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.

Definition

The intersection form

:$Q\_Mcolon\; H^2(M;mathbb\{Z\})\; imes\; H^2(M;mathbb\{Z\})\; o\; mathbb\{Z\}$

is given by

:$Q\_M(a,b)=langle\; asmile\; b,\; [M]\; angle.$

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 $wedge$ is the wedge product, see exterior algebra.

Poincaré duality

Poincaré duality allows a geometric definition of the intersection form. If the Poincaré duals of "a" and "b" are represented by surfaces (or 2-cocycles) "A" and "B" meeting transversely, then each intersection point has a multipicity +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).

Properties and applications

By Wu's formula, a spin 4-manifold must have even intersection form, i.e. "Q(x,x)" is even for every "x". For a simply-connected 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. 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 (possibly both) 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 (scalar 1) intersection form. So Freedman's classification implies there are many non-smoothable 4-manifolds, for example the E8 manifold.

Non-orientable manifolds

Just as there is a version of Poincare duality for Z/2Z coefficients, there is also a version of the intersection form with Z/2Z coefficients, taking values in Z/2Z rather than in Z. In this way non-orientable manifolds get an intersection form as well. Of course one does not see any of this in de Rham cohomology.

References

*citation|first=A. |last=Scorpan|title= The wild world of 4-manifolds|year=2005| publisher= American Mathematical Society|ISBN= 0-8218-3749-4