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