Donaldson theory
{{Short description|Study in mathematical gauge theory}}
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. 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. Important consequences of this theorem include the existence of an exotic R4 and the failure of the smooth h-cobordism theorem in 4 dimensions. The results of Donaldson theory depend therefore on the manifold having a differential structure, and are largely false for topological 4-manifolds.
Many of the theorems in Donaldson theory can now be proved more easily using Seiberg–Witten theory, though there are a number of open problems remaining in Donaldson theory, such as the Witten conjecture and the Atiyah–Floer conjecture.
See also
References
- {{Citation |last=Donaldson |first=Simon |title=An Application of Gauge Theory to Four Dimensional Topology |journal=Journal of Differential Geometry |volume=18 |issue=2 |year=1983 |pages=279–315 |mr=710056 }}.
- {{Citation |first=S. K. |last=Donaldson |first2=P. B. |last2=Kronheimer |author-link2=Peter B. Kronheimer |title=The Geometry of Four-Manifolds |series=Oxford Mathematical Monographs |location=Oxford |publisher=Clarendon Press |year=1997 |isbn=0-19-850269-9 }}.
- {{Citation |first=D. S. |last=Freed |first2=K. K. |last2=Uhlenbeck |author-link2=Karen Uhlenbeck |title=Instantons and four-manifolds |location=New York |publisher=Springer |year=1984 |isbn=0-387-96036-8 }}.
- {{Citation |first=A. |last=Scorpan |title=The wild world of 4-manifolds |location=Providence |publisher=American Mathematical Society |year=2005 |isbn=0-8218-3749-4 }}.
Category:Differential topology
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