P2-irreducible manifold

In mathematics, a P²-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided ${\displaystyle \mathbb {R} P^{2}}$ (real projective plane). An orientable manifold is P²-irreducible if and only if it is irreducible.^[1] Every non-orientable P²-irreducible manifold is a Haken manifold.^[2]