Finding disjoint surfaces in 3-manifolds

TL;DR

This paper characterizes when a 3-manifold contains two disjoint properly embedded surfaces that keep the manifold connected, revealing topological conditions related to homology and link exteriors.

Contribution

It provides a complete characterization of 3-manifolds admitting such disjoint surfaces, using group theory techniques and linking topology with homology conditions.

Findings

Existence of disjoint surfaces is characterized by the manifold's homology properties.

Exterior of links with ≥3 components always contains such surfaces.

Techniques involve discrete and profinite group theory.

Abstract

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-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 Z_2 homology solid torus nor a Z_2 homology cobordism between two tori. In particular, the exterior of a link with at least 3 components always contain two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite.