Donaldson theory

Donaldson theory is the study of smooth 4-manifolds using gauge theory. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold.

Most of the results of Donaldson theory depend on the manifold having a differential structure, and are false for topological 4-manifolds.

Many of the theorems in Donaldson theory can now be proved more easily using Seiberg–Witten theory.

References

• Donaldson, Simon (1983), "An Application of Gauge Theory to Four Dimensional Topology", Journal of Differential Geometry 18 (2): 279–315, MR710056 .
• Donaldson, S. K.; Kronheimer, P. B. (1997), The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford: Clarendon Press, ISBN 0198502699 .
• Freed, D. S.; Uhlenbeck, K. K. (1984), Instantons and four-manifolds, New York: Springer, ISBN 0387960368 .
• Scorpan, A. (2005), The wild world of 4-manifolds, Providence: American Mathematical Society, ISBN 0821837494 .

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