Donaldson's theorem

In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected 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 integers.

History

The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986.

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 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.

Donaldson, S. K. (1983), "An application of gauge theory to four-dimensional topology", Journal of Differential Geometry, 18 (2): 279–315, ISSN 0022-040X, MR 0710056
S. K. Donaldson, P. B. Kronheimer The Geometry of Four-Manifolds (Oxford Mathematical Monographs) ISBN 0-19-850269-9
D.S. Freed, K. Uhlenbeck, Instantons and four-manifolds, Springer (1984)
M. Freedman, F. Quinn, Topology of 4-Manifolds", Princeton University Press (1990)
A. Scorpan,The Wild World of 4-Manifolds, American Mathematical Society (2005)