Foliations II. (English) Zbl 1035.57001
Graduate Studies in Mathematics 60. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0881-8/hbk). xiii, 545 p. (2003).
This is the second volume in a comprehensive treatise on the qualitative theory of foliations. The first volume [Zbl 0936.57001] laid the foundations of foliation theory viewing foliations as generalized dynamical systems and extending to these systems the theory of invariant measures. For the second volume, the authors have selected three important topics where foliation theory interacts with some other highly developed areas of mathematics. Accordingly the book is divided into three parts.
Part 1 has the title “Analysis and Geometry on Foliated Spaces”. It treats \(C^*\)-algebras of foliated spaces and generalizes heat flow and Brownian motion in Riemannian manifolds to such spaces. Chapter 1 provides among others a very nice introduction to the deep theory originated by A. Connes and pursued under the name “noncommutative geometry”.
Part 2 is entitled “Characteristic Classes and Foliations”. The authors present a Chern-Weil type construction of the exotic classes based on the Bott vanishing theorem. As an introduction to Part 2, the authors have a very nice chapter on the Euler class of oriented circle bundles.
Part 3 has the title “Foliated 3-Manifolds” and studies compact 3-manifolds foliated by surfaces. The interest in this topic originates with the famous construction by Reeb of a foliation of the 3-sphere. The results of S.P. Novikov on Reebless foliations are studied and so is the theorem of W.Thurston on compact leaves of Reebless foliations that led D. Gabai to his celebrated work on taut foliations in the study of 3-manifold topology.
As a very good service to the readers, the authors have collected some of the necessary prerequisites from other parts of mathematics in four comprehensive appendices: Appendix A: “\(C^*\)-algebras”, Appendix B: “Riemannian Geometry and Heat Diffusion”; Appendix C: “Brownian Motion”; Appendix D: “Planar Foliations”.
The material covered in the book is contemporary mathematics at the highest level. It is not an easy book, but it is clearly and well written and neither more nor less complicated than necessary. The book will be a pleasure to study for all beginning researchers in subjects involving foliations in their many aspects.
Part 1 has the title “Analysis and Geometry on Foliated Spaces”. It treats \(C^*\)-algebras of foliated spaces and generalizes heat flow and Brownian motion in Riemannian manifolds to such spaces. Chapter 1 provides among others a very nice introduction to the deep theory originated by A. Connes and pursued under the name “noncommutative geometry”.
Part 2 is entitled “Characteristic Classes and Foliations”. The authors present a Chern-Weil type construction of the exotic classes based on the Bott vanishing theorem. As an introduction to Part 2, the authors have a very nice chapter on the Euler class of oriented circle bundles.
Part 3 has the title “Foliated 3-Manifolds” and studies compact 3-manifolds foliated by surfaces. The interest in this topic originates with the famous construction by Reeb of a foliation of the 3-sphere. The results of S.P. Novikov on Reebless foliations are studied and so is the theorem of W.Thurston on compact leaves of Reebless foliations that led D. Gabai to his celebrated work on taut foliations in the study of 3-manifold topology.
As a very good service to the readers, the authors have collected some of the necessary prerequisites from other parts of mathematics in four comprehensive appendices: Appendix A: “\(C^*\)-algebras”, Appendix B: “Riemannian Geometry and Heat Diffusion”; Appendix C: “Brownian Motion”; Appendix D: “Planar Foliations”.
The material covered in the book is contemporary mathematics at the highest level. It is not an easy book, but it is clearly and well written and neither more nor less complicated than necessary. The book will be a pleasure to study for all beginning researchers in subjects involving foliations in their many aspects.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)
MSC:
| 57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |
| 57R30 | Foliations in differential topology; geometric theory |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 53C12 | Foliations (differential geometric aspects) |
