Description

Fields-medal winner Donaldson reviews current work, including his own, on the theory of Floer. The seminal work of Floer has now been placed in a contemporary setting. The author of this monograph writes with the big picture constantly in mind, reviewing current knowledge and predicting future directions. This forms part of the work for which Simon Donaldson was awarded the prestigious Fields Medal. The seminal work of Floer has now been placed in a contemporary setting. The author of this monograph writes with the big picture constantly in mind, reviewing current knowledge and predicting future directions. This forms part of the work for which Simon Donaldson was awarded the prestigious Fields Medal. The concept of Floer homology was one of the most striking developments in differential geometry. It yields rigorously defined invariants which can be viewed as homology groups of infinite-dimensional cycles. The ideas led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory. The first half of this book gives a thorough account of Floer's construction in the context of gauge theory over 3 and 4-dimensional manifolds. The second half works out some further technical developments of the theory, and the final chapter outlines some research developments for the future - including a discussion of the appearance of modular forms in the theory. The scope of the material in this book means that it will appeal to graduate students as well as those on the frontiers of the subject. 1. Introduction; 2. Basic material; 3. Linear analysis; 4. Gauge theory and tubular ends; 5. The Floer homology groups; 6. Floer homology and 4-manifold invariants; 7. Reducible connections and cup products; 8. Further directions. '… relatively short but very infomative, modern and clearly written … I stronly recommend the book to both specialists and graduate students'. S. Merkulov, Proceedings of the Edinburgh Mathematical Society '… a compact but very readable account.' Mathematika '… gives a nice account of the theory of an interesting topic in contemporary geometry and topology. It can be strongly recommended …'. EMS Newsletter