Donaldson's theorem

Idea of proof

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

${\displaystyle \dim \mathcal{M}=8k-3(1-b_1(X)+b_{+}(X)),}$

where ${\displaystyle k=c_2(P)}$ is a Chern class, ${\displaystyle b_1(X)}$ is the first Betti number of ${\displaystyle X}$, and ${\displaystyle b_{+}(X)}$ is the dimension of the positive-definite subspace of ${\displaystyle H_2(X,\mathbb{R})}$ with respect to the intersection form. When ${\displaystyle X}$ is simply-connected with definite intersection form, possibly after changing orientation, one always has ${\displaystyle b_1(X)=0}$ and ${\displaystyle b_{+}(X)=0}$. Thus taking any principal ${\displaystyle \mathrm{SU}(2)}$-bundle with ${\displaystyle k=1}$, one obtains a moduli space ${\displaystyle \mathcal{M}}$ 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 ${\displaystyle b_2(X)}$ many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst ${\displaystyle \mathcal{M}}$ is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of ${\displaystyle \mathcal{M}}$, say ${\displaystyle {\mathcal{M}}_{\epsilon }}$, such that for sufficiently small choices of parameter ${\displaystyle \epsilon }$, there is a diffeomorphism

${\displaystyle {\mathcal{M}}_{\epsilon }\stackrel{\cong }{\to }X\times (0,\epsilon )}$.

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold ${\displaystyle X}$ with curvature becoming infinitely concentrated at any given single point ${\displaystyle x\in X}$. 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 singularity theorem.

Donaldson observed that the singular points in the interior of ${\displaystyle \mathcal{M}}$ corresponding to reducible connections could also be described: they looked like cones over the complex projective plane ${\displaystyle {\mathbb{C}\mathbb{P}}^2}$. Furthermore, we can count the number of such singular points. Let ${\displaystyle E}$ be the ${\displaystyle {\mathbb{C}}^2}$-bundle over ${\displaystyle X}$ associated to ${\displaystyle P}$ by the standard representation of ${\displaystyle SU(2)}$. Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings ${\displaystyle E=L\oplus L^{-1}}$ where ${\displaystyle L}$ is a complex line bundle over ${\displaystyle X}$. Whenever ${\displaystyle E=L\oplus L^{-1}}$ we may compute:

${\displaystyle 1=k=c_2(E)=c_2(L\oplus L^{-1})=-Q(c_1(L),c_1(L))}$,

where ${\displaystyle Q}$ is the intersection form on the second cohomology of ${\displaystyle X}$. Since line bundles over ${\displaystyle X}$ are classified by their first Chern class ${\displaystyle c_1(L)\in H^2(X;\mathbb{Z})}$, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs ${\displaystyle \pm \alpha \in H^2(X;\mathbb{Z})}$ such that ${\displaystyle Q(\alpha ,\alpha )=-1}$. Let the number of pairs be ${\displaystyle n(Q)}$. An elementary argument that applies to any negative definite quadratic form over the integers tells us that ${\displaystyle n(Q)\le \text{rank}(Q)}$, with equality if and only if ${\displaystyle Q}$ is diagonalizable.

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 ${\displaystyle {\mathbb{C}\mathbb{P}}^2}$. Secondly, glue in a copy of ${\displaystyle X}$ itself at infinity. The resulting space is a cobordism between ${\displaystyle X}$ and a disjoint union of ${\displaystyle n(Q)}$ copies of ${\displaystyle {\mathbb{C}\mathbb{P}}^2}$ (of unknown orientations). The signature ${\displaystyle \sigma }$ of a four-manifold is a cobordism invariant. Thus, because ${\displaystyle X}$ is definite:

${\displaystyle \text{rank}(Q)=b_2(X)=\sigma (X)=\sigma (⨆n(Q){\mathbb{C}\mathbb{P}}^2)\le n(Q)}$,

from which one concludes the intersection form of ${\displaystyle X}$ is diagonalizable.

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 definite 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.