Preface to the third edition ................................ xi
Preface to the first edition .............................. xiii
List of notation .......................................... xvii
Introduction .................................................... 1
1 Graphs ....................................................... 9
Abstract graphs and realizations ............................. 9
• Kirchhoff's laws ......................................... 14
Maximal trees and the cyclomatic number ..................... 16
Chains and cycles on an oriented graph ...................... 20
• Planar graphs ............................................ 26
• Appendix on Kirchhoff's equations ........................ 35
2 Closed surfaces ............................................. 38
Closed surfaces and orientability ........................... 39
Polygonal representation of a closed surface ................ 45
• A note on realizations ................................... 47
Transformation of closed surfaces to standard form .......... 49
Euler characteristics ....................................... 55
• Minimal triangulations ................................... 60
3 Simplicial complexes ........................................ 67
Simplexes ................................................... 67
Ordered simplexes and oriented simplexes .................... 73
Simplicial complexes ........................................ 74
Abstract simplicial complexes and realizations .............. 77
Triangulations and diagrams of simplicial complexes ......... 79
Stars, joins and links ...................................... 84
Collapsing .................................................. 88
• Appendix on orientation .................................. 93
4 Homology groups ............................................. 99
Chain groups and boundary homomorphisms ..................... 99
Homology groups ............................................ 104
Relative homology groups ................................... 112
Three homomorphisms ........................................ 121
• Appendix on chain complexes ............................. 124
5 The question of invariance ................................. 127
Invariance under stellar subdivision ....................... 128
• Triangulations, simplicial approximation and topological
invariance .............................................. 133
• Appendix on barycentric subdivision ..................... 136
6 Some general theorems ...................................... 138
The homology sequence of a pair ............................ 138
The excision theorem ....................................... 142
Collapsing revisited ....................................... 144
Homology groups of closed surfaces ......................... 149
The Euler characteristic ................................... 154
7 Two more general theorems .................................. 158
The Mayer-Vietoris sequence ................................ 158
• Homology sequence of a triple ........................... 167
8 Homology modulo 2 .......................................... 171
9 Graphs in surfaces ......................................... 180
Regular neighbourhoods ..................................... 183
Surfaces ................................................... 187
Lefschetz duality .......................................... 191
• A three-dimensional situation ........................... 195
Separating surfaces by graphs .............................. 198
Representation of homology elements by simple closed
polygons ................................................... 200
Orientation preserving and reversing loops ................. 203
A generalization of Euler's formula ........................ 207
• Brussels Sprouts ........................................ 211
Appendix: abelian groups ................................... 215
Basic definitions .......................................... 215
Finitely generated (f.g.) and free abelian groups .......... 217
Quotient groups ............................................ 219
Exact sequences ............................................ 221
Direct sums and splitting .................................. 222
Presentations .............................................. 226
Rank of a f.g. abelian group ............................... 233
References .................................................... 239
Index ......................................................... 243
|
