---
tag: WK8GKRKAVD
kind: concept
name: Intersection form
msc: [57K40, 57N65, 55N45]
sources:
  - url: https://en.wikipedia.org/wiki/Intersection_form_of_a_4-manifold
same_as:
  wikidata: Q6056717
  wikipedia: Intersection form of a 4-manifold
---

## Summary

The intersection form of a closed oriented $4$-manifold $M$ is a symmetric bilinear form on $H^2(M; \Z)$.
It is the cup product followed by evaluation on the [[fundamental class|ZEA0PEM3B2]].
Geometrically, it counts with signs the intersection points of two surfaces in $M$ in general position.
By Poincaré duality it is unimodular on $H^2(M; \Z)$ modulo torsion.
It is the central invariant of $4$-manifold topology: Freedman's classification and Donaldson's theorem are statements about it.

## Explanations

### L1 human

In a $4$-dimensional space, two surfaces in general position meet in isolated points, just as two curves do in the plane.
Each point counts as $+1$ or $-1$ according to orientations, and the total is the intersection number.
The intersection form records this number for every pair of surface classes.
For the complex projective plane, a line meets another line in exactly one point, so the form is the $1 \times 1$ matrix $(1)$.