Let $M_g$ be the moduli space of Riemann surfaces, as described for example in the book of Harris and Morrison - Moduli of curves. As a topological space, or better as orbifold, it has smooth points and singular points. Let $S\subseteq M_g$ be the subset of smooth points. In the article by Cornalba ("On the locus of curves with automorphisms") this subset is identified, in the case $g>3$, with the set of those curves which have just the trivial automorphism.

Is there any idea (or any reference) of how to compute the homotopy groups $$\pi_2(M_g,S) \qquad \mbox{and} \qquad \pi_1(S)?$$ I should have included a base point above, but I intentionally didn't because I wouldn't know which one to choose. Say I would choose it in a codimension 1 component of $S$.

Let $M_g$ be the moduli space of Riemann surfaces, as described for example in the book of Harris and Morrison - Moduli of curves. As a topological space, or better as orbifold, it has smooth points and singular points. Let $S\subseteq M_g$ be the subset of smooth points. In the article by Cornalba ("On the locus of curves with automorphisms") this subset is identified, in the case $g>3$, with the set of those curves which have just the trivial automorphism.

Is there any idea (or any reference) of how to compute the homotopy groups $$\pi_2(M_g,S) \qquad \mbox{and} \qquad \pi_1(S)?$$

Edit: I should have included a base point above, but I intentionally didn't because I wouldn't know which one to choose.