Haken manifold

Remarks

Example: the $3$-torus $T^3 = T^2 \times S^1$ is irreducible, since its universal cover $\R^3$ is irreducible by Alexander's theorem. The torus $T^2 \times \set{1}$ is two-sided, and the inclusion $\Z^2 \to \Z^3$ on fundamental groups is injective, so it is incompressible. Hence $T^3$ is Haken.

Non-example: the $3$-sphere $S^3$ is irreducible by Alexander's theorem, but it is not Haken. A closed surface $S \subseteq S^3$ of positive genus has $\pi_1(S) \ne 1 = \pi_1(S^3)$, so by the loop theorem it is compressible.

Every closed orientable irreducible $3$-manifold with infinite first homology group is Haken. For non-orientable manifolds, irreducible is replaced by $P^2$-irreducible, which also forbids embedded two-sided projective planes.