---
tag: 34B7K77P31
kind: statement
role: theorem
name: Donaldson's theorem
msc: [57K40, 57R57, 57R55]
uses: [8H58Q6YJCH, F45VQ47GPV, S1XK3HXHFX, WK8GKRKAVD]
about: [8H58Q6YJCH, WK8GKRKAVD]
sources:
  - url: https://en.wikipedia.org/wiki/Donaldson%27s_theorem
same_as:
  wikidata: Q5295351
  wikipedia: Donaldson's theorem
---

## Statement

Let $M$ be a closed, oriented, simply connected, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]].
Let $Q_M$ be its [[intersection form|WK8GKRKAVD]] on $H_2(M; \Z) \cong \Z^n$.
If $Q_M$ is negative definite, then $Q_M$ is isomorphic over $\Z$ to the diagonal form $-I_n$, that is, $H_2(M; \Z)$ has a basis $e_1, \ldots, e_n$ with $Q_M(e_i, e_j) = -\delta_{ij}$.

## Remarks

Simon Donaldson proved the theorem in 1983 with gauge theory.
For a suitable Riemannian metric, the moduli space of anti-self-dual [[instantons|YWBTGC7MFA]] on the $SU(2)$-bundle with $c_2 = 1$ is a $5$-dimensional manifold with singular points.
Its singular points are the reducible connections, which correspond to the pairs $\pm e$ with $Q_M(e, e) = -1$, and near each of them it is a cone on $\C P^2$.
One end of the moduli space is a collar on $M$, so $M$ is cobordant to a disjoint union of copies of $\C P^2$ up to orientation.
Comparing signatures forces many vectors of square $-1$, and this makes the form diagonal.
Donaldson later removed the hypothesis that $M$ is simply connected.
Replacing the orientation of $M$ by its opposite gives the positive definite version.
Smoothness is essential.
By [[Freedman's classification|HSD6FKCYRC]], the negative of the $E_8$ form is the intersection form of a closed simply connected topological $4$-manifold.
This form is definite but not diagonalizable.

## Explanations

### L1 human

The intersection form records how surfaces cross inside a four-dimensional space.
Donaldson's theorem says that in a smooth such space, if all self-crossings have the same sign, the table is as simple as possible: a diagonal of $-1$'s.
Many tables allowed by algebra and by topology are forbidden by smoothness.

### L2 human

Combined with Freedman's theorem, it shows that most definite unimodular forms, such as $E_8$ and $E_8 \oplus E_8$, are realized by topological $4$-manifolds that admit no smooth structure.
It was the first major application of the Yang–Mills equations to topology and it started [[Donaldson theory|ZQW6M0S6M0]].
A short proof of the same result was later given with [[Seiberg–Witten invariants|PRDDMSTR4X]].
It also leads to the existence of [[exotic $\R^4$|X44PNC38E2]].