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