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Lectures on Seiberg-Witten invariants. (English) Zbl 0857.58001
Lecture Notes in Mathematics. 1629. Berlin: Springer. vii, 105 p. (1996).
These lecture notes stem from a graduate course given at the University of California in Santa Barbara during the spring quarter of 1995. The purpose of these lecture notes is to make the Seiberg-Witten approach to Donaldson theory accesssible to second-year graduate students who have already taken basic courses in differential geometry and algebraic topology. The author also makes it his aim to use the Seiberg-Witten equations to give the differential geometric parts of the proofs of the well-known theorems of Simon-Donaldson stating the existence of compact topological four-manifolds which have no smooth structure and the existence of pairs of compact smooth four-manifolds which are homeomorphic but not diffeomorphic. In the mathematical preliminaries the reader is familiarized with the tools needed to study spin and \(\text{spin}^c\) structures such as vector bundles, connections, characteristic classes and Hodge theory. The second chapter is devoted to a discussion of spin geometry on four-dimensional manifolds, covering Clifford algebras, spin connections, Dirac operator and the Atiyah-Singer index theorem. The topics treated in the last chapter include the Seiberg-Witten equations, the moduli space of solutions to the Seiberg-Witten equations, Seiberg-Witten invariants and Dirac operators on Kähler surfaces. Many examples make these lecture notes a valuable study text.
Contents: Preliminaries; Spin geometry on four-manifolds; Global analysis of the Seiberg-Witten equations.
Reviewer: V.Abramov (Tartu)

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R17 Symplectic and contact topology in high or arbitrary dimension
53C27 Spin and Spin\({}^c\) geometry
57R55 Differentiable structures in differential topology
58J10 Differential complexes
58D27 Moduli problems for differential geometric structures
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)