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.

Contents: Preliminaries; Spin geometry on four-manifolds; Global analysis of the Seiberg-Witten equations.

Reviewer: V.Abramov (Tartu)

##### MSC:

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.) |