Applications of Algebraic Topology

Graphs and Networks The Picard-Lefschetz Theory and Feynman Integrals

  • S. Lefschetz

Part of the Applied Mathematical Sciences book series (AMS, volume 16)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Application of Classical Topology to Graphs and Networks

    1. Front Matter
      Pages 1-4
    2. S. Lefschetz
      Pages 5-12
    3. S. Lefschetz
      Pages 13-21
    4. S. Lefschetz
      Pages 22-33
    5. S. Lefschetz
      Pages 34-42
    6. S. Lefschetz
      Pages 43-50
    7. S. Lefschetz
      Pages 51-59
    8. S. Lefschetz
      Pages 61-69
    9. S. Lefschetz
      Pages 71-87
    10. S. Lefschetz
      Pages 89-106
  3. The Picard-Lefschetz Theory and Feynman Integrals

    1. Front Matter
      Pages 109-117
    2. S. Lefschetz
      Pages 119-134
    3. S. Lefschetz
      Pages 135-148
    4. S. Lefschetz
      Pages 149-153
    5. S. Lefschetz
      Pages 154-176
    6. S. Lefschetz
      Pages 177-180
  4. Back Matter
    Pages 181-191

About this book

Introduction

This monograph is based, in part, upon lectures given in the Princeton School of Engineering and Applied Science. It presupposes mainly an elementary knowledge of linear algebra and of topology. In topology the limit is dimension two mainly in the latter chapters and questions of topological invariance are carefully avoided. From the technical viewpoint graphs is our only requirement. However, later, questions notably related to Kuratowski's classical theorem have demanded an easily provided treatment of 2-complexes and surfaces. January 1972 Solomon Lefschetz 4 INTRODUCTION The study of electrical networks rests upon preliminary theory of graphs. In the literature this theory has always been dealt with by special ad hoc methods. My purpose here is to show that actually this theory is nothing else than the first chapter of classical algebraic topology and may be very advantageously treated as such by the well known methods of that science. Part I of this volume covers the following ground: The first two chapters present, mainly in outline, the needed basic elements of linear algebra. In this part duality is dealt with somewhat more extensively. In Chapter III the merest elements of general topology are discussed. Graph theory proper is covered in Chapters IV and v, first structurally and then as algebra. Chapter VI discusses the applications to networks. In Chapters VII and VIII the elements of the theory of 2-dimensional complexes and surfaces are presented.

Keywords

Algebraic Algebraic topology Algebraische Topologie Applications Graph Homotopy Sim Topology Vector space

Authors and affiliations

  • S. Lefschetz
    • 1
  1. 1.Princeton UniversityUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4684-9367-2
  • Copyright Information Springer-Verlag New York 1975
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-90137-4
  • Online ISBN 978-1-4684-9367-2
  • Series Print ISSN 0066-5452
  • About this book