We prove that there is no functorial universal finite type invariant for
braids in $\Sigma\times I$ if the genus of $\Sigma$ is positive.Comment: 6p., revised version, C.R.Acad.Sci.Paris, to appea

Bellingeri, Paolo

Funar, Louis

English

arXiv

Paolo Bellingeri

Louis Funar

Comptes Rendus Mathématique

ypeusient déjà’abordz-Menesesou encoren nombreent calcu-u gradué.s, donc lesoeffi-C. R. Acad. Sci. Paris, Ser. I 338 (2004) 157–162Differential TopologyBraids on surfaces and finite type invariantsPaolo Bellingeria, Louis Funarba Mathématiques, cc 051, Univ. Montpellier II, place Eugène Bataillon, 34095 Montpellier cedex 5, Franceb Institut Fourier, BP 74, Univ. Grenoble I, Mathématiques, 38402 Saint-Martin-d’Hères cedex, FranceReceived 6 October 2003; accepted after revision 14 November 2003Presented by Étienne GhysAbstractWe prove that there is no functorial universal finite type invariant for braids inΣ × I if the genus ofΣ is positive.To citethis article: P. Bellingeri, L. Funar, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2003 Published by Elsevier SAS on behalf of Académie des sciences.RésuméTresses sur les surfaces et invariants de type fini.Nous démontrons qu’il n’y a pas d’invariant universel fonctoriel de tfini pour les tresses dansΣ × I , lorsqueΣ est une surface orientable de genre positif.Pour citer cet article : P. Bellingeri,L. Funar, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2003 Published by Elsevier SAS on behalf of Académie des sciences.Version française abrégéeSoit Σ une surface compacte, connexe et orientable. Le groupeB(Σ,n) de tresses àn brins surΣ est unegénéralisation naturelle du groupe de tresses d’ArtinBn et du groupe fondamental deΣ . Ces groupes sont apparpour la première fois dans l’étude des espaces de configurations ([6], voir aussi [4]), bien que certains étaconnus avant (le groupeB(S2, n) a été introduit par Hurwitz). Des présentations pour ces groupes ont été dtrouvées par Scott [17] dans le cas des surfaces fermées. Celles-ci ont été ensuite simplifiées par Gonzále[8], et améliorées et généralisées pour les surfaces à bord dans [3].Le Z-module libre engendré par des objets 1-dimensionels plongés (comme les tresses, les entrelacsles enchevêtrements) admet une filtration naturelle obtenue par la résolution des objets singuliers ayant ufini de points doubles. L’algèbre graduée associée (appelée algèbre de diagrammes) peut être explicitemlée et elle a des propriétés remarquables de finitude. Dans ce contexte, un invariant universel est uneapplicationde notre catégorie dans une complétion de l’algèbre de diagrammes, qui induit un isomorphisme au niveaLe théorème fondamental de la théorie de Vassiliev pour les nœuds (mais aussi pour les enchevêtrementtresses en particulier) dansR3 est la construction, due à Kontsevich, d’un invariant universel de type fini à cE-mail addresses:bellinge@math.univ-montp2.fr (P. Bellingeri), funar@fourier.ujf-grenoble.fr (L. Funar).URL: http://www-fourier.ujf-grenoble.fr/~funar (L. Funar).1631-073X/$ – see front matter 2003 Published by Elsevier SAS on behalf of Académie des sciences.doi:10.1016/j.crma.2003.11.014158 P. Bellingeri, L. Funar / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 157–162s. Uneirstruire unoitfermées)t pas êtreaurcase ofa naturalumberagramsinvariantes ant. A keyiversalsant is notved:nnot berbitraryndary,escients dansQ. Un ingrédient essentiel est l’existence d’un associateur de Drinfel’d à coefficients rationnelpropriété remarquable de l’intégrale de Kontsevich est safonctorialité [14], permettant par exemple de la définpour les entrelacs en utilisant l’invariant pour les tresses. Dans le cas des tresses classiques on sait coninvariant universel de type fini à coefficients dansZ [16], mais on ne sait pas s’il existe un tel invariant qui saussimultiplicatif.González-Meneses et Paris [9] ont construit un invariant universel pour les tresses sur les surfaces (mais qui n’est pas fonctoriel. Le propos de cette Note est de démontrer que en effet leur résultat ne peuamélioré :Théorème 0.1.Il n’existe pas d’invariant universel fonctoriel de type fini pour les tresses dansΣ × I , lorsqueΣest une surface compacte, orientable de genreg  1.La preuve s’appuie sur la forme explicite des relations dansB(Σ,n) et elle ne dépend pas du choix de l’annede base considéré.1. IntroductionLet Σ be a compact, connected and orientable surface. The groupB(Σ,n) of braids onn strands overΣ is anatural generalization of both the classical braid groupBn and the fundamental groupπ1(Σ). It appeared first inthe study of configuration spaces ([6], see also [4]). Presentations forB(Σ,n) were derived by Scott [17], furtheimproved by González-Meneses for closed surfaces [8] and finally given a very simple form in [3]. In theholed spheres the latter have been previously obtained by Lambropoulou [13].Let us consider a category of embedded 1-dimensional objects like braids, links, tangles, etc. There isfiltration on the freeZ-module generated by the objects, coming from the singular objects with a given nof double points. The main feature of this filtration is that the associated grading, which is called the dialgebra, can be explicitly computed, and has some salient finiteness properties. By universal finite typeone generally means amap from our category into some completion of the diagrams algebra, which inducisomorphism at graded level. For instance the celebrated Kontsevich integral is such a universal invarianingredient is the existence of a Drinfel’d associator with rational coefficients. Notice that there exists a uninvariant for usual braids overZ [16], but it is not known whether there exists a multiplicative one overZ. In factan essential feature of the Kontsevich integral is itsfunctoriality [14]: it is precisely this property which enableone to extend the Chen iterated integrals from braids to links [14].González-Meneses and Paris constructed a universal invariant for braids on surfaces [9], but their invarifunctorial (i.e., multiplicative). The purpose of this Note is to show that actually their result cannot be improTheorem 1.1.There does not exist any functorial universal finite type invariant for braids inΣ × I , if the surfaceΣ is of genusg  1.Remark 1. In particular there does not exist any universal invariant for tangles inΣ × I which is functorialwith respect to the vertical composition of tangles. Thus the Andersen–Mattes–Reshetikhin invariant [1] camade tangle functorial. Notice that the functoriality is essential if one seeks for an extension to links in a3-manifolds. The non-existence holds true a fortiori for the category of tangles in 3-manifolds with bouunless these are cylinders over planar surfaces.2. Preliminaries2.1. Surface braids. SetΣg,p for the compact orientable surface of genusg with p boundary components. Wdenote byσ1, . . . , σn−1 the standard generators of the braid group on a disk embedded inΣg,p , viewed as elementP. Bellingeri, L. Funar / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 157–162 159t of theyreverseularizingwithfof B(Σg,p, n). Let alsoa1, . . . , ag, b1, . . . , bg, z1, . . . , zp−1 be the generators ofπ1(Σg,p), wherezi denotes a looparound thei-th boundary component. Assume that the base point of the fundamental group is the startpoinfirst strand. Then eachγ ∈ {a1, . . . , ag, b1, . . . , bg, z1, . . . , zp−1} can be realized by an element, denoted also bγ ,in B(Σg,p, n), by considering the braid whose first strand is describing the curveγ and whose other strands aconstant. We denote[a, b] = aba−1b−1. Following [3] we have:Theorem 2.1.A presentation forB(Σg,p, n) (n  2) is given by:(1) Generators: σ1, . . . , σn−1, a1, . . . , ag, b1, . . . , bg, z1, . . . , zp−1.(2.i) Braid relations:σiσi+1σi = σi+1σiσi+1 (i = 1, . . . , n− 2), σiσj = σjσi (if |i − j |  2).(2.ii) Commutativity relations:[ar , σi] = [br, σi ] = [zk, σi] = 1 (i > 1, 1  r  g, 1  k  p − 1);[ar , σ−11 arσ−11] = [br, σ−11 brσ−11] = [zk, σ−11 zkσ−11] = 1 (1  r  g, 1  k  p − 1);[ar , σ−11 asσ1] = [ar , σ−11 bsσ1] = [br, σ−11 asσ1] = [br, σ−11 bsσ1] = [zi, σ−11 zj σ1] = 1(1 s < r  g, 1  j < i  p − 1);[ar , σ−11 zkσ1] = [br, σ−11 zkσ1] = 1 (1 r  g, 1 k  p − 1);(2.iii) Skew commutativity relations on each handle(wheng > 0):[ar , σ−11 brσ−11] = σ 21 (1  r  g);(2.iv) Whenp = 0 we have the additional relation:[a1, b−11][a2, b−12] · · · [ag, b−1g] = σ1σ2 · · ·σn−2σ 2n−1σn−2 · · ·σ2σ1.2.2. The Vassiliev filtration. Singular braids are obtained from braids by allowing a finite number of transdouble points. One can associate to each singular surface braid a linear combination of braids by desingeach crossing as follows:.We denote byVd the submodule ofZ[B(Σ,n)] generated by the desingularizations of (singular) surface braidsd double points. ThenVd = J d , whereJ ⊂ Z[B(Σ,n)] is the two-sided ideal generated by{σi − σ−1i }i=1,...,n−1.Setgr∗Z[B(Σ,n)] for the associated graded algebra. A morphism ofZ-modulesu :Z[B(Σ,n)] → A is a Vassilievinvariant of degree d if u vanishes onVd+1.Definition 2.2. A universal Vassiliev invariant for braids is a linear mapZ :C[B(,n)] → A such that, for anyfinite type invariantv :C[B(,n)] → A there exists a homomorphismuv :A → A satisfyinguv ◦ Z = v. Theinvariant is functorial ifA has an algebra structure andZ is an algebra homomorphism.2.3. Presentations of the algebra gr∗Z[B(Σ,n)]. Let Sn denote the symmetric group onn elements. The proogiven in [9] forp = 0 extends without essential modifications to the general case:Proposition 2.1.The graded algebra gr∗Z[B(Σ,n)] is the quotient ofZ[Zγij ]1i,jn; γ∈π1(Σ) Sn, the semi-directproduct of a free non-commutative algebra in the given generators withSn, by the following ideal of relationsR:Zγij = Zγ−1ji ,[Zγij ,Zδkl] = 0, [Zγij ,Zδjk + Zγδik] = 0.160 P. Bellingeri, L. Funar / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 157–162dtwodealeofva-This algebra can be described in terms of chord diagrams with beads [7]. Consider the freeZ-module generateby the horizontal chord diagrams with endpoints onn vertical segments, where the vertical intervals betweensuccessive chords endpoints are labeled by elements ofZ[π1(Σ)] called beads. We obtain aZ[π1(Σ)]-algebraD(n,Σ) by imposing the (4T ) relations along with (P ) below, permitting to push beads across a chord:.SetF(Σg,p) = {As,Bs,As−1,Bs−1,Zα,Zα−1}1sg; 1αp−1. The algebraD(n,Σ) is the quotient of the freenon-commutative algebraZ[Zij , γ i ]1i,jn; γ∈F(Σg,p) by the ideal of relations:[γ i, δj] = 0, [γ i + γ j ,Zij] = 0, for i = j, andγ, δ ∈ F(Σg,p),[γ i,Zjk] = 0, Zij = Zji, [Zij ,Zkl] = 0, [Zij ,Zjk +Zik] = 0, if i, j, k are distinct.If Σ is closed one adds also the relations:∑gs=1[Ais,Bis ] = 0.Proposition 2.2.The map gr∗Z[B(Σ,n)]) →D(n,Σ)Sn sendingZγij to γ iZij γ i−1, whereγ i denotes the bealabeledγ on thei-th strand, is an isomorphism.Remark 2.D(n,Σ) is isomorphic to the algebra of trivalent graphs with beads on oriented edges from [7].We denote byH1(A) the abelianization of the graded algebraA, namely the quotient by the homogeneous idgenerated by the elementsab − ba, for a, b ∈ A.Corollary 2.3. H1(gr∗Z[B(Σ,n)]) ∼= Z[Z12,Ai1,Ai1−1,Bj1 ,Bj1−1,Zi1,Zi1−1, τ ]/(τ2 = 1) as graded commutativalgebras. Hereτ corresponds to the signature onSn. The degree ofZ12 is one, while the other generators aredegree zero.3. Proof of the theoremAssume that such a functorial universal invariantZ exists. The algebraM = Z(C[B(Σ,n)]) is filtered by theidealsVjM = Z(Vj ) and we setgr∗(M) for the associated graded algebra.Lemma 3.1.Z induces an isomorphism of graded algebrasZ∗ : gr∗C[B(Σ,n)] → gr∗M.Proof. It suffices to prove thatZ∗ is injective. One introduces the pairing〈, 〉 : grdM×Hom(grdC[B(Σ,n)],C) →C, defined as〈x,λ〉 = uI(λ)(x̂), wherex̂ ∈ VdM is a lift of x andI (λ) :C[B(Σ,n)] → C is an extension ofλ as aVassiliev invariant of degree d .Notice that anyZ-modules homomorphismλ :Vd/Vd+1 → A can be lifted (non-uniquely) to a VassilieinvariantI (λ) of degree d , which coincides withλ onVd/Vd+1.If a ∈ grdC[B(Σ,n)] ∩ kerZ∗, then〈Z∗(a), λ〉 = 0 for all λ ∈ Hom(grdC[B(Σ,n)],C). Choose a representtive â ∈ Vd for a. Then〈Z∗(a), λ〉 = uI(λ)(Z(â)) = I (λ)(â) = λ(a). This impliesa = 0 becausegrdZ[B(Σ,n)] istorsion-free (see [9]) and our claim follows.✷Lemma 3.2.The mapZ∗ab :H1(gr∗C[B(Σ,n)]) → H1(gr∗M) induced byZ∗ is trivial in degree one.P. Bellingeri, L. Funar / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 157–162 161mion ofsoioneents areuotient ofnProof. The image ofσ 21 − 1 in V1V2 is the elementZ12, associated to the diagram . Furthermoreits image underZ∗ is Z(σ 21 ) − 1 ∈ gr1(M). But Z(σ 21 ) = Z([a1, σ−11 b1σ−11 ]) = 1 ∈ H1(gr∗(M)), sinceZis multiplicative and the image of a commutator in the abelianization is trivial. ThereforeZ∗ab(Z12) = 0 andCorollary 2.3 implies our claim. ✷The first lemma implies thatZ∗ab should be an isomorphismH1(gr∗C[B(Σ,n)]) → H1(gr∗M), while the lastlemma shows that this map is trivial in degree one. But the degree one component ofH1(gr∗C[B(Σ,n)]) is nonzerofrom Corollary 2.3, and thus we get a contradiction. This settles the theorem.4. CommentsLet PB(Σ,n) be the pure surface braid group onn strands andK(Σ,n) the kernel of the natural homomorphisθ : PB(Σ,n) → π1(Σ)n which forgets about the braiding and keeps only the fundamental group informateach strand. ThenJ coincides with the two-sided ideal inZ[B(Σ,n)] generated by the augmentation idealIK(Σ,n)One cannot use the Chen iterated integrals for the subgroupK(Σ,n) since this group is not of finite type (see al[5]). An interesting alternative would be to replace the Vassiliev filtrationV∗ by that associated to the augmentatidealIPB(Σ,n). The associated graded algebrag ∗IPB(Σ,n)C[B(Σ,n)] is now isomorphic toUPB(Σ,n)  Sn, whereUG denotes the universal enveloping algebra of the Lie algebra of the groupG verC (see [11]). In this setting oncan find a functorial universal-type invariant (for the new filtration)C[B(Σ,n)] → UPB(Σ,n)Sn using the Cheniterated integrals, which takes values in the completion of an algebra ofsymplectic chord diagrams. SpecificallyUPB(Σ,n) is a chord diagram algebra as defined above, but now the actions of beads on two distinct segmnot anymore assumed to commute, and they are subjected to the following relation:.Nevertheless the grading is different from the previous algebra of chord diagrams since beads generatorsAis , Bjt arenow given the degree 1 while each chord has degree 2. Alternatively one can present this algebra as the qC[Aks ,Bkt ,Zαi,Zij ]1s,tg, 1αp, 1i,j,kn, where degAks = degBkt = 1 and degZmj = 2 (unless the case wheg = 0 and one can renormalize the degree of the generatorsAij to be 1) by the following relations:• The extended infinitesimal braid relations:Zij = Zji, [Zij ,Zkl] = 0, if {i, j } ∩ {k, l} = ∅, [Zij ,Zjk + Zik] = 0, if i, j, k are distinct.[Zαj ,Zkl] = 0, if j /∈ {k, l}, [Zαj ,Zβk] = 0, if {α, j } ∩ {β, k} = ∅, [Zαj ,Zαk + Zjk] = 0, if j = k.• The relations coming from the fundamental group ofΣ :[Ais,Akr] = [Bis ,Bkr] = 0, if i = k, [Ais,Bjr] = 0, if r = s andi = j,g∑s=1[Aks ,Bks] +n∑j=1Zjk +n+p∑α=n+1Zαk = 0.• The mixed relations:[Zjk,Ais] = 0, if i /∈ {j, k}, [Zαk,Ais] = 0, if i = k, [Ajs + Aks ,Zjk] = 0, [Bjs + Bks ,Zjk] = 0.162 P. Bellingeri, L. Funar / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 157–162ased oncticordis]).ethernce, T. Le,ggesting24 (1998)1973.0393.19.r. Knots68 (1995)102.17.• The twist relation (making sense only forg  1):[Ais,Bjs] = Zij , if i = j.The closed case is proved in [2], for the general case see ([11], Theorem 12.6). A different approach bthe weight filtration onPB(Σ,n) is given in [15]. It is not difficult to see that the completion of this symplechord diagrams algebra surjects ontoUK(Σ,n) ⊗U(π1(Σ)) and hence it furnishes an invariant in the usual chdiagrams but whose coefficients are now formal series fromU(π1(Σ)) instead of merely elements ofπ1(Σ). Therelation of the latter with the Vassiliev filtration seems quite obscure.It is worth mentioning that the universal flat Chen connection whose monodromy yields this invariantnotquadraticsince the respective configuration spaces are not formal unlessg = 0 orn = 2 (see [2]).Remark 3. In the case of surface pure braid groups the exact sequence 1→ π1(Σ \ {(n − 1) points}) →PB(Σ,n) → PB(Σ,n − 1) → 1 is not split unlessΣ is a torus orΣ has non-empty boundary (see [10Moreover even when split, the groupsPB(Σ,n) are not almost-direct products ifg  1, n  3. This explainswhy the arguments in [12] fail for higher genus, as already noticed in [2]. In particular it is still unknown whgr∗PB(Σ,n) is torsion-free or ifPB(Σ,n) is residually nilpotent, which would shed some light about the relevaof the filtration considered in this section.AcknowledgementsThe authors are indebted to J. Birman, M. Eisermann, J. González-Meneses, J. Guaschi, T. KohnoS. Lambropoulou, D. Matei, L. Paris and S. Papadima for useful discussions and to the referee for suus the present definition of a functorial universal Vassiliev invariant.References[1] J.E. Andersen, J. Mattes, N. Reshetikhin, Quantization of the algebra of chord diagrams, Math. Proc. Cambridge Philos. Soc. 1451–467.[2] R. Bezrukavnikov, Koszul DG-algebras arising from configuration spaces, Geom. Fumet. Anal. 4 (1994) 119–135.[3] P. Bellingeri, On presentation of surface braid groups, math.GT/0110129.[4] J. Birman, Braids, Links, and Mapping Class Groups, in: Ann. of Math. Stud., vol. 82, Princeton University Press, Princeton, NJ,[5] F.R. Cohen, T. Kohno, M.A. Xicotencatl, Orbit configuration spaces associated to discrete subgroups of PSL(2,R), math.AT/031[6] E. Fadell, L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111–118.[7] S. Garoufalidis, A. Kricker, A rational noncommutative invariant of boundary links, math.GT/0105028.[8] J. González-Meneses, New presentations of surface braid groups, J. Knot Theory Ramifications 10 (2001) 431–451.[9] J. González-Meneses, L. Paris, Vassiliev invariants for braids on surfaces, Trans. Amer. Math. Soc. 356 (2004) 219–243.[10] D.L. Goncalves, J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64.[11] R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597–651.[12] T. Kohno, T. Oda, The lower central series of the pure braid group of an algebraic curve, Adv. Stud. Pure Math. 12 (1987) 201–2[13] S. Lambropoulou, Braid structures in knot complements, handlebodies and 3-manifolds, in: Knots in Hellas ’98 (Delphi), in: SeEverything, vol. 24, World Sci. Publishing, River Edge, NJ, 2000, pp. 274–289.[14] T.Q.T. Le, J. Murakami, Representations of the category of tangles by Kontsevich’s iterated integrals, Commun. Math. Phys. 1535–563.[15] H. Nakamura, N. Takao, Galois rigidity of pro-l pure braid groups of algebraic curves, Trans. Amer. Math. Soc. 350 (1998) 1079–1[16] S. Papadima, The universal finite-type invariant for braids, with integer coefficients, Topology Appl. 118 (2002) 169–185.[17] G.P. Scott, Braid groups and the group of homeomorphisms of a surface, Math. Proc. Cambridge Philos. Soc. 68 (1970) 605–6

Braids on surfaces and finite type invariants

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Braids on surfaces and finite type invariants

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