Let \(M\) be a closed, connected, oriented \(2k\)-manifold with fundamental class \([M]\in H_{2k}(M;\ZZ)\). The intersection form of \(M\) is the bilinear form \[ I\colon H^k(M;\ZZ)\cross H^k(M;\ZZ) \to \ZZ, \qquad I(a, b)\coloneqq \inner{a\smile b}{[M]} ,\] where \(\smile\) is the cup product and \(\inner{\wait}{\wait}\) is the Kronecker pairing.
Graded commutativity of the cup product gives \(I(a, b) = (-1)^k I(b, a)\), so \(I\) is symmetric when \(k\) is even, that is, when \(\dim M\equiv 0 \pmod 4\), and skew-symmetric when \(k\) is odd. By Poincaré duality, \(I\) vanishes on torsion classes and induces a nondegenerate form on \(H^k(M;\ZZ)/\text{torsion}\) whose Gram matrix has determinant \(\pm1\).
[@Hat02].