Finding disjoint surfaces in 3-manifolds

Abstract

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no two-spheres. We investigate the existence of two properly embedded disjoint surfaces \(S_{1}\) and \(S_{2}\) such that \(M - (S_{1} \cup S_{2})\) is connected. We show that there exist two such surfaces if and only if M is neither a \(\mathbb Z _{2}\) homology solid torus nor a \(\mathbb Z _{2}\) homology cobordism between two tori. In particular, the exterior of a link with at least three components always contains two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite.