
Graphs, Surfaces and Homology
An Introduction to Algebraic Topology
P. Giblin(Author)
Chapman and Hall (Publisher)
2nd Edition
Published on 29. October 1981
Book
Paperback/Softback
XVII, 329 pages
978-0-412-23900-7 (ISBN)
Description
viii homology groups. A weaker result, sufficient nevertheless for our purposes, is proved in Chapter 5, where the reader will also find some discussion of the need for a more powerful in variance theorem and a summary of the proof of such a theorem. Secondly the emphasis in this book is on low-dimensional examples the graphs and surfaces of the title since it is there that geometrical intuition has its roots. The goal of the book is the investigation in Chapter 9 of the properties of graphs in surfaces; some of the problems studied there are mentioned briefly in the Introduction, which contains an in formal survey of the material of the book. Many of the results of Chapter 9 do indeed generalize to higher dimensions (and the general machinery of simplicial homology theory is avai1able from earlier chapters) but I have confined myself to one example, namely the theorem that non-orientable closed surfaces do not embed in three-dimensional space. One of the principal results of Chapter 9, a version of Lefschetz duality, certainly generalizes, but for an effective presentation such a gener- ization needs cohomology theory. Apart from a brief mention in connexion with Kirchhoff's laws for an electrical network I do not use any cohomology here. Thirdly there are a number of digressions, whose purpose is rather to illuminate the central argument from a slight dis tance, than to contribute materially to its exposition.
More details
Series
Edition
2nd Revised edition
Language
English
Place of publication
Dordrecht
Netherlands
Target group
Professional and scholarly
Research
Edition type
Revised edition
Illustrations
41 s/w Abbildungen
XVII, 329 p. 41 illus.
Dimensions
Height: 216 mm
Width: 140 mm
Thickness: 19 mm
Weight
439 gr
ISBN-13
978-0-412-23900-7 (9780412239007)
DOI
10.1007/978-94-009-5953-8
Schweitzer Classification
Content
1 Graphs.- 2 Closed Surfaces.- 3 Simplicial Complexes.- 4 HomoLogy Groups.- 5 The Question of Invariance.- 6 Some General Theorems.- 7 Two More General Theorems.- 8 Homology Modulo 2.- 9 Graphs In Surfaces.- Appendix: Abelian Groups.- Basic definitions.- Finitely generated (f.g.) and free abelian groups.- Quotient groups.- Exact sequences.- Direct sums and splitting.- Presentations.- Rank of a f.g. abelian group.- References.- List of Notation.