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.
Define the intersection form modulo
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).
nd homology group Using the notion of transversality, one can state the following results (which constitute an equivalent definition of the intersection form).
, one can give a dual (and so an equivalent) definition as follows.
nd 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.
, then the intersection form can be expressed by the integral where
The definition using cup product has a simpler analogue modulo
Poincare duality states that the intersection form is unimodular (up to torsion).
By Wu's formula, a spin 4-manifold must have even intersection form, i.e.,
For a simply-connected smooth 4-manifold (or more generally one with no 2-torsion residing in the first homology), the converse holds.
The signature of the intersection form is an important invariant.
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.
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 (scalar 1) intersection form.
So Freedman's classification implies there are many non-smoothable 4-manifolds, for example the E8 manifold.