Intersection form (4-manifold)

In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd cohomology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure.

The intersection form

is given by

where ${\displaystyle \smile }$ is the cup product.

When the 4-manifold is also smooth, then in de Rham cohomology, if a and b are represented by 2-forms α and β, then the intersection form can be expressed by the integral

where ${\displaystyle \wedge }$ is the wedge product.

Poincaré duality allows a geometric definition of the intersection form. If the Poincaré duals of a and b are represented by surfaces (or 2-cycles) A and B meeting transversely, then each intersection point has a multiplicity +1 or −1 depending on the orientations, and Q_M(a, b) is the sum of these multiplicities.

Thus the intersection form can also be thought of as a pairing on the 2nd homology group. Poincare duality also implies that the form is unimodular (up to torsion).

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