Inclusion maps of $3$-manifolds which induce monomorphisms of fundamental groups

Abstract:

The main result is the following “duality” theorem. Let M be a 3-manifold, P a compact and connected polyhedral 3-submanifold of $\int M$, and X a compact and connected polyhedron in $\int P$. If ${\pi _1}(X) \to {\pi _1}(P)$ is onto, then ${\pi _1}(M - P) \to {\pi _1}(M - X)$ is one-to-one. Some related results are proved, for instance: we can allow P to be noncompact if also X satisfies a certain noncompactness condition: if M lies in a 3-manifold W with ${H_1}(W) = 0$, then the condition that ${\pi _1}(X) \to {\pi _1}(P)$ is onto can be replaced by the weaker one that ${H_1}(X) \to {H_1}(P)$ is onto.