Nonsingular cup product pairing

Let \(M\) be a closed orientable \(n\)-manifold. The cup product pairing \[ H^k(M;\ZZ)/\mathrm{torsion}\times H^{n-k}(M;\ZZ)/\mathrm{torsion}\to\ZZ,\qquad (a,b)\mapsto\langle a\smile b, [M]\rangle \] is nonsingular, and for a field \(F\) the pairing \(H^k(M;F)\times H^{n-k}(M;F)\to F\) is nonsingular for every closed \(F\)-orientable \(n\)-manifold \(M\) [@Hat02]. Under Poincaré duality this is the intersection pairing on \(H_{n-k}(M)\times H_k(M)\) modulo torsion.