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Homotopy K3 surfaces and gluing results in Seiberg-Witten theory. (English) Zbl 0870.57022
Lecture Notes Series, Seoul. 37. Seoul: Seoul National University, Research Inst. of Math., Global Analysis Research Center, iii, 53 p. (1996).
These lecture notes are not an introduction into Seiberg-Witten equations. They rather give a treatment of the gluing and blow down formulae which are crucial for computations of the Seiberg-Witten invariants. The text consists of three lectures given by the author in Seoul, 1995. The first lecture discusses the topology of elliptic surfaces, log-transforms and rational blow downs. There are many examples given in terms of handlebody pictures. In the second lecture the Seiberg-Witten invariants of several homotopy K3 surfaces are computed. It is shown that there are infinitely many differential structures on the K3 surface and that not every 4-manifold is built out of complex parts. The third lecture discusses Floer theory for Seiberg-Witten invariants. It gives a homology theory for 3-manifolds with \(b_1\geq 2\).

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)