This question has been cross posted on MathOverflow with some very interesting answers and discussion.
I'm currently writing a project on the braid groups and their analogues on closed surfaces. It's an easy exercise to show that if $B_n$ is Artin's classical braid group on $n$ strings, then $B_n$ can be embedded in $B_{n+1}$ (the homomorphism is given by 'adding a string' on to the end of any braid in $B_n$ and this can be shown to be a monomorphism). A similar statement can be proved for the pure braid group $P_n$.
Let $P\mathcal{S}_n$ be the pure $n$-string braid group on the sphere $S^2$. Fox's definition of this group is the fundamental group of the configuration space $F_{n}S^2=\prod_n S^2\setminus\{(x_1,\ldots,x_n)|\exists i\neq j, x_i=x_j\}$ with basepoint $\hat{x}=(\hat{x}_1,\ldots,\hat{x}_n)$. The full braid group $\mathcal{S}_n$ is then the fundamental group of the configuration space $B_nS^2=F_nS^2/\sim$ where $x \sim y$ if the coordinates of $y$ are a permutation of the coordinates of $x$. It follows that $\mathcal{S}_n/P\mathcal{S}_n=\Sigma_n$ where $\Sigma_n$ is the symmetric group on $n$ elements.
It's well known that for $n\geq 3$, $P\mathcal{S}_n$ (and so $\mathcal{S}_n$) has torsion elements (given by the solution to the Dirac string problem for instance).
With that framework now built up, my question is, can $\mathcal{S}_n$ be embedded in to $\mathcal{S}_{n+1}$ for $n\geq 3$ (and similarly for their pure counter parts)? The naive 'add a string on the end' map will not work because, for instance, the following trivial braid $\gamma$ here (see image below) becomes non-trivial when a string is added on the end, and so any such map would not respect the equivalence class of isotopic braids.
I would think that the answer is no, but a proof eludes me. I've not been able to find any answer in the extensive literature which leads me to believe that the question is difficult. If a solution does exist, I would prefer a geometric proof as opposed to an algebraic proof, but any proof would be welcomed.
