---
tag: NFTS5A3NXE
kind: definition
name: Intersection form
msc: [57K40, 57N65, 55N45]
defines: WK8GKRKAVD
uses: [QQQ9B1DK4P, 913WEQERBE, ZEA0PEM3B2, JX47MS1R1H, 6N7BC70CX0, 0AXZNE5XNH, 9VY90HJVZX, R6RBQHW191, 07EZA80NEH]
sources:
  - url: https://en.wikipedia.org/wiki/Intersection_form_of_a_4-manifold
---

## Definition

Let $M$ be a closed, connected, [[oriented|913WEQERBE]] [[topological manifold|QQQ9B1DK4P]] of dimension $4$, with [[fundamental class|ZEA0PEM3B2]] $[M] \in H_4(M; \Z)$.
Let $H^2$ be [[singular cohomology|6N7BC70CX0]] and $\smile$ the [[cup product|JX47MS1R1H]].
Let $\langle \cdot, \cdot \rangle$ be the Kronecker pairing of $H^4(M; \Z)$ with $H_4(M; \Z)$.
The *intersection form* of $M$ is the map
$$Q_M \colon H^2(M; \Z) \times H^2(M; \Z) \to \Z, \qquad Q_M(a, b) = \langle a \smile b, [M] \rangle.$$

## Remarks

$Q_M$ is a [[symmetric bilinear form|0AXZNE5XNH]], since $a \smile b = b \smile a$ for classes of even degree.
It vanishes if $a$ or $b$ is a torsion class, so it induces a form on the free abelian group $H^2(M; \Z)/\text{torsion}$.
By [[Poincaré duality|9VY90HJVZX]] this induced form is unimodular: its matrix in any basis is a [[unimodular matrix|R6RBQHW191]], with determinant $\pm 1$.
Its rank is $b_2(M)$, and the [[signature|07EZA80NEH]] of its real extension is the signature of $M$.

Example: for the complex projective plane $\C P^2$ with its complex orientation, $H^2 \cong \Z$ is generated by $x$ with $\langle x^2, [\C P^2] \rangle = 1$, so $Q = (1)$.
For $S^2 \times S^2$ the form has matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ in the basis given by the two factors.

Non-example: on a closed oriented surface $\Sigma$, the analogous form $H^1(\Sigma) \times H^1(\Sigma) \to \Z$ is skew-symmetric, not symmetric.
The reason is that $a \smile b = -\,b \smile a$ in degree $1$.
A symmetric form needs dimension $4$, or more generally $4k$ with middle degree $2k$.