Let $X_1,...,X_m$ be connected, simply-connected CW sub-complexes of a CW complex $X$. Let the symmetric group on $m$ letters, $S_m$, act on $P:=X_1\times\cdots\times X_m$ in $X^m$ by permuting components. Let $Y$ be the smallest subspace of $X^m$ that contains $P$ that is also stable under this action of the symmetric group; the saturation. Is $Y/S_m$ simply-connected?
If $X=X_1=X_2=...=X_m$, then $Y=P$ and $Y/S_m$ is the usual symmetric product construction. Then the answer is yes by Dold and Puppe since the $m$-symmetric product of $X$ has its fundamental group equal to $H_1(X)$; which is trivial since we are assuming $X$ is simply-connected.
This newly published paper addresses a different direction, namely, restricting the action of the symmetric group to a smaller subgroup.
I have some ideas of how to proceed subject to certain conditions: first show the saturation is simply-connected using van Kampen, and then using some kind of path-lifting property show the quotient is $\pi_1$-surjective. However, I would like to know about references, interesting examples, or if someone has a solid proof under some specific conditions.
