This book is based on a series of seminars on the topic held at Oxford University. It has two different goals. Firstly, it is an exposition of Floer’s original work. Secondly, the author develops further aspects of the theory, which did not appear in the literature before. The Floer homology groups are new topological invariants of three-dimensional manifolds. They fit very nicely into a broader scheme inspired by topological quantum field theories. Intuitively speaking, they are middle dimensional holomogy groups of the infinite dimensional space of connections modulo gauges. There are very important relations of the Floer homology groups with invariants of four dimensional manifolds, instantons and Yang-Mills theory. The introduction describes motivations for the theory and its evolution. The first part of the book (Chapt. 2 - 5) contains a systematic treatment of the Floer homology groups of a homology 3-sphere. The second part of the book starts (Chapt. 6) with a description of the relation between the Floer homology groups and the invariants of 4-manifolds defined by Yang-Mills instantons. Chapt. 7 includes a description of a product structure on the Floer homology groups and a discussion showing how the Floer groups fit into topological field theory for a special class of 4-manifolds. In the last chapter, further possible research directions are described. The book gives a nice account of the theory of an interesting topic in contemporary geometry and topology. It can be strongly recommended to anybody interested in new ideas coming from recent important interactions between mathematics and modern theoretical physics.

Reviewer:

jbu