Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research.
"Thoroughly recommended" by "The Physics Bulletin, " this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.

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."

INTRODUCTION . . . . . . xiii 1. LINEAR EQUATIONS. BASIC NOTIONS . 3 2. EQUATIONS WITH A CLOSED OPERATOR 6 3. THE ADJOINT EQUATION . . . . . . 10 4. THE EQUATION ADJOINT TO THE FACTORED EQUATION. 17 5. AN EQUATION WITH A CLOSED OPERATOR WHICH HAS A DENSE DOMAIN 18 NORMALLY SOLVABLE EQUATIONS WITH FINITE DIMENSIONAL KERNEL. 22 6. A PRIORI ESTIMATES .. . . . . . 24 7. EQUATIONS WITH FINITE DEFECT . . . 27 8. 9. SOME DIFFERENT ADJOINT EQUATIONS . 30 10. LINEAR TRANSFORMATIONS OF EQUATIONS 33 TRANSFORMATIONS OF d-NORMAL EQUATIONS . 38 11. 12. NOETHERIAN EQUATIONS. INDEX. . . . . . 42 13. EQUATIONS WITH OPERATORS WHICH ACT IN A SINGLE SPACE 44 14. FREDHOLM EQUATIONS. REGULARIZATION OF EQUATIONS 46 15. LINEAR CHANGES OF VARIABLE . . . . . . . . 50 16. STABILITY OF THE PROPERTIES OF AN EQUATION 53 OVERDETERMINED EQUATIONS 59 17. 18. UNDETERMINED EQUATIONS 62 19. INTEGRAL EQUATIONS . . . 65 DIFFERENTIAL EQUATIONS . 80 20. APPENDIX. BASIC RESULTS FROM FUNCTIONAL ANALYSIS USED IN THE TEXT 95 LITERATURE CITED . . . . . . . . . . . . . . . . . . .. . . . 99 . . PRE F ACE The basic material appearing in this book represents the substance v of a special series of lectures given by the author at Voronez University in 1968/69, and, in part, at Dagestan University in 1970."

Bruhat-Tits theory that suffices for the main applications. Part III treats modern topics that have become important in current research. Part IV provides a few sample applications of the theory. The appendices contain further details on the topic of integral models.