The proof of Birman's conjecture on singular braid monoids

Let B_n be the Artin braid group on n strings with standard generators sigma_1, …, sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{±1}, …, sigma_{n-1}^{±1}, tau_1, …, tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{±1}) =_i^{±1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove Birman’s conjecture, namely, that the desingularization map eta is injective.