Intuition behind boundary-incompressible surfaces

Question

Can anybody please give me some intuition behind ∂-incompressible surfaces? I'm really a beginner in 3-manifold topology but as far as I've studied, incompressible surfaces seem important in the concepts of Hierarchies as we can cut along those surfaces to get simpler 3-manifolds. But what attribute does ∂-incompressible surface bring in? Thanks for any suggestion in advance.

Answer 1

Suppose you take a surface of genus one with one boundary component $F$ and take its cartesian product with an interval. If $\alpha$ is a nonseparating simple closed curve on $F$ then $\alpha\times [0,1]$ is an incompressible surface in $F\times [0,1]$ that is not boundary incompressible. Exercise: Prove that $F\times [0,1]$ cut along $\alpha \times [0,1]$ is homeomorphic to $F\times [0,1]$. Hence the result of cutting along $\alpha \times [0,1]$ is not any simpler. Cutting along incompressible, boundary incompressible surfaces yields a simpler manifold. You need this to get hierarchies to terminate.