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Graphs, Surfaces and Homology

An Introduction to Algebraic Topology

  • P. J. Giblin

Part of the Chapman and Hall Mathematics Series book series (CHMS)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. P. J. Giblin
    Pages 1-10
  3. P. J. Giblin
    Pages 11-48
  4. P. J. Giblin
    Pages 49-84
  5. P. J. Giblin
    Pages 85-125
  6. P. J. Giblin
    Pages 127-162
  7. P. J. Giblin
    Pages 163-175
  8. P. J. Giblin
    Pages 177-200
  9. P. J. Giblin
    Pages 201-215
  10. P. J. Giblin
    Pages 217-228
  11. P. J. Giblin
    Pages 229-307
  12. Back Matter
    Pages 277-329

About this book

Introduction

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.

Keywords

Algebraic topology cohomology cohomology theory homology topology

Authors and affiliations

  • P. J. Giblin
    • 1
  1. 1.Department of Pure MathematicsUniversity of LiverpoolUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-009-5953-8
  • Copyright Information Springer Science+Business Media B.V. 1977
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-412-23900-7
  • Online ISBN 978-94-009-5953-8
  • Buy this book on publisher's site