# Donaldson's theorem

In mathematics, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the integers.

## History

The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986.

## Idea of proof

Donaldson's proof utilizes the moduli space ${\mathcal {M}}_{P}$  of solutions to the anti-self-duality equations on a principal $\operatorname {SU} (2)$ -bundle $P$  over the four-manifold. By the Atiyah-Singer index theorem, the dimension of the moduli space is given by

$\dim {\mathcal {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),$

where $c_{2}(P)=k$ , $b_{1}(X)$  is the first Betti number of $X$  and $b_{+}(X)$  is the dimension of the positive-definite subspace of $H_{2}(X,\mathbb {R} )$  with respect to the intersection form. When $X$  is simply-connected with definite intersection form, possibly after changing orientation, one always has $b_{1}(X)=0$  and $b_{+}(X)=0$ . Thus taking any principal $\operatorname {SU} (2)$ -bundle with $k=1$ , one obtains a moduli space ${\mathcal {M}}$  of dimension five.

This moduli space is non-compact and generically smooth, with singularities occurring only at the points corresponding to reducible connections, of which there are exactly $b_{2}(X)$  many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst ${\mathcal {M}}$  is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of ${\mathcal {M}}$ , say ${\mathcal {M}}_{\varepsilon }$ , such that for sufficiently small choices of parameter $\varepsilon$ , there is a diffeomorphism

${\mathcal {M}}_{\varepsilon }{\xrightarrow {\quad \cong \quad }}X\times (0,\varepsilon )$ .

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold $X$  with curvature becoming infinitely concentrated at any given single point $x\in X$ . For each such point, in the limit one obtains a unique singular ASD connection, which becomes a well-defined smooth ASD connection at that point using Uhlenbeck's removable singularity theorem.

Donaldson observed that the singular points in the interior of ${\mathcal {M}}$  corresponding to reducible connections could also be described: they looked like cones over the complex projective plane $\mathbb {CP} ^{2}$ , with its orientation reversed.

It is thus possible to compactify the moduli space as follows: First, cut off each cone at a reducible singularity and glue in a copy of $\mathbb {CP} ^{2}$ . Secondly, glue in a copy of $X$  itself at infinity. The resulting space is a cobordism between $X$  and a disjoint union of $b_{2}(X)$  copies of $\mathbb {CP} ^{2}$  with its orientation reversed. The intersection form of a four-manifold is a cobordism invariant up to isomorphism of quadratic forms, from which one concludes the intersection form of $X$  is diagonalisable.

## Extensions

Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold. Combining this result with the Serre classification theorem and Donaldson's theorem, several interesting results can be seen:

1) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure (so cannot be smoothed).

2) Two smooth simply-connected 4-manifolds are homeomorphic, if and only if, their intersection forms have the same rank, signature, and parity.