Donaldson's theorem

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In mathematics, Donaldson's theorem states that a positive definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable to the identity matrix. It was proved by Simon Donaldson.

Michael Freedman had previously shown that any positive definite unimodular symmetric bilinear form is realized as the intersection form of some four-manifold; combining his and Donaldson's result, any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure (so cannot be smoothed).

[edit] References

• Donaldson, Simon. An Application of Gauge Theory to Four Dimensional Topology. Journal of Differential Geometry vol. 18, 1983, 279-315.
• S. K. Donaldson, P. B. Kronheimer The Geometry of Four-Manifolds (Oxford Mathematical Monographs) ISBN 0198502699
• D.S. Freed, K.K. Uhlenbeck, Instantons and four-manifolds , Springer (1984)