Let \(M\) be a closed connected oriented \(n\)-manifold with fundamental class \([M]\in H_n(M;\ZZ)\), and let \(D\colon H^k(M;\ZZ)\to H_{n-k}(M;\ZZ)\), \(D(\varphi)=[M]\frown\varphi\), be the Poincaré duality isomorphism. The intersection product is the bilinear map \[ H_i(M;\ZZ)\times H_j(M;\ZZ)\to H_{i+j-n}(M;\ZZ),\qquad \alpha\cdot\beta\coloneqq[M]\frown\qty{\inverseof{D}(\alpha)\smile \inverseof{D}(\beta)}, \] using the cup product and the cap product. For \(j=n-i\), composing with \(H_0(M;\ZZ)\cong\ZZ\) gives the intersection pairing \(H_i(M;\ZZ)\times H_{n-i}(M;\ZZ)\to\ZZ\).
The intersection pairing vanishes when either argument is a torsion class, and the induced pairing \[ H_i(M;\ZZ)/\mathrm{torsion}\times H_{n-i}(M;\ZZ)/\mathrm{torsion}\to\ZZ \] is nondegenerate: its adjoint is an isomorphism onto \(\Hom(H_{n-i}(M;\ZZ)/\mathrm{torsion},\ZZ)\). This is the cup product pairing of [@Hat02] transported by \(D\).
If \(A,B\subseteq M\) are closed oriented submanifolds of dimensions \(i\) and \(j\) that intersect transversely, with fundamental classes representing \(\alpha\) and \(\beta\), then \(\alpha\cdot\beta\) is represented by the submanifold \(A\cap B\) of dimension \(i+j-n\) with its induced orientation. When \(i+j=n\), the intersection \(A\cap B\) is a finite set of points and \(\alpha\cdot\beta\in\ZZ\) is the number of intersection points counted with signs given by the orientations.