Lectures on the Topology of 3-Manifolds

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.

Table of Contents

1 Heegaard splittings

2 Dehn surgery

3 Kirby calculus

4 Even surgeries

5 Review of 4-manifolds

6 Four-manifolds with boundary

7 Invariants of knots and links

8 Fibered knots

9 The Art-invariant

10 Rohlin’s theorem

11 The Rohlin invariant

12 The Casson invariant

13 The group SU(2)

14 Representation spaces

15 The local properties of representation spaces

16 Casson’s invariant for Heegaard splittings

17 Casson’s invariant for knots

18 An application of the Casson invariant

19 The Casson invariant of Seifert manifolds

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

“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

About the Author

Nikolai Saveliev, University of Miami, Florida, USA.