Genus two 3–manifolds are built from handle number one pieces

Abstract

Let $M$ be a closed, irreducible, genus two 3–manifold, and $F$ a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in $M$. Then each component manifold $M_i$ of $M-F$ has handle number at most one, ie admits a Heegaard splitting obtained by attaching a single 1–handle to one or two components of $\partial M_i$. This result also holds for a decomposition of $M$ along a maximal collection of incompressible tori.