Presentations for the monoids of singular braids on closed surfaces

arXiv:math/0112313v1 [math.GT] 31 Dec 2001 Presentations for the monoids of singular braids on closed surfaces Juan Gonz´alez-Meneses November, 2000 Abstract We give presentations, in terms of generators and relations, for the monoids SBn(M) of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by Birman for the monoids of Singular Artin braids. 1 Introduction In this paper we deal with the braid groups of a closed surface M. These groups are a natural generalization of Artin braid groups [A] and of the fundamental group of M. They are also subgroups of some Mapping Class groups of M, and finally they are fundamental groups of the so called Configuration spaces of M(see [B] for a general exposition). They can be defined as follows. Fix n(n≥1) distinct points {P1,...,Pn} ∈ M. A n-braid on Mis an n-tuple b= (b1, . . . , bn) of disjoint smooth paths biin M×[0,1], such that for all i, the path biruns, monotonically on t∈[0,1], from (Pi,0) to some (Pj,1). These n-braids are considered modulo isotopy (deformation of braids fixing the ends), and there exists a multiplication of braids, given by concatenation of paths. The set of isotopy classes of n-braids on M, along with this multiplication, forms the braid group with nstrings on M, denoted by Bn(M). The following is a simple presentation of Bn(M), in terms of generators and relations, where Mis a closed, orientable surface of genus g[G-M]: •Generators: σ1, . . . , σn−1, a1,...,a2g. •Relations: (R1) σiσj=σjσi(|i−j| ≥ 2) (R2) σiσi+1σi=σi+1σiσi+1 (1 ≤i≤n−2) (R3) a1···a2ga−1 1···a−1 2g=σ1···σn−2σ2 n−1σn−2···σ1 (R4) arA2,s =A2,sar(1 ≤r, s ≤2g;r6=s) (R5) (a1···ar)A2,r =σ2 1A2,r (a1···ar) (1 ≤r≤2g) (R6) arσi=σiar(1 ≤r≤2g;i≥2) where A2,r =σ−1 1a1···ar−1a−1 r+1 ···a−1 2gσ−1 1. The generators are represented in Figure 1, where we have drawn the the canonical projections on Mof the considered braids, and Mis represented as a polygon of 4gsides, pairwise identified. Keywords: Braid - Singular Braid - Surface - Monoid - Presentation. Mathematics Subject Classification: Primary: 20F36. Secondary: 20F05. Partially supported by DGESIC-PB97-0723 and by the european network TMR Sing. Eq. Diff. et Feuill. 1 a2k σi α2k+1 α2k+1 α2k α2k P1Pn P1Pn PiPi+1 P1 α1 α2 α2g α2g α1 α2 a2k+1 Pn Figure 1: The generators of Bn(M). We can also find in [G-M] a similar presentation, when Mis a non-orientable, closed surface. In the same way that singular Artin braids were defined (see [B2]) to study Vassiliev invariants for these braids, we can define singular braids on M. Their definition is the same that the one of non-singular braids, but this time we allow a finite number of singular points (transverse intersection of two strings). The isotopy classes of these singular braids, with the analogous multiplication, form the monoid of singular braids with nstrings on M, denoted by SBn(M). This monoid is used in [G-MP] to define the Vassiliev invariants of braids on closed, orientable surfaces, proving, among other results, that these invariants classify these braids. In [B2] we can find presentations for SBn, the monoids of singular Artin braids, in terms of generators and relations. The main result of this paper is to give presentations for SBn(M). We will see, as well, that the proof of this results furnishes a new proof of the validity of the presentations in [B2]. 2 Presentation of SBn(M) We shall now give a presentation of Bn(M), when Mis a closed, orientable surface of genus g≥0. The non-orientable case is completely analogous, and is treated in a final remark at the end of this paper. We define, for all i= 1,...,n−1, the singular braid τias in Figure 2, where the only non-trivial strings are the i-th and the (i+ 1)-th ones, which intersect to form a singular point. The result is the following: Theorem 2.1. The monoid SBn(M)admits the following presentation: •Generators: σ1, . . . , σn−1, a1,...,a2g, τ1,...,τn−1. •Relations: (R1-R6) Relations of Bn(M) (R7) σiτj=τjσi(|i−j| ≥ 2) (R8) τiτj=τjτi(|i−j| ≥ 2) (R9) σiτi=τiσi(i= 1,...,n−1) (R10) σiσjτi=τjσiσj(|i−j|= 1) (R11) (ai,rai+1,r)τi(a−1 i+1,ra−1 i,r ) = τi(i= 1, . . . , n −1; r= 1,...,2g) (R12) τiaj,r =aj,rτi(j6=i, i + 1; r= 1,...,2g) where ai,r =(σ−1 i−1···σ−1 1)ar(σ−1 1···σ−1 i−1)if ris odd, (σi−1···σ1)ar(σ1···σi−1)if ris even. 2 P1PiPi+1 Pn M× {1} M× {0} Figure 2: The singular braid τi. Pn Pi αr Pn Pi P1 αr αrαr P1 Figure 3: The braid ai,r = (σ−1 i−1···σ−1 1)ar(σ−1 1···σ−1 i−1), when ris odd. Remark that ai,r can be thought of as the i-th string crossing the “wall” αr, as we can see in Figure 3 for the case when ris odd. Proof of Theorem 2.1: First, it is evident that {σ1,...,σn−1, a1,...,a2g, τ1,...,τn−1}is a set of generators of SBn(M), once that we know (by [G-M]) that {σ1,...,σn−1, a1,...,a2g}generates Bn(M). It is also easy to prove that the proposed relations hold: (R1-R6) hold in Bn(M), which is a sub-monoid of SBn(M). (R7-R10) are known to hold in SBn, so they hold in a cylinder D×[0,1], where Dis a disk containing the npoints P1,...,Pn. We have just to extend the corresponding isotopy to all M×[0,1] by the identity. (R11) can be seen to hold in Figure 4, and finally (R12) is clear, since the only nontrivial strings of τiand aj,r can be isotoped to have disjoint projections on M, so these braids commute. PiPi+1 P1Pi P1Pi+1 PnPn αr αr αrαr Figure 4: The braids ai,rai+1,rτiand τiai,rai+1,r are isotopic, when ris odd. In order to show that the relations are sufficient, we need the following lemma: Lemma 2.2. The monoid SBn(M)is left-cancelative. That is, for all a, b, c ∈SBn(M), one has: c a =c b ⇒a=b. Proof: Since σ1,...,σn−1, a1,...,a2gare invertible, for they belong to Bn(M), we just need to prove that τia=τib⇒a=bfor all a, b ∈SBn(M) and all i= 1,...,n−1. Thus, let us suppose that there exists an isotopy Htof M×[0,1], such that H0= idM×[0,1] 3 and H1(τia) = τib. Call pthe first singular point of τia(the one corresponding to τi), and let pt=Ht(p). One has p0=p1=p. Let Vbe the interior of a sphere of radius εcentered at p. We take εsmall enough, such that V∩(τia) is as follows: V p Denote st=Ht(τia) and Vt=Ht(V). We can suppose, without loss of generality, that Vtis the interior of the sphere of radius εcentered at pt, and that Vt∪stis as in the above picture. Now, for t∈[0,1], denote by estthe braid which is obtained by modifying st, only inside Vt, as follows: We observe that es0=aand es1=b, so Htis an isotopy which transforms ainto b. Therefore, a=b. Let us then show that Relations (R1-R12) are sufficient. Let b, b′∈SBn(M) be two isotopic singular braids, written in the generators of Theorem 2.1. We must show that we can transform binto b′by using Relations (R1-R12). Being isotopic, both braids have the same number of singular points, say k. If k= 0, the result follows from [G-M], since (R1-R6) are sufficient relations for Bn(M). Suppose that k > 0, and the result holds for braids with less than ksingular points. We can assume that the first letter of bis τi, for some i(otherwise we can multiply band b′on the left by the greatest nonsingular “prefix” of b). We will show that, using (R1-R12), we can transform b′into a braid whose first letter is τi. The result then follows from Lemma 2.2, and by induction hypothesis. Let pbe the point of b′corresponding (via isotopy) to the first singular point of b. This point p must correspond to some τj, letter of b′. By (R10) and the braid relations (R1-R2), we can easily deduce the following: τj=(σj−1σj−2···σi)(σjσj−1···σi+1)τi(σ−1 i+1 ···σ−1 j)(σ−1 i···σ−1 j+1) if i < j, (σj+1σj+2 ···σi)(σjσj+1 ···σi−1)τi(σ−1 i−1···σ−1 j)(σ−1 i···σ−1 j+1) if i > j. Hence, we can assume that the letter corresponding to pis τi. Let us then write b′=u τiv, where u, v ∈SBn(M) and τiis the above letter. Since bis isotopic to b′, we can assume, up to replacing τiby σiτiσ−1 i(using (R9)), that the i-th string of u ends at the point (Pi, s), for some s∈[0,1]. Hence, its canonical projection on Mis a loop in M based at Pi, which induces an element µ∈π1(M, P1). This element can be modified as desired: it suffices to use (R11), replacing τiby aε i,raε i+1,rτia−ε i+1,ra−ε i,r (ε=±1), to have µtransformed into µaε i,r, where aε i,r is the projection on Mof the i-th string of aε i,r. Since {ai,1, . . . , ai,2g}is a set of generators of π1(M, Pi), we can assume that µ= 1. Now notice that uhas less than ksingular points, hence any braid isotopic to ucan be obtained from it by applying (R1-R12), by induction hypothesis. We can then deform uin such a way that its i-th string will not go through the “walls” α1,...,α2g(we can do this since µ= 1). Let us go back to b, and consider a “band” Γ, determined by the i-th and the (i+ 1)-th strings of b, and which goes from s= 0 to the first singular point of b, as in the figure below. 4 PiPi+1 b Γ Consider also an isotopy Htwhich transforms binto b′. Recall that b′=u τiv, where the i-th string of udoes not go through the walls. We can now consider Γ1=H1(Γ), and deform the (i+ 1)-th string of ualong this band, in such a way that it will be as close to the i-th string as desired (recall that we are allowed to deform u). We can then assume that neither the i-th nor the (i+ 1)-th string of ugoes through the “walls” of the cylinder M×[0,1]. Moreover, using (R9) we can modify the number of crossings of these two strings, as desired (just replacing τiby σr iτiσ−r i, r∈Z). Therefore, we can assume that they do not cross, i.e. there is no σjin uinvolving the i−th and the (i+ 1)-th strings. We can also assume that these two strings are so close that one has the following property: if there is a letter σε j(ε=±1) of uwhich involves the i-th or the (i+ 1)-th string, then this letter, together with either the previous or the following one, forms a sub-word of uof one of the following four types: σjσj+1 σ−1 j+1σ−1 jσ−1 jσ−1 j+1 σj+1σj Γ1Γ1 Γ1Γ1 But in this case it is easy to see that, using relations (R7), (R8), (R10) and (R12), we can “raise” the point p, until we get τias the first letter of b′. So by Lemma 2.2 we can cancel τi, and by induction hypothesis the resulting braids are equivalent by means of Relations (R1-R12). This ends the proof of Theorem 2.1 Remark 2.3. There is an analogous presentation of SBn(M), when Mis a non-orientable, closed surface. We just need to consider the presentation given in [G-M] for Bn(M). Then replace, in the presentation of Theorem 2.1, the generators a1,...,a2gby the corresponding generators on the non-orientable surface, and Relations (R1-R6) by the relations given in [G-M]. The same proof remains valid. Remark 2.4. The presentation given in Theorem 2.1 can be easily simplified. It suffices to eliminate the generators τ2, . . . , τn−1, replacing in the relations τ3by (σ2σ1σ3σ2)τ1(σ−1 2σ−1 3σ−1 1σ−1 2), and eliminating all relations containing some τj(j6= 1,3), since they are obtained from the remaining ones. We proposed the presentation above since it is more useful for handling singular braids. Remark 2.5. We can also replace (R1-R6) by any other set of sufficient relations for the given generators of Bn(M). Remark 2.6. The above proof of Theorem 2.1, after eliminating every allusion to π1(M), is a new proof of the validity of the presentation for SBnproposed in [B2]. Acknowledgements: I wish to thank Prof. Orlando Neto, for giving me the opportunity to enjoy the excellent working conditions I found at the Centro de Matem´atica e Aplica¸coes Fundamentais 5 of the University of Lisbon, where this paper was written down. References [A] E. ARTIN, Theory of braids, Annals of Math. 48 (1946) 101-126. [B] J. S. BIRMAN, “Braids, Links and Mapping Class Groups”, Annals of Math. Studies 82, Princeton University Press, 1973. [B2] J. S. BIRMAN New points of view in knot theory, Bull. Amer. Math. Soc. 28 (1993), no. 2, 253-287. [G-M] J. GONZ ´ ALEZ-MENESES, New presentations of surface braid groups, J. of Knot Theory and its Ramifiactions. To appear. [G-MP] J. GONZ ´ ALEZ-MENESES and L. PARIS, Vassiliev invariants of surface braid groups, Preprint. J. GONZ´ ALEZ-MENESES Departamento de Matem´atica Aplicada I Escuela T´ecnica Superior de Arquitectura Avda. Reina Mercedes, 2 41012-Sevilla (Spain) [email protected] 6