# Toral decomposition

I have a couple of questions on the following theorem:

Theorem. (Jaco, Shalen)

Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection $\mathcal{C}$ of disjoint, embedded incompressible tori such that $M\setminus \bigcup \mathcal{C}$ consists only of Seifert manifolds and atoroidal manifolds.

My questions are:

1. Is each component of $M\setminus \bigcup \mathcal{C}$ above orientable?

2. If the boundary of $M$ is made of incompressible tori, what can we say about the boundary of each component in the theorem? Are all the boundary tori incompressible, too?

• 1. Yes, a codimension-0 submanifold of an orientable manifold is orientable; 2. It should be incompressible, yes. If it was compressible in the component, it would be compressible in M as well.. – Marco Golla Oct 9 '13 at 9:08
• a component atoroidal com boundary incomprensible toral é haken, therefhore is hiperbolic manifold of volume finite , is true ?? – jhoel Oct 10 '13 at 3:30
• Yes, it admits a hyperbolic geometry on its interior. – Steve D Oct 11 '13 at 0:53