Donaldson's theorem
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 diagonalizable.
If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the integers.
The original version[1] of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.
[2] 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
By the Atiyah–Singer index theorem, the dimension of the moduli space is given by where
is the dimension of the positive-definite subspace of
with respect to the intersection form.
is simply-connected with definite intersection form, possibly after changing orientation, one always has
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
[3] Results of Clifford Taubes and Karen Uhlenbeck show that whilst
is non-compact, its structure at infinity can be readily described.
, 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.
[6][3] Donaldson observed that the singular points in the interior of
corresponding to reducible connections could also be described: they looked like cones over the complex projective plane
Furthermore, we can count the number of such singular points.
Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings
is the intersection form on the second cohomology of
, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs
An elementary argument that applies to any negative definite quadratic form over the integers tells us that
[3] 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
, from which one concludes the intersection form of
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 indefinite 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.
