# gt.geometric topology – Classifying nested 3-manifolds with fundamental group property

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.