Ordinary-coefficient Poincaré duality

Example 11.4.3. For \(E=H\F _2\), every real vector bundle is \(E\)-orientable because \(\Line _{H\F _2}\simeq *\). Thus every closed smooth manifold satisfies mod-\(2\) Poincaré duality. For \(E=H\Z \), Example 10.4.8 identifies an \(H\Z \)-orientation with a classical orientation, or equivalently with a nullhomotopy of \(w_1(T_M)\). This recovers integral Poincaré duality for orientable manifolds.