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). It was proved by Simon Donaldson.

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.

References

• Donaldson, S. K. (1983), "An application of gauge theory to four-dimensional topology", Journal of Differential Geometry 18 (2): 279–315, MR710056, ISSN 0022-040X, http://projecteuclid.org/euclid.jdg/1214437665 
• S. K. Donaldson, P. B. Kronheimer The Geometry of Four-Manifolds (Oxford Mathematical Monographs) ISBN 0-19-850269-9
• D.S. Freed, K.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)

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