Is there a manifold with no embedded incompressible surface?

Question

I mean, in any 3-manidold there is an embedded disk (since locally we have $\mathbb{R}^3$ and one can find a disk in $\mathbb{R}^3$) and disk will always be an incompressible surface. Why am I wrong?

Thank you.

Answer 1

Wikipedia article does not do a very good job here, which, I think, is the source of your confusion.

1. Suppose that $S$ is a closed (compact, with empty boundary), 2-sided surface embedded in a (boundaryless) 3-manifold $M$. Such a surface $S$ is called compressible in $M$, if it is either a 2-dimensional sphere bounding a 3-ball in $M$, or $S$ contains a simple loop $c$ which bounds an embedded disk $D$ in $M$ such that $D\cap S= c$. Accordingly, $S$ is called incompressible if it is not compressible. In particular, a 2-dimensional disk in $M$ will fail the definition of an incompressible surface.

2. The notion of an incompressible surface also makes sense in the category of manifolds with boundary, namely, assuming that $M$ be a 3-dimensional manifold (possibly with boundary) and $S$ a (for simplicity, compact) 2-sided properly embedded, surface in $M$ (properly embedded here means that $\partial S= S\cap \partial M$). The definition is the same as above, or it can be strengthened to the one of a boundary-incompressible surface.

As for your example of a 2-dimensional disk $D$ in a 3-manifold $M$: It will never be incompressible (it is a good exercise to see why). However, it can be boundary-incompressible, if it is properly embedded in $M$ and there is no 3-dimensional ball $B\subset M$ whose boundary is a union of two disks: One is $D$ and the other is in $\partial M$.

As for your title question: Yes, there are many manifolds (even closed ones) without incompressible surfaces, for instance, $S^3$, small Seifert manifolds, small hyperbolic manifolds...