If $X$ is a simply connected 4-manifold and $\Sigma\subset X$ is a connected and oriented surface, then can we say anything about the index 2 subgroups of $\pi_1(X\setminus\Sigma)$?
(Best case for me would be to say that there are no index 2 subgroups. I'm pretty sure that's false though since Alexander duality implies that $H_1(S^4\setminus\Sigma)=H^2(\Sigma)=\mathbb Z$. So $\pi_1(S^4\setminus\Sigma)$ is nontrivial and probably is at least sometimes $\mathbb Z$, which obviously has an index 2 subgroup.)