Monopoles and Three-Manifolds
The goal of this book is to provide a complete foundation, with detailed proofs, for the Seiberg-Witten or monopole Floer homology. The book explains Floer homology theory of 3-dimensional manifolds based on the Seiberg-Witten equations (monopole equations) instead of Floer’s construction, which uses the anti-self-dual Yang-Mills equations (instanton equations). The first half of the book is devoted to a detailed study of moduli space (as a Hilbert manifold) and to a definition of monopole Floer homology as Morse theory of the Chern-Simons-Dirac functional. The authors develop analytic properties of the Seiberg–Witten equations. The Floer groups are defined here for any compact, connected, oriented 3-manifold, which is remarkable in a theory known for requiring special conditions to achieve applicability. The next chapter is dedicated to cobordism/topological invariance, exploring implications of the circle-valued Morse theory. Two final chapters are devoted to a calculation of Floer groups and to applications of the theory in topology. The book will be more understandable to readers with experience of gauge theory, Hilbert manifolds and slice theorems. It should be of interest to any mathematician faced with an infinite-dimensional moduli space of some sort.