Poincaré duality for the real projective plane

Example 11.4.5. The manifold \(\R P^2\) is not integrally orientable, and integral Poincaré duality indeed fails: \(H_0(\R P^2;\Z )\cong \Z \), whereas \(H^2(\R P^2;\Z )\cong \Z /2\). With \(\F _2\)-coefficients, it is orientable and Poincaré duality holds.