (ε(w )) are in position {i ,...,i }, and let b = n−d. Since this matrix is in i k 1 d row-echelon form, the vector space H (K ;Q) = Qb has basis y = {y ,...,y }, where y = e1 1 G 1 b j id+j for 1 ≤ j ≤ b. Furthermore, the vector space H (K ;Q) = Qm−d has basis {e2 ,...,e2}. We will 2 G d+1 m chooseasbasisfor H1(K ;Q)theset{u ,...,u },whereu istheKronecker dualtoy. G 1 b i i Remark 3.2. Suppose G admits a commutator-relators presentation of the form P = F/R, with R ⊂ [F,F]. Then the augmented Fox Jacobian matrix (ε(r )) is the zero matrix, and thus the i k presentation P isan echelon presentation. In this case, the integer (co)homology groups of K are G torsion-free, andsotheaforementioned choices ofbasesworkforinteger(co)homology, aswell. More generally, the next proposition shows that for any finitely presented group, we can con- struct agroup with anechelon presentation such that the cohomology groups of the corresponding presentation 2-complexes areisomorphic. Proposition 3.3. Let G be a finitely presented group. There exists then a group G with echelon e presentation, and a map f: K → K between the respective presentation 2-complexes such that Ge G theinducedhomomorphismonfundamentalgroups,ρ = f : G ։G,issurjective, andtheinduced ♯ e homomorphism incohomology, f∗: H∗(K ;Z)→ H∗(K ;Z),inanisomorphism. G Ge Proof. SupposeG has presentation hx ,...,x | r ,...,r i. Asin theabove discussion, the matrix 1 n 1 m of the boundary map d∗: C1(K ;Z) → C2(K ;Z) is the transpose of the m × n Jacobian matrix 2 G G CUPPRODUCTS,LOWERCENTRALSERIES,ANDHOLONOMYLIEALGEBRAS 9 (ε(r )). ByGaussianeliminationoverZ,thereexistsamatrixC = (c )∈ GL(m;Z)suchthatC·d∗ i k l,k 2 isinrow-echelon form(alsoknownasHermitenormalform). Wedefineanewgroup, (16) G = hx ,...,x | w ,...,w i, e 1 n 1 m bysettingw = rc1,krc2,k ···rcm,k for1≤ k ≤ m. k 1 2 m Leth: K(1) → K(1) bethehomeomorphism betweenthe1-skeleta oftherespective 2-complexes Ge G obtained by matching 1-cells. If ψ : S1 → K(1) denotes the attaching map of the 2-cell in K k Ge Ge corresponding to the relator w , then by construction h ◦ ψ is null-homotopic in K . Thus, h k k G extends to a cellular map f: K → K . Clearly, the induced homomorphism ρ = f : G → G Ge G ♯ e is surjective. Furthermore, the map f induces a chain map between the respective cellular chain complexes, f : C (K ;Z) → C (K ;Z), with f given by the matrixC. It is now straightforward ∗ ∗ Ge ∗ G 2 toseethatthemap f induces anisomorphism inhomology, andthus, bytheUniversalCoefficients theorem,anisomorphism incohomology, too. (cid:3) Note that the group G constructed above depends on the given (finite) presentation for G, not e justontheisomorphismtypeofG. Ontheotherhand,ifGisacommutator-relators group,then,by Remark3.2,thegroupG isisomorphictoG. e 3.3. Atransferredbasis. Onceagain,letGbeagroupadmittingafinitepresentationhx | ri,where x = {x ,...,x } and r = {r ,...,r }, and let K be the corresponding presentation 2-complex. 1 n 1 m G Using an echelon approximation for the given presentation, we describe now convenient bases for the Q-vector spaces H1(K ,Q) and H2(K ,Q), which will be used extensively in the next two G G sections. By Proposition 3.3, there exists a group G with echelon presentation hx | wi, where w = e {w ,...,w },and amap f: K → K inducing anisomorphism in(co)homology. Asin§3.2, we 1 m Ge G maychooseabasisy = {y ,...,y }fortheQ-vectorspaceH (K ;Q) (cid:27) H (K ;Q);let{u ,...,u } 1 b 1 G 1 Ge 1 b be the dual basis for H1(K ;Q) (cid:27) H1(K ;Q). We also choose a basis {z ,...,z } for C (K ;Q) G Ge 1 m 2 G andabasis{e2,...,e2}forC (K ;Q)corresponding to{1⊗e˜2,...,1⊗e˜2}. Finally,ifweset 1 m 2 Ge 1 m m (17) γ := f (e2)= c z , k ∗ k l,k l Xl=1 then{γ ,...γ }isanotherbasisforC (K ;Q). Furthermore,{e2 ,...,e2}isabasisforH (K ;Q) 1 m 2 G d+1 m 2 Ge and{γ ,...,γ }isabasisforH (K ;Q). Thus, H2(K ;Q)hasdualbasis{β ,...,β }. d+1 m 2 G G d+1 m 4. Grouppresentationsand(co)homology Wecomputeinthissectionthecup-productinthecohomologyringofthe2-complexofafinitely presented groupintermsoftheMagnusexpansion associated tothepresentation. 4.1. A chain transformation. We start by reviewing the classical bar construction. Let G be a discrete group, and let B (G)be the normalized bar resolution (see e.g. [4, 8]), where B (G)is the ∗ p free left ZG-module on generators [g |...|g ], withg ∈ G and g , 1, and B (G) = ZG is free on 1 p i i 0 onegenerator, []. Theboundary operators areG-modulehomomorphisms, δ : B (G) → B (G), p p p−1 10 ALEXANDERI.SUCIUANDHEWANG definedby p−1 (18) δ [g |...|g ]= g [g |...|g ]+ (−1)i[g |...|gg |...|g ]+(−1)p[g |...|g ]. p 1 p 1 2 p 1 i i+1 p 1 p−1 Xi=1 In particular, δ [g] = (g−1)[ ] and δ [g |g ] = g [g ]−[g g ]+[g ]. Let ε: B (G) → Z be the 1 2 1 2 1 2 1 2 1 0 ε augmentation map. Wethenhaveafreeresolution B (G)→− ZofthetrivialG-moduleZ. ∗ WeviewhereZasarightZG-module,withactioninducedbytheaugmentationmap. Anelement of the cochain group Bp(G) = Hom (B (G),Z) may be viewed as a set function u: Gp → Z ZG p satisfying the normalization condition u(g ,...,g ) = 0 if some g = 1. The cup-product of two 1 p i 1-dimensional classesu,u′ ∈ H1(G;Z)(cid:27) Hom(G,Z)isgivenby (19) u∪u′[g |g ]= u(g )u′(g ). 1 2 1 2 Forfutureuse,werecordaresultduetoFennandSjerve([8,Thm.2.1andp.327]). Lemma 4.1 ([8]). There exists a chain transformation T: C (K ) → B (G) of augmented chain ∗ G ∗ complexes, f 0 oo Z oo ε C (K ) oo d˜1 C (K ) oo d˜2 C (K ) oo 0 oo ··· 0 G 1 G 2 G fT0 fT1 fT2 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) 0 oo Z oo ε B (G) oo δ1 B (G) oo δ2 B (G) oo B (G) oo ··· . 0 1 2 3 definedbyT (λ) := λ[], 0 (20) T (e˜1) = [x] and T (e˜2) = τ T d˜ (e˜2), 1 i i 2 k 1 1 2 k whereτ : B (G)→ B (G)isthehomomorphism definedby 1 1 2 (21) τ (g[g ]) = [g|g ], 1 1 1 forallg,g ∈G. 1 4.2. Cup products for echelon presentations. Now let G be a group with echelon presentation G = hx | wi,where x = {x ,...,x }and w = {w ,...,w },as inDefinition 3.1. Welet B (G;Q) = 1 n 1 m ∗ Q⊗B (G)and B∗(G;Q) = Q⊗B∗(G). ∗ Lemma 4.2. For each basis element u ∈ H1(K ;Q) (cid:27) H1(G;Q) as above, and each r ∈ F, we i G havethat n u([ϕ(r)]) = ε (r)a = κ(r), i s i,s i Xs=1 where(a )istheb×nmatrixfortheprojection mapπ: F →G . i,s Q Q Proof. Ifr ∈ F,thenϕ(r) ∈Gand[ϕ(r)] ∈ B (G). Hence, 1 n n (22) u([ϕ(r)]) = ε (r)u([x ]) = ε (r)a = κ(r). i s i s s i,s i Xs=1 Xs=1 Since H1(G;Q) (cid:27) B1(G;Q) (cid:27) Hom(G,Q), we may view u as a group homomorphism. This i yields the first equality in (22). Since π(x ) = b a y and u = y∗, the second equality follows. s j=1 i,s i i i Thelastequality followsfromLemma2.5. P (cid:3)