An Introduction to Contact Topology
This is a fundamental monograph on the topology of contact structures and its applications. It presents almost all the important results in the field and it simultaneously introduces the reader to contemporary research. There is also a lot about geometry of contact structures. Chapter 1 has an introductory character and represents an invitation to the subject. Chapter 2 is a basis for further reading, containing the Gray stability theorem with the Moser trick and the contact disc theorem. Chapter 3 is devoted to knots in contact three manifolds and surgical constructions of contact manifolds appears here for the first time. Chapter 4 deals with 3-manifolds. It includes the proof of existence of a contact structure on any 3-manifold, the Lutz twist - a topologically trivial Dehn surgery, the Eliashberg classification of overtwisted contact structures, the proof of the Cerf theorem Γ4=0 and an introduction to convex surface theory in contact 3-manifolds. The remaining four chapters study higher dimensional contact manifolds. The main aim of chapters 5 and 6 is to introduce contact surgery. These results are then applied in chapter 8 to contact 5-manifolds. The final chapter brings together various topological constructions useful in a study of contact manifolds. The book is very well-written. The author mentions that the intended reader is an advanced graduate student. But even a less advanced graduate student, who is really interested, will be attracted by this book. The author has not included exercises in the book but on the other hand there are a lot of examples. There is a list of 246 references, a notation index, an author index and a subject index. All of them are quite helpful. The book can be strongly recommended for graduate students and is indispensable for specialists in the field.