Braids on Surfaces and Finite Type Invariants

We prove that there is no functorial universal finite type invariant for braids in Σ× I if the genus of Σ is positive. MSC: 53 C 23, 57 N 15. Let Σ be a compact, connected and orientable surface. The group B(Σ, n) of braids on n strands over Σ is a natural generalization of both the classical braid group Bn and the fundamental group π1(Σ). It appeared first in the study of configuration spaces ([5], see also [4]). Presentations for B(Σ, n) were derived by Scott ([17]), further improved by González-Meneses for closed surfaces ([8]) and finally given a very simple form in [3]. In the case of holed spheres the latter have been previously obtained by Lambropoulou ([12]). Let us consider a category of embedded 1-dimensional objects like braids, links, tangles etc. There is a natural filtration on the free Z-module generated by the objects, coming from the singular objects with a given number of double points. The main feature of this filtration is that the associated grading, which is called the diagrams algebra, can be explicitly computed, and has some salient finiteness properties. By universal finite type invariant one generally means a map from our category into some completion of the diagrams algebra, which induces an isomorphism at graded level. For instance the celebrated Kontsevich integral is such a universal invariant. A key ingredient is the existence of a Drinfel’d associator with rational coefficients. Notice that there exists a universal invariant for usual braids over Z ([15]), but it is not known whether there exists a multiplicative one over Z. In fact an essential feature of the Kontsevich integral is its functoriality ([13]): it is precisely this property which enables one to extend the Chen iterated integrals from braids to links. González-Meneses and Paris constructed a universal invariant for braids on surfaces ([9]), but their invariant is not functorial (i.e. multiplicative). The purpose of this note is to show that actually their result cannot be improved: Theorem 1. There does not exist a functorial universal finite type invariant for braids in Σ × I, if the surface Σ is of genus g ≥ 1. Remark 1. The non-existence of the multiplicative invariant is not related to the ground ring, which will be considered to be Q in the sequel, but it might as well be C. Remark 2. In particular there does not exist any universal invariant for tangles in Σ× I which is functorial with respect to the vertical composition of tangles. The same holds true a fortiori for the category of tangles in 3-manifolds with boundary when the family of 3-manifolds allowed is different from cylinders over planar surfaces. 1. Preliminaries Surface braids. Set Σg,p for the compact orientable surface of genus g with p boundary components. We denote by σ1, ..., σn−1 the standard generators of the braid group on a disk embedded in Σg,p, viewed as elements of B(Σg,p, n). Let also a1, ..., ag, b1, ..., bg , z1, ..., zp−1 be the generators of π1(Σg,p), where zi denotes a loop around the i-th boundary component. Assume that the base point of the fundamental group is the startpoint of the first strand. Then each γ ∈ {a1, ..., ag , b1, ..., bg, z1, ..., zp−1} can be realized by an element denoted also by γ in B(Σg,p, n), by considering the braid whose first strand is describing the curve γ and other strands are constant. We denote [a, b] = aba−1b−1. Following [3] we have: Theorem 2. A presentation for B(Σg,p, n) (n ≥ 2) is given by: Preprint available at http://www-fourier.ujf-grenoble.fr/~ funar. 1 2 P.BELLINGERI AND L.FUNAR • Generators: σ1, ..., σn−1, a1, ..., ag , b1, ..., bg , z1, ..., zp−1. • Braid relations: σiσi+1σi = σi+1σiσi+1 (i = 1, . . . , n− 2) σiσj = σjσi (if | i− j |≥ 2). • Commutativity relations: [ar, σi] = [br, σi] = [zk, σi] = 1, (i > 1, 1 ≤ r ≤ g, 1 ≤ k ≤ p− 1) ; [ar, σ −1 1 arσ −1 1 ] = [br, σ −1 1 brσ −1 1 ] = [zk, σ −1 1 zkσ −1 1 ] = 1, (1 ≤ r ≤ g, 1 ≤ k ≤ p− 1); [ar, σ −1 1 asσ1] = [ar, σ −1 1 bsσ1] = [br, σ −1 1 asσ1] = [br, σ −1 1 bsσ1] = [zi, σ −1 1 zjσ1] = 1, (1 ≤ s < r ≤ g, 1 ≤ j < i ≤ p− 1) ; [ar, σ −1 1 zkσ1] = [br, σ −1 1 zkσ1] = 1, (1 ≤ r ≤ g, 1 ≤ k ≤ p− 1) ; • Skew commutativity relations on each handle (when g > 0): [ar, σ −1 1 brσ −1 1 ] = σ 2 1, (1 ≤ r ≤ g) ; • When p = 0 we have an additional relation: [a1, b −1 1 ][a2, b −1 2 ]...[ag , b −1 g ] = σ1σ2...σn−2σ 2 n−1σn−2...σ2σ1 . This presentation is not symmetric. One might consider one more generator zp (satisfying the same relations as the other zj) and add a global relation (similar to that for p = 0): z1...zp[a1, b −1 1 ][a2, b −1 2 ]...[ag , b −1 g ] = σ1σ2...σn−2σ 2 n−1σn−2...σ2σ1. Remark 3. Although Σg,1 and Σ0,2g+1 are homotopically equivalent their braid groups are not isomorphic. In fact H1(B(Σ0,2g+1, n)) = Z 2g+1 while H1(B(Σg,p, n)) = Z/2Z⊕H1(Σg,p), if g ≥ 1. The Vassiliev filtration. Singular braids are obtained from braids by allowing a finite number of transverse double points. One can associate to each singular surface braid a linear combination of braids by desingularizing each crossing as follows: