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Concept
Donaldson's theorem
Summary
In mathematics, and especially differential topology and gauge theory, 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 . 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.
The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986.
Donaldson's proof utilizes the moduli space of solutions to the anti-self-duality equations on a principal -bundle over the four-manifold . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by
where , is the first Betti number of and is the dimension of the positive-definite subspace of with respect to the intersection form. When is simply-connected with definite intersection form, possibly after changing orientation, one always has and . Thus taking any principal -bundle with , one obtains a moduli space 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 many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of , say , such that for sufficiently small choices of parameter , there is a diffeomorphism
The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold with curvature becoming infinitely concentrated at any given single point . 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.
Official source
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