Gauge theory and related areas of geometry have been an important tool for the study of 4-dimensional manifolds since the early 1980s, when Donaldson introduced ideas from Yang–Mills theory to solve long-standing problems in topology. In dimension 3, the same techniques formed the basis of Floer's construction of his “instanton homology” groups of 3-manifolds [32]. Today, Floer homology is an active area, and there are several varieties of Floer homology theory, all with closely related structures. While Floer's construction used the anti-self-dual Yang–Mills (or instanton) equations, the theory presented in this book is based instead on the Seiberg–Witten equations (or monopole equations).

We have aimed to lay a secure foundation for the study of the Seiberg–Witten equations on a general 3-manifold, and for the construction of the associated Floer groups. Our goal has been to write a book that is complete in its coverage of several aspects of the theory that are hard to find in the existing literature, providing at the same time an introduction to the techniques from analysis and geometry that are used. We have omitted some background topics that are now well treated in several good sources: in particular, the Seiberg–Witten invariants of closed 4-manifolds and related topics in gauge theory are given only a brief exposition here.