Incompressible surface

Remarks

Example: let $M = T^2 \times [0, 1]$ and $S = T^2 \times \set{1/2}$. Let $D$ be a disk in $M$ with $D \cap S = \partial D$. The inclusion $S \to M$ is a homotopy equivalence, so the loop $\partial D$, null-homotopic in $M$, is null-homotopic in $S$. A null-homotopic simple closed curve in a surface bounds a disk in it, so $D$ is not a compressing disk, and $S$ is incompressible.

Non-example: let $M = D^2 \times S^1$ be a solid torus and $S = \partial M$. The meridian disk $D = D^2 \times \set{1}$ meets $S$ in $\partial D$, and $\partial D$ represents a nonzero class in $\pi_1(S) \cong \Z^2$, so it bounds no disk in $S$. Hence $D$ is a compressing disk and $S$ is compressible.

If $S$ is two-sided, the loop theorem shows that $S$ is incompressible if and only if the inclusion induces an injective map $\pi_1(S) \to \pi_1(M)$. Some authors also call a $2$-sphere incompressible when it does not bound a $3$-ball in $M$.