Cambridge Tracts in Mathematics. 147. Cambridge: Cambridge University Press. vii, 236 p. £50.00; $ 75.00 (2002).
The book is devoted to the concept of Floer homology and its applications in Yang-Mills theory. It originated from a series of seminars on this subject held in Oxford. The aim was to give a thorough exposition of Floer’s original work and to develop aspects of the theory which have not appeared in detail in literature before. It is emphasized that the Floer homology yields rigorously defined invariants of different manifolds which are viewed as homology groups of infinite-dimensional cycles. Also, it is argued that the ideas from Floer homology are intimately related to developments in quantum field theory.
The first part of the book contains a presentation of the geometrical and analytical techniques in the context of gauge theory over 3- and 4-dimensional manifolds. The Yang-Mills theory is reviewed and the Floer homology groups are studied in details. The invariants for closed 4-manifolds are also summarised.
In the second half of the book some further technical developments of the theory, mainly involving ideas from algebraic topology, are given. Reducible connections and the cup products, instanton solutions over 4-dimensional manifolds or the Floer homology of connected sums are presented in detail. Other topics like the Casson invariant of homology spheres, Floer’s exact surgery sequence or the links between Floer’s theory and the moduli spaces of flat connections over surfaces are not included in this book. However, in the final chapter, open problems are discussed and further developments are mentioned.
The book is of a great interest for graduate students as well as for researches working on the frontiers of the subject.