Manifolds with Compressible Boundary

Question

I recently stumbled over the following terminology, but since I am not really familiar with geometric topology I having a hard time to understand it correctly. So, lets start with the following definitions:

Let $\mathcal{M}$ be a (sufficiently nice) $3$-manifold.

1. A properly embedded surface $\mathcal{S}$ is called "compressible", if there is a "compression disk" D, which is a disk embedded in $\mathcal{M}$ such that $D\cap\mathcal{S}=\partial D$, such that the intersection is transverse and such that $\partial D$ does not bound a disk inside $\mathcal{S}$.
2. There is also the notion of $\partial$-compressible surfaces, which is a properly embedded surface $\mathcal{S}$ such that there is a $\partial$-compression disk", which is a disk embedded in $\mathcal{M}$ such that $D\cap\mathcal{S}=:\alpha$ and $D\cap\partial\mathcal{M}=:\beta$ are arcs in $\partial D$ with the property $\partial D=\alpha\cup\beta$ and $\alpha\cap\beta=\partial\alpha=\partial\beta$ and such that $\alpha$ does not bound a disk in $\mathcal{S}$ together with another arc in $\partial\mathcal{S}$.

(Are this definitions correc?) I have two question to this terminology:

1. Is there any relation between these two different concepts?
2. What does an author mean when he say "a manifold has a compressible boundary"? So, which one of the two concepts mentioned above does one mean and with respect to which surface? (I originally thought they just take $\partial\mathcal{M}=\mathcal{S}$ in this case, but this surface is certainly not properly embedded...

Does maybe someone has some good reference clarifying these definitions?

Answer 1

i. Both definitions you gave are correct with few minor remarks:

(a) "Sufficiently nice" should be removed. Instead you should say that you are working either with topological manifolds and locally flat submanifolds, or that you are working with PL manifolds and PL submanifolds, or you are working with smooth manifolds and smooth submanifolds.

(b) Sometimes the definition of incompressibility is applied differently to 2-spheres in 3-manifolds.

(ii) If you want some intuition of how the two notions (incompressible and $\partial$-incompressible) are related, the best way that I know is to use the notion of the double of a manifold along its boundary. Let $M$ be a manifold with boundary. Form the product $P=M\times \{0, 1\}$ and consider an equivalence relation on $P$: $(x, 0)\sim (x,1)$ for all $x\in \partial M$. The quotient space $P/\sim$ is called the double of $M$ and (sometimes) denoted $DM$. This is a manifold with empty boundary. Now, if you have a properly embedded surface $S\subset M$, then you can apply the doubling construction to $S$ (inside $M$) as well and obtain a natural embedding $$ DS\to DM. $$ It is a pleasant exercise to check that if $S$ is $\partial$-compressible in $M$, then $DS$ is compressible in $DM$. A harder result (I do not have a reference) is that the converse is true as well.

(iii) Lastly, regarding incompressible boundary: yes, indeed, $\partial M$ is not properly embedded in $M$. However, you can repeat the same definition of an incompressible surface for $\partial M$ and thus, obtain the definition of a manifold with incompressible boundary.