Let M be a closed 4-manifold. Take a triangulation T of M. Denote by the dual cell subdivision. Represent classes by 2-cycles A and Bmodulo 2 viewed as unions of 2-simplices of T and of, respectively. Define the intersection form modulo 2 by the formula This is well-defined because the intersection of a cycle and a boundary consists of an even number of points. If M is oriented, analogously one defines the intersection form on the 2nd homology group Using the notion of transversality, one can state the following results.
If classes are represented by closed surfaces A and B meeting transversely, then
If M is oriented and classes are represented by closed oriented surfaces A and B meeting transversely, then every intersection point in has the sign +1 or −1 depending on the orientations, and is the sum of these signs.
Using the notion of the cup product, one can give a dual definition as follows. Let M be a closed oriented 4-manifold. Define the intersection form on the 2nd cohomology group by the formula The definition of a cup product is dual to the above definition of the intersection form on homology of a manifold, but is more abstract. However, the definition of a cup product generalizes to complexes and topological manifolds. This is an advantage for mathematicians who are interested in complexes and topological manifolds. When the 4-manifold is smooth, then in de Rham cohomology, if a and b are represented by 2-forms and, then the intersection form can be expressed by the integral where is the wedge product. The definition using cup product has a simpler analogue modulo 2. Of course one does not have this in de Rham cohomology.
Poincare duality states that the intersection form is unimodular. By Wu's formula, a spin 4-manifold must have even intersection form, i.e., is even for every x. For a simply-connected 4-manifold, the converse holds. The signature of the intersection form is an important invariant. A 4-manifold bounds a 5-manifold if and only if it has zero signature. Van der Blij's lemma implies that a spin 4-manifold has signature a multiple of eight. In fact, Rokhlin's theorem implies that a smooth compact spin 4-manifold has signature a multiple of 16. Michael Freedman used the intersection form to classify simply-connected topological 4-manifolds. Given any unimodular symmetric bilinear form over the integers, Q, there is a simply-connected closed 4-manifold M with intersection form Q. If Q is even, there is only one such manifold. If Q is odd, there are two, with at least one having no smooth structure. Thus two simply-connected closed smooth 4-manifolds with the same intersection form are homeomorphic. In the odd case, the two manifolds are distinguished by their Kirby–Siebenmann invariant. Donaldson's theorem states a smooth simply-connected 4-manifold with positive definite intersection form has the diagonal intersection form. So Freedman's classification implies there are many non-smoothable 4-manifolds, for example the E8 manifold.
