Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant

Nikolai Saveliev

Published by De Gruyter 2012

Lectures on the Topology of 3-Manifolds

An Introduction to the Casson Invariant

Nikolai Saveliev

In the series De Gruyter Textbook

https://doi.org/10.1515/9783110250367

Overview
Contents

About this book

Progress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant.

The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments.

The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.

A beginner's introduction to low-dimensional topology
2^nd revised and extended edition
New material is added to provide a link to recent research in Heegaard Floer homology and contact topology

Author / Editor information

Nikolai Saveliev, University of Miami, Florida, USA.

Reviews

"This is an excellent introduction to the Rokhlin and Casson invariants for homology 3-spheres [...], and in particular also to the necessary background material from the theory of 3- and 4-manifolds [...], so the book may serve also as a reasonable short and efficient introduction to some important parts of low-dimensional topology. It grew out of a course for second year graduate students and concentrates 19 lectures on less than 200 pages, including also a glossary on back-ground material from algebraic topology, a collection of exercises, open problems and comments on recent developments [...] To conclude, the author has succeeded in presenting a lot of material in a clear and efficient way, and the book is interesting and stimulating to read."
Birge Zimmermann-Huisgen, Zentralblatt MATH

Supplementary Materials

Contents

Topics

Frontmatter Requires Authentication Unlicensed Licensed Download PDF	i
Preface Nikolai Saveliev Requires Authentication Unlicensed Licensed Download PDF	v
Contents Requires Authentication Unlicensed Licensed Download PDF	ix
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Glossary Requires Authentication Unlicensed Licensed Download PDF	3
Lecture 1. Heegaard splittings Requires Authentication Unlicensed Licensed Download PDF	16
Lecture 2. Dehn surgery Requires Authentication Unlicensed Licensed Download PDF	32
Lecture 3. Kirby calculus Requires Authentication Unlicensed Licensed Download PDF	43
Lecture 4. Even surgeries Requires Authentication Unlicensed Licensed Download PDF	58
Lecture 5. Review of 4-manifolds Requires Authentication Unlicensed Licensed Download PDF	63
Lecture 6. Four-manifolds with boundary Requires Authentication Unlicensed Licensed Download PDF	72
Lecture 7. Invariants of knots and links Requires Authentication Unlicensed Licensed Download PDF	81
Lecture 8. Fibered knots Requires Authentication Unlicensed Licensed Download PDF	98
Lecture 9. The Arf-invariant Requires Authentication Unlicensed Licensed Download PDF	109
Lecture 10. Rohlin’s theorem Requires Authentication Unlicensed Licensed Download PDF	116
Lecture 11. The Rohlin invariant Requires Authentication Unlicensed Licensed Download PDF	123
Lecture 12. The Casson invariant Requires Authentication Unlicensed Licensed Download PDF	135
Lecture 13. The group SU(2) Requires Authentication Unlicensed Licensed Download PDF	142
Lecture 14. Representation spaces Requires Authentication Unlicensed Licensed Download PDF	148
Lecture 15. The local properties of representation spaces Requires Authentication Unlicensed Licensed Download PDF	159
Lecture 16. Casson’s invariant for Heegaard splittings Requires Authentication Unlicensed Licensed Download PDF	163
Lecture 17. Casson’s invariant for knots Requires Authentication Unlicensed Licensed Download PDF	172
Lecture 18. An application of the Casson invariant Requires Authentication Unlicensed Licensed Download PDF	181
Lecture 19. The Casson invariant of Seifert manifolds Requires Authentication Unlicensed Licensed Download PDF	184
Conclusion Requires Authentication Unlicensed Licensed Download PDF	191
Bibliography Requires Authentication Unlicensed Licensed Download PDF	195
Index Requires Authentication Unlicensed Licensed Download PDF	205

(Deutsch)

Language: English
Publisher: De Gruyter
Edition: 2nd rev. ed.
Audience: Graduate Students, Lecturers, and Researchers in Mathematics and Theoretical Physic; Academic Libraries
Pages
- Front matter: 11
- Main content: 207
Illustrations
- Illustrations: 175
Keywords: Topology; Manifold; Casson; Invariant

eBook

Published: December 23, 2011
ISBN: 9783110250367

Hardcover

Published: December 19, 2011
ISBN: 9783110250350