Geometry of surfaces

Table of contents......Page 8
List of special symbols......Page 12
Authors\' preface......Page 14
Preface to the second edition......Page 16
1.1 INTRODUCTION......Page 18
1.2 PRELIMINARY SKIRMISH......Page 19
1.3 MODELS......Page 21
1.4 CONNECTED SETS......Page 22
1.5 PROBLEM SURFACES......Page 23
1.6 HOMEOMORPHIC SURFACES......Page 24
1. 7 SOME BASIC SURFACES......Page 27
1.8 ORIENT ABILITY......Page 30
1.9 THE CONNECTED SUM CONSTRUCTION......Page 31
1.11 EXERCISES......Page 32
2.1 THE BASIC PLANE MODELS......Page 35
2.2 PAPER MODELS OF THE BASIC SURFACES......Page 40
2.5 SUMMARY......Page 41
2.6 COMMENTS......Page 42
2.7 EXERCISES......Page 45
3.1 PLANE MODELS AND THE CONNECTED SUM CONSTRUCTION......Page 52
3.2 ALGEBRAIC DESCRIPTION OF SURFACES......Page 54
3.3 ORIENTABLE 2n-GONS......Page 56
3.4 NON-ORIENT ABLt: 2n-GONS......Page 60
3.5 THE WORKING DEFINITION OF A SURFACE......Page 62
3.6 THE CLASSIFICATION THEOREM......Page 63
3.8 EXERCISES......Page 64
4.1 INTRODUCING X(M)......Page 67
4.2 X(M) AND THE CONNECTED SUM CONSTRUCTION......Page 68
4.3 HOW TO TELL THE DIFFERENCE......Page 71
4.4 CAN YOU TELL THE DIFFERENCE?......Page 72
4.5 COMMENTS......Page 73
4.6 EXERCISES......Page 74
5.1 PATTERNS AND X(M)......Page 80
5.2 COMPLEXES......Page 85
5.3 REGULAR COMPLEXES......Page 89
5.4 b-VALENT COMPLEXES......Page 93
5.5 COMMENTS......Page 96
5.6 EXERCISES......Page 97
6.1 COLOURING MAPS ON SURFACES......Page 104
6.2 EMBEDDING GRAPHS IN SURFACES......Page 108
6.3 PLANAR GRAPHS......Page 111
6.5 EMBEDDING THE COMPLETE GRAPHS......Page 112
6.6 SPROUTS......Page 115
6.8 COMMENTS......Page 117
6.9 EXERCISES......Page 119
7.1 A WATERPROOF......Page 123
7.2 HAIRY SURFACES......Page 125
7.3 INTERPRETATIONS OF THE INDEX THEOREM......Page 132
7.4 LAKES......Page 133
7.5 ISLANDS IN LAKES......Page 134
7.7 VECTOR FIELDS AND DIFFERENTIAL EQUATIONS......Page 137
7.8 COMMENTS......Page 140
7.9 EXERCISES......Page 141
8.1 PLANE EUCLIDEAN GEOMETRY......Page 144
8.2 GROUPS......Page 147
8.3 PLANE HYPERBOLIC GEOMETRY......Page 153
8.4 PLANE TESSELLATIONS......Page 157
8.5 COMMENTS......Page 178
8.6 EXERCISES......Page 179
9.2 TESSELLATIONS AND PATTERNS......Page 184
9.3 TESSELLATIONS AND MAP COLOURING......Page 188
9.4 TESSELLATIONS ;.\\ND VECTOR FIELDS......Page 190
9.6 EXERCISES......Page 199
10.1 INTRODUCTION......Page 202
10.2 THE FUNDAMENTAL GROUP......Page 203
10.3 ISOMORPHIC GROUPS......Page 204
10.4 COMMENTS......Page 205
10.5 EXERCISES......Page 206
Outline solutions to the exercises......Page 208
Further reading and references......Page 219
Index......Page 221