About this Title
John Hempel, Rice University, Houston, TX
Publication: AMS Chelsea Publishing
Publication Year: 1976; Volume 349
ISBNs: 978-0-8218-3695-8 (print); 978-1-4704-3025-2 (online)
A careful and systematic development of the theory of the topology of 3-manifolds, focusing on the critical role of the fundamental group in determining the topological structure of a 3-manifold … self-contained … one can learn the subject from it … would be very appropriate as a text for an advanced graduate course or as a basis for a working seminar.
For many years, John Hempel's book has been a standard text on the topology of 3-manifolds. Even though the field has grown tremendously, the book remains one of the best and most popular introductions to the subject.
The theme of this book is the role of the fundamental group in determining the topology of a given 3-manifold. The essential ideas and techniques are covered in the first part of the book: Heegaard splittings, connected sums, the loop and sphere theorems, incompressible surfaces, free groups, and so on. Along the way, many useful and insightful results are proved, usually in full detail. Later chapters address more advanced topics, including Waldhausen's theorem on a class of 3-manifolds that is completely determined by its fundamental group. The book concludes with a list of problems that were unsolved at the time of publication.
Hempel's book remains an ideal text to learn about the world of 3-manifolds. The prerequisites are few and are typical of a beginning graduate student. Exercises occur throughout the text.
Graduate students and research mathematicians interested in low-dimensional topology.
Table of Contents
- Chapter 1. Preliminaries
- Chapter 2. Heegaard splittings
- Chapter 3. Connected sums
- Chapter 4. The loop and sphere theorems
- Chapter 5. Free groups
- Chapter 6. Incompressible surfaces
- Chapter 7. Kneser’s conjecture on free products
- Chapter 8. Finitely generated subgroups
- Chapter 9. More on connected sums; Finite and abelian subgroups
- Chapter 10. I-bundles
- Chapter 11. Group extensions and fibrations
- Chapter 12. Seifert fibered spaces
- Chapter 13. Classification of $P^2$-irreducible, sufficiently large 3-manifolds
- Chapter 14. Some approaches to the Poincaré conjecture
- Chapter 15. Open problems