  Let $M$ be a closed oriented surface of genus $g\ge 1$, let $B_n(M)$ be the
braid group of $M$ on $n$ strings, and let $SB_n(M)$ be the corresponding
singular braid monoid. Our purpose in this paper is to prove that the
desingularization map $\eta: SB_n(M) \to \Z [B_n(M)]$, introduced in the
definition of the Vassiliev invariants (for braids on surfaces), is injective.
