Applied Mathematical Sciences EDITORS Fritz John Lawrence Sirovich Courant Institute of Division of Mathematical Sciences Applied Mathematics New York University Brown University New York, N.Y. 10012 Providence, R.1. 02912 Joseph P. LaSalle Gerald B. Whitham Division of Applied Mathematics Applied Mathematics Firestone Laboratory Brown University California Institute of Technology Providence, R.1. 02912 Pasadena, CA. 91109 EDITORIAL STATEMENT The mathematization of all sciences, the fading of traditional scientific boun daries, the impact of computer technology, the growing importance of mathemati cal-computer modelling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scien tist. In particular, this series will provide an outlet for material less formally pre sented and more anticipatory of needs than finished texts or monographs, yet of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. The aim of the series is, through rapid publication in an attractive but inex pensive format, to make material of current interest widely accessible. This implies the absence of excessive generality and abstraction, and unrealistic idealization, but with quality of exposition as a goal. Many of the books will originate out of and will stimUlate the development of new undergraduate and graduate courses in the applications of mathematics. Some of the books will present introductions to new areas of research, new appli cations and act as signposts for new directions in the mathematical sciences. This series willi often serve as an intermediate stage of the publication of material which, through exposure here, will be further developed and refined and appear later in one of Springer-Verlag's other mathematical series. MANUSCRIPTS The Editors welcome all inquiries regarding the submission of manuscripts for the series. Final preparation of all manuscripts will take place in the editorial offices of the series in the Division of Applied Mathematics, Brown University, Providence, Rhode Island. SPRINGER-VERLAG NEW YORK INC., 175 Fifth Avenue, New York, N. Y. 10010 Printed in U.S.A. Applied Mathematical Sciences I Volume 16 s. Lefschetz Applications of Algebraic Topology Graphs and Networks The Picard-Lefschetz Theory and Feynman Integrals With 52 Illustrations Springer-Verlag New York· Heidelberg· Berlin 1975 S. Lefschetz Formerly of Princeton University AMS Classifications: 55-01, 55A 15,81 A 15 Library of Congress Cataloging in Publication Data Lefschetz, Solomon, 1884-1972. Applications of algebraic topology. (Applied mathematical sciences; v.16) Bibliography: p. Includes index. 1. Algebraic topology. 2. Graph theory. 3. Elec tric networks. 4. Feynman integrals. I. Title. II. Series. QA1.A647 vol. 16 [QA611) 510'.8 [514'.2) 75-6924 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1975 by Springer-Verlag New York Inc. ISBN 978-0-387-90137-4 ISBN 978-1-4684-9367-2 (eBook) DOl 10.1007/978-1-4684-9367-2 v Solomon Lefschetz (1884-1972) was one of the great mathematicians of his generation. This volume published posthumously and completed shortly before his death is in his own unique and vigorous style. Were he alive there are many people whom he would thank. Among them are Sandra Spinacci for the careful typing of his manuscript, Mauricio Peixoto for his constant encouragement, and John Mallet-Paret for his careful reading of the manuscript. January 1975 J. P. LaSalle vi TABLE OF CONTENTS PART I APPLICATION OF CLASSICAL TOPOLOGY TO GRAPHS AND NETWORKS INTRODUCTION 4 CHAPTER I. A RESUME OF LINEAR ALGEBRA 5 1. Matrices 5 2. Vector and Vector Spaces 7 3. Column Vectors and Row Vectors 10 4. Application to Linear Equations l1 CHAPTER II. DUALITY IN VECTOR SPACES 13 1. General Remarks on Duality 13 2. Questions of Nomenclature 14 3. Linear Functions on Vector Spaces. Multiplication 15 4. Linear Transformations. Duality 16 5. Vector Space Sequence of Walter Mayer 18 CHAPTER III. TOPOLOGICAL PRELIMINARIES 22 1. First Intuitive Notions of Topology 22 2. Affine and Euclidean Spaces 25 3. Continuity, Mapping, Homeomorphism 26 4. General Sets and Their Combinations 27 5. Some Important Subsets of a Space 28 6. Connectedness 29 7. Theorem of Jordan-Schoenflies 30 CHAPTER IV. GRAPHS. GEOMETRIC STRUCTURE 34 1. Structure of Graphs 34- 2. Subdivision. Characteristic Betti Number 37 CHAPTER V. GRAPH ALGEBRA 43 1. Preliminaries 43 2. Dimensional Calculations 46 3. Space Duality. Co-theory 48 CHAPTER VI. ELECTRICAL NETWORKS 51 1. Kirchoff I,S Laws 51 2. Different Types of Elements in the Branches 53 3. A Structural Property 54 4. Differential Equations of an Electrical Network 56 CHAPTER VII. COMPLEXES 61 1. Complexes 61 2. Subdivision 64 3. Complex Algebra 66 4. Subdivision Invariance 68 CHAPTER VIII. SURFACES 71 1. Def±nition of Surfaces 71 2. Orientable and Nonorientable Surfaces 72 3. Cuts 76 4. A Property of the Sphere 79 vii 5. Reduction of Orientable Surfaces to a Normal. Form 83 6. Reduction of Nonorientable Surfaces to a Normal Form 84 7. Duality in Surfaces 86 CHAPTER IX. PLANAR GRAPHS 89 l~ Preliminaries 89 2. Statement and Solution of the Spherical Graph Problem 91) 3. Generalization 95 4. Direct Characterization of Planar Graphs by Kuratowski 96 5. Reciprocal Networks 103 6. Duality of Electrical Networks 104 PART II THE PICARD-LEFSCHETZ THEORY AND FEYNMAN INTEGRALS INTRODUCTION 113 CHAPTER I. TOPOLOGICAL AND ALGEBRAIC CONSIDERATIONS 119 l. Complex Analytic and Projective Spaces 119 2. Application to Complex Projective n-space gin 119 3. Algebraic Varieties 121 4. A Resume of Standard Notions of Algebraic Topology 124 5. Homotopy. Simplicial Mappings 128 6. Singular Theory 129 7. The Poincare Group of Paths 130 8. Intersection Properties for Orientable M2 n Complex 131 9. Real Manifolds 133 CHAPTER II. THE PICARD-LEFSCHETZ THEORY 135 1. Genesis of the Problem 135 2. Method 136 3. Construction of the Lacets of 138 Surface <l>z 4. Cycles of <l>z' Variations of Integrals Taken On <l>z 140 5. An Alternate Proof of the Picard-Lefschetz Theorem 140 6. The Al-manifold M. Its Cycles and Their Relation to Variations 146 CHAPTER III. EXTENSION TO HIGHER VARIETIES 149 1. Preliminary Remarks 149 2. First Application 150 3. Extension to Multiple Integrals 151 viii Page 4. The 2-Cycles of an Algebraic Surface 152 CHAPTER IV. FEYNMAN INTEGRALS 154 1. On Graphs 154 2. Algebraic Properties 156 3. Feynman Graphs 160 4. Feynman Integrals 162 5. Singularities 163 6. Polar Loci 164 7. More General Singularities 168 8. On the Loop-Complex 170 9. Some Complements 170 10. Examples 171 11. Calculation of an Integral 174 12. A Final Observation 175 CHAPTER V. FEYNMAN INTEGRALS. B. 177 1. Introduction 177 2. General Theory 177 3. Relative Theory 178 4. Application to Graphs 178 5. On Certain Transformations 180 BIBLIOGRAPHY 181 SUBJECT INDEX PART I 183 SUBJECT INDEX PART II 187 1 PART I APPLICATION OF CLASSICAL TOPOLOGY TO GRAPHS AND NETWORKS PREFACE 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