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The Seiberg-Witten equations and 4-manifold topology. (English) Zbl 0872.57023
Since 1982 the use of gauge theory, in the shape of the Yang-Mills instanton equations, has permeated research in 4-manifold theory. In the last part of the year 1994 this research area was turned on its head by the introduction of a new kind of differential-geometric equation by Seiberg and Witten. In the space of a few weeks long-standing problems were solved, new and unexpected results were found, along with simpler new proofs of extending ones, and new vistas for research opened up.
The paper under review is a report on some of these developments specially due to P. B. Kronheimer and T. S. Mrowka [Bull. Am. Math. Soc., New Ser. 30, 215-221 (1994; Zbl 0815.57010); Math. Res. Lett. 1, 797-808 (1994; Zbl 0851.57023)] and C. H. Taubes [Math. Res. Lett. 1, 809-822 (1994; Zbl 0853.57019); 2, 9-13 (1995; Zbl 0854.57019)] building on the seminal work of N. Seiberg [Phys. Lett. B 318, 469 (1993)] and N. Seiberg and E. Witten [Nucl. Phys. B 431, 581-640 (1994)]. It is written as an attempt to take stock of the progress stemming from this initial period of intense activity.
In conclusion a wonderful work which will be useful to all those interested in 4-manifold theory.

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 81T13 Yang-Mills and other gauge theories in quantum field theory
##### Keywords:
Seiberg-Witten invariants; gauge theory; 4-manifold theory
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