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.



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> **Question.** What is the topological fundamental group $\pi\_1(X - \Delta)?$
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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. 



Fundamental group of $\mathrm{Sym}^2 (C_g)$ minus the diagonal
$\newcommand{\Sym}{\operatorname{Sym}}$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. 


