Applications of Algebraic Topology: Graphs and Networks. the Picard-Lefschetz Theory and Feynman Integrals

ISBN-13: 9780387901374 / Angielski / Miękka / 1975 / 191 str.

Solomon Lefschetz

Opis

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.

Spis treści

I Application of Classical Topology to Graphs and Networks.- I. A Résumé of Linear Algebra.- 1. Matrices.- 2. Vector and Vector Spaces.- 3. Column Vectors and Row Vectors.- 4. Application to Linear Equations.- II. Duality in Vector Spaces.- 1. General Remarks on Duality.- 2. Questions of Nomenclature.- 3. Linear Functions on Vector Spaces. Multiplication.- 4. Linear Transformations. Duality.- 5. Vector Space Sequence of Walter Mayer.- III. Topological Preliminaries.- 1. First Intuitive Notions of Topology.- 2. Affine and Euclidean Spaces.- 3. Continuity, Mapping, Homeomorphism.- 4. General Sets and Their Combinations.- 5. Some Important Subsets of a Space.- 6. Connectedness.- 7. Theorem of Jordan-Schoenflies.- IV. Graphs. Geometric Structure.- 1. Structure of Graphs.- 2. Subdivision. Characteristic Betti Number.- V. Graph Algebra.- 1. Preliminaries.- 2. Dimensional Calculations.- 3. Space Duality. Co-theory.- VI. Electrical Networks.- 1. Kirchoff’s Laws.- 2. Different Types of Elements in the Branches.- 3. A Structural Property.- 4. Differential Equations of an Electrical Network.- VII. Complexes.- 1. Complexes.- 2. Subdivision.- 3. Complex Algebra.- 4. Subdivision Invariance.- VIII. Surfaces.- 1. Definition of Surfaces.- 2. Orientable and Nonorientable Surfaces.- 3. Cuts.- 4. A Property of the Sphere.- 5. Reduction of Orientable Surfaces to a Normal Form.- 6. Reduction of Nonorientable Surfaces to a Normal Form.- 7. Duality in Surfaces.- IX. Planar Graphs.- 1. Preliminaries.- 2. Statement and Solution of the Spherical Graph Problem.- 3. Generalization.- 4. Direct Characterization of Planar Graphs by Kuratowski.- 5. Reciprocal Networks.- 6. Duality of Electrical Networks.- II The Picard-Lefschetz Theory and Feynman Integrals.- I. Topological and Algebraic Considerations.- 1. Complex Analytic and Projective Spaces.- 2. Application to Complex Projective n-space Pn.- 3. Algebraic Varieties.- 4. A Resume of Standard Notions of Algebraic Topology.- 5. Homotopy. Simplicial Mappings.- 6. Singular Theory.- 7. The Poincaré Group of Paths.- 8. Intersection Properties for Orientable M2n Complex.- 9. Real Manifolds.- II. The Picard-Lefschetz Theory.- 1. Genesis of the Problem.- 2. Method.- 3. Construction of the Lacets of Surface ?z.- 4. Cycles of ?z. Variations of Integrals Taken On ?z.- 5. An Alternate Proof of the Picard-Lefschetz Theorem.- 6. The ?1-manifold M. Its Cycles and Their Relation to Variations.- III. Extension to Higher Varieties.- 1. . Preliminary Remarks.- 2. First Application.- 3. Extension to Multiple Integrals.- 4. The 2-Cycles of an Algebraic Surface.- IV. Feynman Integrals.- 1. On Graphs.- 2. Algebraic Properties.- 3. Feynman Graphs.- 4. Feynman Integrals.- 5. Singularities.- 6. Polar Loci.- 7. More General Singularities.- 8. On the Loop-Complex.- 9. Some Complements.- 10. Examples.- 11. Calculation of an Integral.- 12. A Final Observation.- V. Feynman Integrals. B.- 1. Introduction.- 2. General Theory.- 3. Relative Theory.- 4. Application to Graphs.- 5. On Certain Transformations.- Subject Index Part I.- Subject Index Part II.