Intersection |

**Intersection Form** > s.a. 4-manifolds.

$ __Def__: A (symmetric and unimodular?) bilinear pairing

*ω*: H_{2}(*M*; \(\mathbb Z\)) × H_{2}(*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*])
= ∑_{p ∈ α ∩ β}
(–1)^{n(p)} ,

where *α* ∩ *β* can
be made finite by transversality and *n*(*p*) = 0 or 1,
depending on whether the orientations agree or not in T_{p}*M* ≅
T_{p}*α* ⊕ T_{p}*β*.

* __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].

> __Online resources__:
see Wikipedia page.

**Intersection Theory**

* __Idea__: The algebraic geometric theory of two schemes meeting in
a third, founded in 1720 by MacLaurin.

@ __References__: Fulton 84.

> __Online resources__:
see Wikipedia page.

**Related Topics** > see set theory [ordinary set
intersection].

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