Graphs, surfaces, and homology : an introduction to algebraic topology
著者
書誌事項
Graphs, surfaces, and homology : an introduction to algebraic topology
（Chapman and Hall mathematics series）
Chapman and Hall, 1981
2nd ed
- pbk.
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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.
目次
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.
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