Lectures on the Topology of 3-Manifolds (eBook)

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Nikolai Saveliev, University of Miami, Florida, USA.

Preface 6
Contents 10
Introduction 14
Glossary 16
1 Heegaard splittings 29
1.1 Introduction 29
1.2 Existence of Heegaard splittings 30
1.3 Stable equivalence of Heegaard splittings 31
1.4 The mapping class group 34
1.5 Manifolds of Heegaard genus < _ 1
1.6 Seifert manifolds 39
1.7 Heegaard diagrams 41
1.8 Exercises 44
2 Dehn surgery 45
2.1 Knots and links in 3-manifolds 45
2.2 Surgery on links in S3 46
2.3 Surgery description of lens spaces and Seifert manifolds 48
2.4 Surgery and 4-manifolds 52
2.5 Exercises 55
3 Kirby calculus 56
3.1 The linking number 56
3.2 Kirby moves 58
3.3 The linking matrix 67
3.4 Reversing orientation 68
3.5 Exercises 69
4 Even surgeries 71
4.1 Exercises 75
5 Review of 4-manifolds 76
5.1 Definition of the intersection form 76
5.2 The unimodular integral forms 80
5.3 Four-manifolds and intersection forms 81
5.4 Exercises 84
6 Four-manifolds with boundary 85
6.1 The intersection form 85
6.2 Homology spheres via surgery on knots 90
6.3 Seifert homology spheres 90
6.4 The Rohlin invariant 92
6.5 Exercises 93
7 Invariants of knots and links 94
7.1 Seifert surfaces 94
7.2 Seifert matrices 96
7.3 The Alexander polynomial 98
7.4 Other invariants from Seifert surfaces 102
7.5 Knots in homology spheres 104
7.6 Boundary links and the Alexander polynomial 106
7.7 Exercises 109
8 Fibered knots 111
8.1 The definition of a fibered knot 111
8.2 The monodromy 113
8.3 More about torus knots 115
8.4 Joins 116
8.5 The monodromy of torus knots 118
8.6 Open book decompositions 119
8.7 Exercises 121
9 The Arf-invariant 122
9.1 The Arf-invariant of a quadratic form 122
9.2 The Arf-invariant of a knot 125
9.3 Exercises 128
10 Rohlin’s theorem 129
10.1 Characteristic surfaces 129
10.2 The definition of q~ 130
10.3 Representing homology classes by surfaces 135
11 The Rohlin invariant 136
11.1 Definition of the Rohlin invariant 136
11.2 The Rohlin invariant of Seifert spheres 136
11.3 A surgery formula for the Rohlin invariant 140
11.4 The homology cobordism group 142
11.5 Exercises 146
12 The Casson invariant 148
12.1 Exercises 154
13 The group SU (2) 155
13.1 Exercises 160
14 Representation spaces 161
14.1 The topology of representation spaces 161
14.2 Irreducible representations 162
14.3 Representations of free groups 163
14.4 Representations of surface groups 163
14.5 Representations for Seifert homology spheres 166
14.6 Exercises 171
15 The local properties of representation spaces 172
15.1 Exercises 175
16 Casson’s invariant for Heegaard splittings 176
16.1 The intersection product 176
16.2 The orientations 179
16.3 Independence of Heegaard splitting 181
16.4 Exercises 184
17 Casson’s invariant for knots 185
17.1 Preferred Heegaard splittings 185
17.2 The Casson invariant for knots 186
17.3 The difference cycle 190
17.4 The Casson invariant for boundary links 191
17.5 The Casson invariant of a trefoil 192
18 An application of the Casson invariant 194
18.1 Triangulating 4-manifolds 194
18.2 Higher-dimensional manifolds 195
18.3 Exercises 196
19 The Casson invariant of Seifert manifolds 197
19.1 The space R(S (p, q, r)) 197
19.2 Calculation of the Casson invariant 200
19.3 Exercises 203
Conclusion 204
Bibliography 208
Index 218

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"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