Lectures on the Topology of 3-Manifolds,9783110162721

Lectures on the Topology of 3-Manifolds

by
ISBN13: 9783110162721
ISBN10: 3110162725
Format: Hardcover
Pub. Date: 1999-09-01
Publisher(s): De Gruyter
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Author Biography

Nikolai Saveliev is an assistant professor of Mathematics at Tulane University in New Orleans, U.S.A.

Table of Contents

Preface v
Introduction 1(2)
Glossary 3(12)
Heegaard splittings
15(12)
Introduction
15(1)
Existence of Heegaard splittings
16(1)
Stable equivalence of Heegaard splittings
16(4)
The Mapping class group
20(1)
Manifolds of Heegaard genus ≤ 1
21(4)
Seifert Manifolds
25(2)
Dehn Surgery
27(10)
Knots and links in 3-manifolds
27(1)
Surgery on links in S3
28(2)
Surgery description of lens spaces and Seifert manifolds
30(4)
Surgery and 4-Manifolds
34(3)
Kirby calculus
37(14)
The linking number
37(2)
Kirby moves
39(9)
The linking matrix
48(1)
Reversing orientation
49(2)
Even surgeries
51(6)
Review of 4-manifolds
57(8)
Definition of the intersection form
57(3)
The unimodular integral forms
60(2)
Four-manifolds and intersection forms
62(3)
Four-manifolds with boundary
65(8)
The intersection form
65(4)
Homology spheres via surgery on knots
69(1)
Seifert homology spheres
70(2)
The Rohlin invariant
72(1)
Invariants of Knots and links
73(14)
Seifert surfaces
73(2)
Seifert matrices
75(2)
The Alexander polynomial
77(4)
Other invariants from Seifert surfaces
81(2)
Knots in homology spheres
83(1)
Boundary links and the Alexander polynomial
84(3)
Fibered knots
87(10)
The definition of a fibered knot
87(1)
The monodromy
88(3)
More about tours knots
91(1)
Joins
92(2)
The monodromy of tours knots
94(3)
The Arf-invariant
97(6)
The Art-invariant of a quadratic form
97(2)
The Art-invariant of a knot
99(4)
Rohlin's theorem
103(8)
Characteristic surfaces
103(1)
The definition of q
104(4)
Representing homology classes by surfaces
108(3)
The Rohlin invariant
111(10)
Definition of the Rohlin invariant
111(1)
The Rohlin invariant of Seifert spheres
111(3)
A surgery formula for the Rohlin invariant
114(3)
The homology cobordism group
117(4)
The Casson invariant
121(8)
The group SU(2)
129(6)
Representation spaces
135(10)
The topology of representation spaces
135(1)
Irreducible representations
136(1)
Representations of free groups
136(1)
Representations of surface groups
137(2)
Representations of Seifert homology spheres
139(6)
The local properties of representation spaces
145(4)
Casson's invariant for Heegaard splittings
149(8)
The intersection product
149(2)
The orientations
151(3)
Independence of Heegaard splitting
154(3)
Casson's invariant for knots
157(10)
Preferred Heegaard splittings
157(1)
The Casson invariant for knots
158(3)
The difference cycle
161(2)
The Casson invariant of a unlinks
163(1)
The Casson invariant of a trefoil
164(3)
An application of the Casson invariant
167(2)
Triangulating 4-manifolds
167(1)
Higher-dimensional manifolds
168(1)
The Casson invariant of Seifert manifolds
169(8)
The Space R(p,q,r)
169(3)
Calculation of the Casson invariant
172(5)
Conclusion 177(6)
Exercises 183(4)
Bibliography 187(10)
Index 197

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