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

by Nikolai Saveliev

Overview

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.

Product Details

ISBN-13:	9783110250350
Publisher:	De Gruyter
Publication date:	12/19/2011
Series:	De Gruyter Textbook
Edition description:	2nd rev. ed.
Pages:	218
Product dimensions:	6.69(w) x 9.45(h) x 0.02(d)
Age Range:	18 Years

About the Author

Nikolai Saveliev, University of Miami, Florida, USA.

Table of Contents

Preface
Introduction
Glossary

1 Heegaard splittings
1.1 Introduction
1.2 Existence of Heegaard splittings
1.3 Stable equivalence of Heegaard splittings
1.4 The mapping class group
1.5 Manifolds of Heegaard genus ≤ 1
1.6 Seifert manifolds
1.7 Heegaard diagrams

2 Dehn surgery
2.1 Knots and links in 3-manifolds
2.2 Surgery on links in S³
2.3 Surgery description of lens spaces and Seifert manifolds
2.4 Surgery and 4-manifolds

3 Kirby calculus
3.1 The linking number
3.2 Kirby moves
3.3 The linking matrix
3.4 Reversing orientation

4 Even surgeries

5 Review of 4-manifolds
5.1 Definition of the intersection form
5.2 The unimodular integral forms
5.3 Four-manifolds and intersection forms

6 Four-manifolds with boundary
6.1 The intersection form
6.2 Homology spheres via surgery on knots
6.3 Seifert homology spheres
6.4 The Rohlin invariant

7 Invariants of knots and links
7.1 Seifert surfaces
7.2 Seifert matrices
7.3 The Alexander polynomial
7.4 Other invariants from Seifert surfaces
7.5 Knots in homology spheres
7.6 Boundary links and the Alexander polynomial

8 Fibered knots
8.1 The definition of a fibered knot
8.2 The monodromy
8.3 More about torus knots
8.4 Joins
8.5 The monodromy of torus knots
8.6 Open book decompositions

9 The Arf-invariant
9.1 The Arf-invariant of a quadratic form
9.2 The Arf-invariant of a knot

10 Rohlin’s theorem
10.1 Characteristic surfaces
10.2 The definition of ˜q
10.3 Representing homology classes by surfaces

11 The Rohlin invariant
11.1 Definition of the Rohlin invariant
11.2 The Rohlin invariant of Seifert spheres
11.3 A surgery formula for the Rohlin invariant
11.4 The homology cobordism group

12 The Casson invariant

13 The group SU(2)

14 Representation spaces
14.1 The topology of representation spaces
14.2 Irreducible representations
14.3 Representations of free groups
14.4 Representations of surface groups
14.5 Representations for Seifert homology spheres

15 The local properties of representation spaces

16 Casson’s invariant for Heegaard splittings
16.1 The intersection product
16.2 The orientations
16.3 Independence of Heegaard splitting

17 Casson’s invariant for knots
17.1 Preferred Heegaard splittings
17.2 The Casson invariant for knots
17.3 The difference cycle
17.4 The Casson invariant for boundary links
17.5 The Casson invariant of a trefoil

18 An application of the Casson invariant
18.1 Triangulating 4-manifolds
18.2 Higher-dimensional manifolds

19 The Casson invariant of Seifert manifolds
19.1 The space R(p; q; r)
19.2 Calculation of the Casson invariant

Conclusion
Bibliography
Index

Editorial 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

From the Publisher