Let \(R\) be a commutative ring and \(M\) a closed $R$-orientable \(n\)-manifold with fundamental class \([M]\in H_n(M;R)\). The Poincaré duality map is \[ D\colon H^k(M;R)\to H_{n-k}(M;R),\qquad D(\alpha)\coloneqq[M]\frown\alpha, \] given by the cap product with \([M]\).
Let \(R\) be a commutative ring and \(M\) a closed \(R\)-orientable \(n\)-manifold with fundamental class \([M]\in H_n(M;R)\). Then the Poincaré duality map \(D\colon H^k(M;R)\to H_{n-k}(M;R)\), \(\alpha\mapsto[M]\frown\alpha\), is an isomorphism for every \(k\) [@Hat02].