In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a closed, oriented, smooth manifold of dimension 4 is diagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the . The original version of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.
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 Yang–Mills moduli space <math>\mathcal{M}_P</math> of solutions to the anti-self-duality equations on a principal <math>\operatorname{SU}(2)</math>-bundle <math>P</math> over the four-manifold <math>X</math>. By the Atiyah–Singer index theorem, the dimension of the moduli space is given by
:<math>\dim \mathcal{M} = 8k - 3(1-b_1(X) + b_+(X)),</math>
where <math>k=c_2(P) </math> is a Chern class, <math>b_1(X)</math> is the first Betti number of <math>X</math>, and <math>b_+(X)</math> is the dimension of the positive-definite subspace of <math>H_2(X,\mathbb{R})</math> with respect to the intersection form. When <math>X</math> is simply-connected with definite intersection form, possibly after changing orientation, one always has <math>b_1(X) = 0</math> and <math>b_+(X)=0</math>. Thus taking any principal <math>\operatorname{SU}(2)</math>-bundle with <math>k=1</math>, one obtains a moduli space <math>\mathcal{M}</math> of dimension five.
thumb|right|Cobordism given by the [[Yang–Mills moduli space 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 <math>b_2(X)</math> many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst <math>\mathcal{M}</math> is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of <math>\mathcal{M}</math>, say <math>\mathcal{M}_{\varepsilon}</math>, such that for sufficiently small choices of parameter <math>\varepsilon</math>, there is a diffeomorphism
:<math>\mathcal{M}_{\varepsilon} \xrightarrow{\quad \cong\quad} X\times (0,\varepsilon)</math>.
The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold <math>X</math> with curvature becoming infinitely concentrated at any given single point <math>x\in X</math>. 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.