Intersection form

Summary

The intersection form of a closed oriented $4$-manifold $M$ is a symmetric bilinear form on $H^2(M; \Z)$. It is the cup product followed by evaluation on the fundamental class. Geometrically, it counts with signs the intersection points of two surfaces in $M$ in general position. By Poincaré duality it is unimodular on $H^2(M; \Z)$ modulo torsion. It is the central invariant of $4$-manifold topology: Freedman's classification and Donaldson's theorem are statements about it.