Graphs, Surfaces and Homology

by Peter Giblin

• ISBN13: 9780521154055
• ISBN10: 0521154057
• Edition: 3rd
• Format: Paperback
• Copyright: 2010-09-20
• Publisher: Cambridge University Press

Summary

Homology theory is a powerful algebraic tool that is at the centre of current research in topology and its applications. This accessible textbook will appeal to mathematics students interested in the application of algebra to geometrical problems, specifically the study of surfaces (sphere, torus, Mobius band, Klein bottle). In this introduction to simplicial homology - the most easily digested version of homology theory - the author studies interesting geometrical problems, such as the structure of two-dimensional surfaces and the embedding of graphs in surfaces, using the minimum of algebraic machinery and including a version of Lefschetz duality. Assuming very little mathematical knowledge, the book provides a complete account of the algebra needed (abelian groups and presentations), and the development of the material is always carefully explained with proofs given in full detail. Numerous examples and exercises are also included, making this an ideal text for undergraduate courses or for self-study.

Author Biography

Peter Giblin is a Professor of Mathematics (Emeritus) at the University of Liverpool.

Table of Contents

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