Let $C_g$ be a compact Riemann surface of genus $g$, and let $X:=\mathrm{Sym}^2(C_g)$ be its double symmetric product and $\Delta \subset X$ the diagonal.
Question. What is the topological fundamental group $\pi_1(X - \Delta)?$
I'm primarily interested to the case $g=2$, but I would like to know the general answer. I'm aware that this is probably standard material for the experts, so any reference will be greatly appreciated.
$\newcommand{\Sym}{\operatorname{Sym}}$The answer is above, in the comment by Stefan Behrens -- I am just adding this to make some robot happy.
The $k$-fold symmetric product $\Sym^k$ of a space $C$ (minus the big diagonal $\Delta$) is the "configuration space" of distinct unlabelled $k$-tuples of points in $C$. After fixing a basepoint $(c_i)$, a loop in $\Sym^k(C)$ is a one-parameter family of $k$-tuples - so it is a union of $k$ non-intersecting arcs in $C$ that start and end at the $c_i$.
Accordingly we call $\pi_1(\Sym^k(C) - \Delta)$ the $k$-strand braid group in $C$. Note that there is a homomorphism from $\pi_1(\Sym^k(C) - \Delta)$ to the symmetric group on the indices $\{1, 2, \ldots, k\}$.
If $C$ is a surface, then sometimes the phrase surface braid group is used. If $C$ is the complex plane, then we obtain the standard braid group.
