Intersection

Intersection Form > s.a. 4-manifolds.
\$ Def: A (symmetric and unimodular?) bilinear pairing

ω: H2(M; $$\mathbb Z$$) × H2(M; $$\mathbb Z$$) → $$\mathbb Z$$ ,

where M is a (simply connected, compact, oriented?) 4-manifold.
* Calculation: A pair can be represented by smoothly embedded oriented submanifolds α and β, and the form

ω([a], [b]) = ∑pαβ (–1)n(p) ,

where αβ can be made finite by transversality and n(p) = 0 or 1, depending on whether the orientations agree or not in TpM ≅ Tpα ⊕ Tpβ.
* Even intersection form: One such that ω(α, α) is even for all α.
* Result: Every symmetric unimodular ω is the intersection form of some compact, simply connected 4-manifold (up to homeomorphism, only 1 if ω is even, and only 2 if ω is not even).
* Isomorphic intersection forms: A pair related by conjugation by GL( · , $$\mathbb Z$$).
@ References: Donaldson JDG(86).
> In physics: see types of quantum field theories [on simplicial complexes, as action].