Graphs, surfaces and homology : an introduction to algebraic topology / P. J. Giblin

Chapters. 1. Graphs (Abstract graphs and realizations; Kirchhoff's laws; Maximal trees and the cyclomatic number; Chain and cycles on an oriented graph; Planar graphs; Appendix on Kirchhoff's equations), 2. Closed surfaces (Closed surfaces and orientability; Polygonal representation of a closed surface; A note on realizations; Transformation of a closed surface to standard form; Euler characteristics; Minimal triangulations), 3. Simplicial complexes (Simplexes; Ordered simplexes and oriented simplexes; Simplicial complexes; Abstract simplicial complexes and realizations; Triangulations and diagrams of simplicial complexes; Stars, joins and links; Collapsing; Appendix on orientation), 4. Homology groups (Chain groups and boundary homomorphisms; Homology groups; Relative homology groups; Three homomorphisms; Appendix on chain complexes), 5. The question of invariance (Invariance under stellar subdivision; Triangulations, simplicial approximation and topological invariance; Appendix on barycentric subdivision), 6. Some general theorems (The homology sequence of a pair; The excision theorem; Collapsing revisited; Homology groups of closed surfaces; The Euler characteristic), 7. Two more general theorems (The Mayer-Vietoris sequence; Homology sequence of a triple), 8. Homology modulo 2, 9. Graphs in surfaces (Regular neighborhoods; Surfaces; Lefschetz duality; A three-dimensional situation; Separating surfaces by graphs; Representation of homology elements by simple closed polygons; Orientation preserving and reversing loops; A generalization of Euler's formula; Brussels Sprouts), Appendix: Abelian groups, References, Index.

Bibliogr. p. 309-312. Index