Intersection form of a 4-manifold

In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure. Let M be a closed 4-manifold (PL or smooth). Take a triangulation T of M. Denote by the dual cell subdivision. Represent classes by 2-cycles A and B modulo 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 (by definition of a cycle and a boundary). If M is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2nd homology group Using the notion of transversality, one can state the following results (which constitute an equivalent definition of the intersection form). If classes are represented by closed surfaces (or 2-cycles modulo 2) A and B meeting transversely, then If M is oriented and classes are represented by closed oriented surfaces (or 2-cycles) 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 (and so an equivalent) definition as follows. Let M be a closed oriented 4-manifold (PL or smooth). Define the intersection form on the 2nd cohomology group by the formula The definition of a cup product is dual (and so is analogous) 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 (not only in PL and smooth 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.