Birman’s conjecture for singular braids on closed surfaces
نویسنده
چکیده
Let M be a closed oriented surface of genus g ≥ 1, let Bn(M) be the braid group of M on n strings, and let SBn(M) be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map η : SBn(M) → Z[Bn(M)], introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective. AMS Subject Classification: Primary 20F36; Secondary 57M27.
منابع مشابه
Presentations for the monoids of singular braids on closed surfaces
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.
متن کاملOn Singular Braids
In Vassiliev theory, there is a natural monoid homo-morphism from n-strand singular braids to the group algebra of n-strand braid group. J. Birman conjectured that this monoid homomorphism is injective. We show that the monoid homomor-phism is injective on braids with up to three singularities and that Birman's conjecture is equivalent to that singular braids are dis-tinguishable by Vassiliev b...
متن کاملThe proof of Birman’s conjecture on singular braid monoids
Let Bn be the Artin braid group on n strings with standard generators σ1, . . . , σn−1 , and let SBn be the singular braid monoid with generators σ ±1 1 , . . . , σ ±1 n−1, τ1, . . . , τn−1 . The desingularization map is the multiplicative homomorphism η : SBn → Z[Bn] defined by η(σ ±1 i ) = σ ±1 i and η(τi) = σi − σ −1 i , for 1 ≤ i ≤ n − 1. The purpose of the present paper is to prove Birman’...
متن کاملOn the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W . More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of s...
متن کاملFilling Area Conjecture and Ovalless Real Hyperelliptic Surfaces
We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu’s result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbif...
متن کامل