Separating, incompressible surfaces in 3-manifolds

Abstract

Incompressible surfaces (see definition below) are one of the fundamental tools in the topological theory of 3-manifolds. The elementary fact that a compact, connected, orientable 3-manifold M with non-empty boundary contains a nonseparating, two-sided (definition below) incompressible surface is the starting point for Haken's theorem ([5], Theorem 13.3) that if M is also irreducible (w then it has a "hierarchy", i.e. can be reduced to a ball by successive splittings along two-sided incompressible surfaces. Most of the deeper known results on 3manifolds depend on this. For many purposes, separating incompressible surfaces seem more useful than non-separating ones. This is apparent, for example, to anyone who has studied fundamental groups of 3-manifolds by using hierarchies: a separating incompressible surface in M yields a decomposition of zrl(M ) as a free product with amalgamation, whereas a non-separating one gives only an " H N N decomposition" of nl(M), from which group-theoretical experience shows that it is much harder to extract information. On the other hand, separating incompressible surfaces which are non-triviali.e. not boundary-parallel (see below) have, until now, seemed very hard to come by, except in examples. The main theorem of this paper, Theorem 1 of Sect. 4, guarantees that a compact connected orientable 3-manifold M will have a non-trivial separating incompressible surface provided that H I (M;Q) is carried by the boundary of M and that some boundary component of M has genus > 1. This is used to prove Theorem 2 of Sect. 5, which asserts that if in addition M is irreducible and every incompressible torus in M is boundary-parallel, then M can be reduced to a disjoint union of solid tori and products of 2-tori with closed intervals by successive splittings along separating incompressible surfaces. As a corollary we get an analogous result for a large class of homology 3-spheres. Theorem 1 is quite deep, for it depends on William Thurston's recent work on hyperbolic structures in 3-manifolds. The proof combines one of Thurston's results with an algebraic result of Hyman Bass's (based on work of Serre's) and some elementary algebraic geometry. This material is reviewed in Sects. 1 and 3.