Donaldson's theorem

Remarks

Simon Donaldson proved the theorem in 1983 with gauge theory. For a suitable Riemannian metric, the moduli space of anti-self-dual instantons 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, 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.

Going deeper

For a student

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. A short proof of the same result was later given with Seiberg–Witten invariants. It also leads to the existence of exotic $\R^4$.