Donaldson's proof utilizes the
Yang–Mills 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 X. 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 k=c_2(P) is a
Chern class, 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. in Donaldson's theorem 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 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. Furthermore, we can count the number of such singular points. Let E be the \mathbb{C}^2-bundle over X associated to P by the standard representation of SU(2). Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings E = L\oplus L^{-1} where L is a complex line bundle over X. Whenever E = L\oplus L^{-1} we may compute: 1 = k = c_2(E) = c_2(L\oplus L^{-1}) = - Q(c_1(L), c_1(L)), where Q is the intersection form on the second cohomology of X. Since line bundles over X are classified by their first Chern class c_1(L)\in H^2(X; \mathbb{Z}), we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs \pm\alpha\in H^2(X; \mathbb{Z}) such that Q(\alpha, \alpha) = -1. Let the number of pairs be n(Q). An elementary argument that applies to any negative definite quadratic form over the integers tells us that n(Q)\leq\text{rank}(Q), with equality if and only if 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 \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 n(Q) copies of \mathbb{CP}^2 (of unknown orientations). The signature \sigma of a four-manifold is a cobordism invariant. Thus, because X is definite: \text{rank}(Q) = b_2(X) = \sigma(X) = \sigma(\bigsqcup n(Q) \mathbb{CP}^2) \leq n(Q), from which one concludes the intersection form of X is diagonalizable. ==Extensions==