Intersection Form

The intersection form of a closed oriented four-manifold is the symmetric bilinear pairing on its second cohomology given by cup product and evaluation on the fundamental class. Over the real numbers it is

It is nondegenerate by Poincaré duality. With integral coefficients, it is defined on cohomology modulo torsion. The integers and are the dimensions of maximal positive and negative subspaces of the real form. Reversing orientation exchanges them.