Spectral purity

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

Template:See The intersection form

$Q_M:H^2(M;\mathbb{\mathbb{Z}})\times H^2(M;\mathbb{\mathbb{Z}})\to \mathbb{\mathbb{Z}}$

is given by

$Q_M(a,b)=⟨a⌣b,[M]⟩.$

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

$Q(a,b)=\underset{M}{\int }\alpha \wedge \beta$

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-cycles) 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

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