Giblin P.J. Graphs, surfaces and homology. - 3rd ed. - Cambridge; New York: Cambridge University Press, 2010. - xx, 251 p.: ill. - Ref.: p.239-242. - Ind.: p.243-251. - ISBN 978-0-521-76665-4; ISBN 978-0-521-15405-5  Место хранения:   01 | ГПНТБ СО РАН | Новосибирск

Оглавление / Contents

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