Let $M_1subseteq M_2subseteqmathbb R^3$ be closed connected subsets with smooth boundary. Suppose that every closed loop in $M_1$ is freely homotopic inside $M_2$ to a closed loop inside $M_2setminus M_1^circ$. Equivalently, every element of $pi_1(M_1)$ is conjugate inside $pi_1(M_2)$ to an element in the image of $pi_1(M_2setminus M_1)topi_1(M_2)$. Also assume that the complements of $M_1$ and $M_2$ are connected (by Alexander’s theorem this implies that $M_1$ and $M_2$ are both irreducible). What can be said about $M_1$ and $M_2$ in this situation?
My goal here is to force $pi_1(M_1)topi_1(M_2)$ to be zero. Unfortunately, this is not guaranteed in the situation above, since we could take $M_1=M_2$ to be a handlebody. Indeed, any loop inside a handlebody can be pushed to its boundary (use general position to make it disjoint from a graph $Gsubseteq M_1$ of which $M_1$ is a regular neighborhood). But maybe this is “as bad as it gets”?
Without the assumption on the connectedness of $mathbb R^3setminus M_i$, there are more counterexamples such as $M_1=B(2)setminus B(1)^circ$ and $M_2=B(2)$, but I don’t care much about this relaxation of the question.
The significance of the hypothesis that $M_2subseteqmathbb R^3$ (rather than just being any compact connected orientable irreducible three-manifold with boundary) is that I know it holds in the situation I am interested in. It may be end up being irrelevant as far as the question asked is concerned.