Intersection pairing ($H_k(W)\times H_{n-k}(W,\partial W)\to F$)

An intersection pairing counts signed intersections of transverse representatives of complementary homology classes, using one relative representative when the manifold has boundary. With ${\mathbb{F}}_2$ coefficients it is defined without orientation choices. Poincare-Lefschetz duality makes this a perfect pairing for compact manifolds.

The intersection matrix records an intersection pairing on chosen classes. Its nonsingularity proves linear independence: a relation between the classes, paired with each class in turn, gives a vector in the matrix's kernel.

In the product $(S^{2d}{)}^{2\ell }$, for $d,\ell \ge 1$, vary the factors indexed by an $\ell$-element subset $I$ and fix points in the other factors. Orient the resulting submanifold $S_I$ in increasing factor order. These homology classes form an integral middle-dimensional basis by the Künneth theorem. Complementary subsets give a transverse intersection at one point, with positive sign since all factor dimensions are even. For noncomplementary subsets some factor is fixed in both representatives; moving that point makes them disjoint. In particular every basis class has zero self-intersection number. Complementary pairs give hyperbolic blocks in the intersection form.

For a compact three-manifold $W$, let $L=\ker (H_1(\partial W;{\mathbb{F}}_2)\to H_1(W;{\mathbb{F}}_2))$. The boundary intersection pairing satisfies $L^{\perp }=L$: the long exact sequence in relative homology identifies $L$ with boundaries of relative surfaces, and the boundary-interior adjoint identity identifies its orthogonal complement with the same kernel. Hence $2\dim L=\dim H_1(\partial W;{\mathbb{F}}_2)$. This also implies that a closed surface with odd first mod-two Betti number, such as ${\mathbb{RP}}^2$, cannot be the entire boundary of a compact three-manifold.