I'm currently trying to understand the proof of the (elementary) JSJ-Decomposition in Allen Hatcher's notes on 3-Manifolds.
Specifically, I have trouble understanding the application of proposition 1.7 to corollary 1.8.
Proposition 1.7 states:
For a compact irreducible 3-Manifold M there is a bound on the number of components in a system $S = S_1 \cup \dots \cup S_n$ of disjoint closed incompressible surfaces $S_i \subset M$ such that no component of $M | S$ ("M split along S") is a product $T \times I$ with $T$ a closed surface.
Corollary 1.8 states:
In a compact connected irreducible 3-Manifold M there exists a (possibly empty) collection T of disjoint incompressible tori, such that each component of $M|T$ is atoroidal.
In the proof, one starts with $T=\emptyset$, and then applies the following steps repeatedly:
- If $M|T$ is atoroidal, we are done.
- Otherwise, there exists an incompressible torus in $M|T$ which is not boundary-parallel (by definition of atoroidal). Add this torus to T. Go to step 1.
To show that this procedure will finish in a finite number of steps, the following argument is given:
Assume otherwise. Then there exist arbitrarily large collections T of disjoint incompressible tori, such that no component of $M|T$ is a product of $I$ and a torus, in contradiction to proposition 1.7.
I don't see why proposition 1.7 applies here - the condition in 1.7 is stronger, namely that no component of $M|T$ is a product of $I$ and a closed surface.
So my question is: why and how is prop. 1.7 applicable here?
I hope this question is suitable for MathOverflow. If not, a hint where I might get an answer would be nice.