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].

@ __References__: Weinzierl a2011-proc [applications of intersection numbers in physics].

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